A300835 -id:A300835

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A300837

a(n) is the total number of terms (1-digits) in Zeckendorf representation of all divisors of n.

+10
11

1, 2, 2, 4, 2, 5, 3, 5, 4, 5, 3, 10, 2, 6, 5, 7, 4, 9, 4, 10, 5, 6, 3, 13, 5, 5, 7, 11, 3, 13, 4, 10, 8, 6, 6, 16, 3, 8, 5, 14, 4, 12, 4, 11, 10, 8, 3, 18, 6, 11, 9, 10, 5, 16, 5, 14, 7, 6, 4, 23, 4, 8, 9, 13, 6, 16, 5, 10, 7, 14, 4, 23, 4, 8, 12, 12, 8, 13, 4, 20, 10, 9, 5, 23, 9, 9, 8, 17, 2, 22, 6, 12, 8, 6, 8, 24, 3, 12, 13, 19, 5, 15, 4, 14, 13

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

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10946

FORMULA

a(n) = Sum_{d|n} A007895(d).

a(n) = A300836(n) + A007895(n).

For all n >=1, a(n) >= A005086(n).

EXAMPLE

For n=12, its divisors are 1, 2, 3, 4, 6 and 12. Zeckendorf-representations (A014417) of these numbers are 1, 10, 100, 101, 1001 and 10101. Total number of 1's present is 10 (ten), thus a(12) = 10.

PROG

(PARI)

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649

A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); };

A300837(n) = sumdiv(n, d, A007895(d));

CROSSREFS

Cf. A000045, A007895, A014417, A072649, A300835, A300836.

Cf. also A005086, A093653.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 18 2018

STATUS

approved

A300834

a(n) = Product_{d|n, d<n} A019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.

+10
7

1, 2, 2, 6, 2, 30, 2, 60, 10, 42, 2, 4200, 2, 126, 70, 660, 2, 9240, 2, 13860, 210, 330, 2, 5082000, 14, 78, 220, 32760, 2, 3783780, 2, 42900, 550, 780, 294, 924924000, 2, 1092, 130, 41621580, 2, 3898440, 2, 112200, 60060, 306, 2, 28078050000, 42, 235620, 1300, 92820, 2, 200119920, 770, 128648520, 1820, 1122, 2, 424964656116000, 2, 3366

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

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..3194

FORMULA

a(n) = Product_{d|n, d<n} A019565(A003714(d)).

For n >= 1, A001222(a(n)) = A300836(n).

PROG

(PARI)

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649

A003714(n) = { my(s=0, w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }

A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565

A300834(n) = { my(m=1); fordiv(n, d, if(d < n, m *= A019565(A003714(d)))); m; };

CROSSREFS

Cf. A003714, A019565, A300835 (rgs-transform of this sequence), A300836.

Cf. also A293214, A293216, A293221, A293222, A293231.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 18 2018

STATUS

approved

A304103

Restricted growth sequence transform of A304102, a filter sequence related to the proper divisors of n expressed in Fibonacci number system.

+10
5

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

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

OFFSET

1,2

COMMENTS

For all i, j: a(i) = a(j) => b(i) = b(j), where b can be any of {A000005, A293435, A304095 or A300836} for example.

LINKS

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

PROG

(PARI)

\\ Needs also code from A304101.

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; };

write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }

A304102(n) = { my(m=1); fordiv(n, d, if(d<n, m *= prime(A304101(d)-1))); (m); };

write_to_bfile(1, rgs_transform(vector(up_to, n, A304102(n))), "b304103.txt");

CROSSREFS

Cf. A293435, A304095, A300836, A304102.

Cf. also A300835, A304105, A305800.

Cf. A305793 (analogous filter for base 2).

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 13 2018

STATUS

approved

A318835

Restricted growth sequence transform of A318834, product_{d|n, d<n} A019565(A000010(d)).

+10
5

1, 2, 2, 3, 2, 4, 2, 4, 5, 6, 2, 7, 2, 8, 9, 8, 2, 10, 2, 11, 12, 13, 2, 14, 15, 16, 12, 14, 2, 17, 2, 18, 19, 20, 21, 22, 2, 23, 24, 25, 2, 26, 2, 27, 28, 29, 2, 30, 9, 31, 32, 33, 2, 34, 24, 35, 36, 37, 2, 38, 2, 39, 40, 39, 41, 42, 2, 43, 44, 45, 2, 46, 2, 47, 48, 49, 50, 51, 2, 52, 53, 54, 2, 55, 56, 57, 58, 59, 2, 60, 61, 62, 63, 64, 65, 66, 2

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

OFFSