A336466 -id:A336466

A336470

Lexicographically earliest infinite sequence such that a(i) = a(j) => A336466(i) = A336466(j) and A336158(i) = A336158(j), for all i, j >= 1.

+20
12

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 5, 2, 3, 3, 6, 1, 2, 4, 7, 2, 8, 5, 9, 2, 4, 3, 10, 3, 11, 6, 12, 1, 13, 2, 8, 4, 7, 7, 8, 2, 5, 8, 14, 5, 15, 9, 16, 2, 17, 4, 6, 3, 18, 10, 13, 3, 19, 11, 20, 6, 12, 12, 21, 1, 8, 13, 22, 2, 23, 8, 24, 4, 7, 7, 15, 7, 25, 8, 26, 2, 27, 5, 28, 8, 6, 14, 29, 5, 9, 15, 19, 9, 25, 16, 19, 2, 3, 17, 30, 4, 31, 6, 32, 3, 33

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Restricted growth sequence transform of the ordered pair [A336466(n), A336158(n)].

For all i, j:

A336460(i) = A336460(j) => a(i) = a(j),

a(i) = a(j) => A329697(i) = A329697(j),

a(i) = a(j) => A336471(i) = A336471(j).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

PROG

(PARI)

up_to = 65537;

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };

A000265(n) = (n>>valuation(n, 2));

A336466(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k, 1], 1, (A000265(f[k, 1]-1))^f[k, 2])); };

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523

A336158(n) = A046523(A000265(n));

Aux336470(n) = [A336466(n), A336158(n)];

v336470 = rgs_transform(vector(up_to, n, Aux336470(n)));

A336470(n) = v336470[n];

CROSSREFS

Cf. A336158, A336466.

Cf. also A329697, A336160, A336471.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 24 2020

STATUS

approved

A336460

Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(n), A336158(n), A336466(n)], for all i, j >= 1.

+20
6

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 3, 5, 9, 3, 10, 6, 11, 2, 12, 7, 13, 4, 14, 8, 15, 1, 16, 3, 17, 5, 18, 9, 19, 3, 20, 10, 21, 6, 22, 11, 23, 2, 24, 12, 25, 7, 26, 13, 27, 4, 28, 14, 29, 8, 30, 15, 31, 1, 32, 16, 33, 3, 34, 17, 35, 5, 18, 18, 22, 9, 36, 19, 37, 3, 38, 20, 39, 10, 40, 21, 41, 6, 42, 22, 43, 11, 44, 23, 45, 2, 7, 24, 46, 12, 47, 25, 48, 7, 49

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Restricted growth sequence transform of the ordered triple [A278222(n), A336158(n), A336466(n)].

For all i, j:

A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j),

a(i) = a(j) => A336159(i) = A336159(j),

a(i) = a(j) => A336470(i) = A336470(j) => A336471(i) = A336471(j),

a(i) = a(j) => A336472(i) = A336472(j),

a(i) = a(j) => A336473(i) = A336473(j).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

PROG

(PARI)

up_to = 65537;

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };

A000265(n) = (n>>valuation(n, 2));

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523

A278222(n) = A046523(A005940(1+n));

A336158(n) = A046523(A000265(n));

A336466(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k, 1], 1, (A000265(f[k, 1]-1))^f[k, 2])); };

Aux336460(n) = [A278222(n), A336158(n), A336466(n)];

v336460 = rgs_transform(vector(up_to, n, Aux336460(n)));

A336460(n) = v336460[n];

CROSSREFS

Cf. A278222, A336158, A336466.

Cf. A003602, A324400, A336159, A336470, A336471, A336472, A336473.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 24 2020

STATUS

approved

A340085

a(n) = A336466(n) / gcd(n-1, A336466(n)); Odd part of A340082(n).

+20
6

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 9, 3, 1, 1, 3, 1, 5, 1, 11, 1, 1, 3, 1, 1, 1, 1, 1, 5, 3, 9, 7, 1, 1, 1, 15, 3, 1, 3, 1, 1, 1, 11, 1, 1, 1, 1, 9, 1, 3, 15, 3, 1, 1, 1, 5, 1, 3, 1, 21, 7, 5, 1, 1, 1, 11, 15, 23, 9, 1, 1, 9, 5, 1, 1, 1, 1, 3, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,14

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8191

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

FORMULA

a(n) = A000265(A340082(n)).

a(n) = A336466(n) / A340084(n) = A336466(n) / gcd(n-1, A336466(n)).

For all n >= 0, a(A003961(A019565(n))) = a(A019565(2*n)) = A339901(n).

MATHEMATICA

Array[#2/GCD[#1 - 1, #2] & @@ {#, Times @@ Map[If[# <= 2, 1, (# - 1)/2^IntegerExponent[# - 1, 2]] &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]]} &, 105] (* Michael De Vlieger, Dec 29 2020 *)

PROG

(PARI)

A000265(n) = (n>>valuation(n, 2));

A336466(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k, 1], 1, (A000265(f[k, 1]-1))^f[k, 2])); };

A340085(n) = { my(u=A336466(n)); u/gcd(n-1, u); };

CROSSREFS

Cf. A000265, A003958, A003961, A019565, A336466, A339901, A340082, A340084, A340086.

Cf. also A160595.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 28 2020

STATUS

approved

A336472

Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A336466(i) = A336466(j), for all i, j >= 1.

+20
5

1, 1, 2, 1, 3, 2, 4, 1, 3, 3, 5, 2, 6, 4, 7, 1, 3, 3, 8, 3, 9, 5, 10, 2, 11, 6, 12, 4, 13, 7, 14, 1, 15, 3, 6, 3, 16, 8, 17, 3, 18, 9, 19, 5, 20, 10, 21, 2, 8, 11, 12, 6, 22, 12, 23, 4, 24, 13, 25, 7, 26, 14, 27, 1, 28, 15, 29, 3, 30, 6, 31, 3, 16, 16, 20, 8, 32, 17, 33, 3, 34, 18, 35, 9, 36, 19, 37, 5, 38, 20, 39, 10, 40, 21, 41, 2, 6, 8, 42, 11, 43, 12, 44, 6, 45

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Restricted growth sequence transform of the ordered pair [A278222(n), A336466(n)].

For all i, j: A336460(i) = A336460(j) => a(i) = a(j).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

PROG

(PARI)

up_to = 65537;

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523

A278222(n) = A046523(A005940(1+n));

A000265(n) = (n>>valuation(n, 2));

A336466(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k, 1], 1, (A000265(f[k, 1]-1))^f[k, 2])); };

Aux336472(n) = [A278222(n), A336466(n)];

v336472 = rgs_transform(vector(up_to, n, Aux336472(n)));

A336472(n) = v336472[n];

CROSSREFS

Cf. A278222, A336466.

Cf. also A336162, A336460, A336470, A336471, A336473.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 24 2020

STATUS

approved

A365426

a(n) = A336466(A163511(n)).

+20
5

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 3, 1, 3, 3, 5, 1, 1, 1, 1, 1, 1, 1, 27, 1, 1, 1, 9, 1, 9, 9, 25, 1, 1, 1, 3, 1, 3, 3, 15, 1, 3, 3, 5, 3, 5, 5, 3, 1, 1, 1, 1, 1, 1, 1, 81, 1, 1, 1, 27, 1, 27, 27, 125, 1, 1, 1, 9, 1, 9, 9, 75, 1, 9, 9, 25, 9, 25, 25, 9, 1, 1, 1, 3, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,16

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..16385

FORMULA

For all n >= 1, a(n) = a(2*n) = a(A000265(n)).

PROG

(PARI)

A000265(n) = (n>>valuation(n, 2));

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));

A336466(n) = { my(f=factor(n)); prod(k=1, #f~, A000265(f[k, 1]-1)^f[k, 2]); };

A365426(n) = A336466(A163511(n));

CROSSREFS

Cf. A000265, A163511, A336466.

Cf. also A365427.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Sep 03 2023

STATUS

approved

A340084

a(n) = gcd(n-1, A336466(n)); Odd part of A340081(n).

+20
4

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 11, 1, 1, 1, 1, 3, 7, 1, 15, 1, 1, 1, 1, 1, 9, 1, 1, 1, 5, 1, 21, 1, 1, 1, 23, 1, 3, 1, 1, 3, 13, 1, 1, 1, 1, 1, 29, 1, 15, 1, 1, 1, 1, 5, 33, 1, 1, 3, 35, 1, 9, 1, 1, 3, 1, 1, 39, 1, 1, 1, 41, 1, 1, 1, 1, 1, 11, 1, 9, 1, 1, 1, 1, 1, 3, 1, 1, 1, 25, 1, 51, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,7

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

FORMULA

a(n) = gcd(n-1, A336466(n)).

a(n) = A000265(A340081(n)) = A336466(n) / A340085(n).

For n >= 2, a(n) = A000265(n-1) / A340086(n).

For n >= 1, a(A000040(n)) = A057023(n).

For n >= 0, a(A019565(2*n)) = A339899(n).

MATHEMATICA

Array[GCD[#1 - 1, #2] & @@ {#, Times @@ Map[If[# <= 2, 1, (# - 1)/2^IntegerExponent[# - 1, 2]] &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]]} &, 105] (* Michael De Vlieger, Dec 29 2020 *)

PROG

(PARI)

A000265(n) = (n>>valuation(n, 2));

A336466(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k, 1], 1, (A000265(f[k, 1]-1))^f[k, 2])); };

A340084(n) = { my(u=A336466(n)); gcd(n-1, u); };

CROSSREFS

Cf. A000265, A003958, A003961, A019565, A057023, A336466, A339899, A340081, A340085, A340086.

Cf. also A049559.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 29 2020

STATUS

approved

A340086

a(1) = 0, for n > 1, a(n) = A000265(n-1) / gcd(n-1, A336466(n)).

+20
4

0, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 21, 1, 23, 3, 25, 13, 9, 1, 29, 1, 31, 1, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 1, 47, 1, 49, 25, 17, 1, 53, 27, 55, 7, 57, 1, 59, 1, 61, 31, 63, 1, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 5, 81, 1, 83, 21, 85, 43, 87, 1, 89

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

From the second term onward, the odd part of A340083.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8191

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

FORMULA

a(1) = 0; for n > 1, a(n) = A000265(n-1) / A340084(n) = A000265(A340083(n)).

PROG

(PARI)

A000265(n) = (n>>valuation(n, 2));

A336466(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k, 1], 1, (A000265(f[k, 1]-1))^f[k, 2])); };

A340086(n) = if(1==n, 0, my(u=A336466(n)); A000265(n-1)/gcd(n-1, u));

CROSSREFS

Cf. A000265, A003958, A003961, A019565, A336466, A339901, A340083, A340084, A340085.

Cf. also A160596.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 29 2020

STATUS

approved

A336468

a(n) = A336466(phi(n)), where A336466 is fully multiplicative with a(p) = A000265(p-1) for prime p, with A000265(k) giving the odd part of k.

+20
3

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 11, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 3, 7, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 3, 3, 1, 5, 1, 1, 5, 1, 11, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,23

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

FORMULA

a(n) = A336466(A000010(n)).

Multiplicative with a(p^e) = A336466(p-1) * A336466(p)^(e-1).

PROG

(PARI)

A000265(n) = (n>>valuation(n, 2));

A336468(n) = { my(f=factor(eulerphi(n))); prod(k=1, #f~, A000265(f[k, 1]-1)^f[k, 2]); };

\\ Alternatively, as follows, requiring also code from A336466:

A336468(n) = { my(f=factor(n)); prod(k=1, #f~, A336466(f[k, 1]-1) * A336466(f[k, 1])^(f[k, 2]-1)); };

CROSSREFS

Cf. A000010, A000265, A336466, A336469.

KEYWORD

nonn,mult

AUTHOR

Antti Karttunen, Jul 22 2020

STATUS

approved

A339876

a(n) = A336466(A122111(n)).

+20
3

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 9, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,16

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

FORMULA

a(1) = 1, for n > 1, a(n) = A000265(A105560(n)-1) * a(A064989(n)).

a(n) = A336466(A122111(n)).

PROG

(PARI)

A000265(n) = (n>>valuation(n, 2));

A122111(n) = if(1==n, n, my(f=factor(n), es=Vecrev(f[, 2]), is=concat(apply(primepi, Vecrev(f[, 1])), [0]), pri=0, m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));

A336466(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k, 1], 1, (A000265(f[k, 1]-1))^f[k, 2])); };

A339876(n) = A336466(A122111(n));

(PARI)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};

A105560(n) = if(1==n, n, prime(bigomega(n)));

A339876(n) = if(1==n, n, A000265(A105560(n)-1) * A339876(A064989(n)));

CROSSREFS

Cf. A000265, A064989, A105560, A122111, A336466.

Cf. also A334107, A339877.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 25 2020

STATUS

approved

A342666

a(n) = A336466(A156552(n)).

+20
3

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 9, 1, 5, 1, 11, 1, 3, 3, 3, 1, 3, 1, 15, 1, 21, 1, 1, 1, 1, 5, 3, 1, 9, 1, 33, 5, 9, 1, 23, 1, 1, 3, 65, 1, 7, 1, 35, 21, 5, 1, 21, 1, 341, 9, 3, 1, 11, 1, 27, 1, 5, 1, 5, 1, 15, 3, 51, 1, 27, 1, 39, 1, 1365, 1, 1, 5, 49, 9, 1, 1, 1, 1, 117, 5, 825, 3, 9, 1, 9, 3, 1, 1, 7, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,7

LINKS

Antti Karttunen, Table of n, a(n) for n = 2..10000 (based on Hans Havermann's factorization of A156552)

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = A336466(A156552(n)) = A336466(A322993(n)).

a(p) = 1 for all primes p.

a(A003961(n)) = a(n).

PROG

(PARI)

A000265(n) = (n>>valuation(n, 2));

A336466(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k, 1], 1, (A000265(f[k, 1]-1))^f[k, 2])); };

A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};

A342666(n) = A336466(A156552(n));

(PARI)

\\ Version using the factorization file available at https://oeis.org/A156552/a156552.txt

v156552sigs = readvec("a156552.txt");

A000265(n) = (n>>valuation(n, 2));

A342666(n) = if(isprime(n), 1, my(prsig=v156552sigs[n], ps=prsig[1], es=prsig[2]); prod(i=1, #ps, A000265(ps[i]-1)^es[i])); \\ Antti Karttunen, Jan 29 2022

CROSSREFS

Cf. A000265, A003961, A156552, A322993, A336466.

Cf. also A324104, A342651, A342656, A350067.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 18 2021

STATUS

approved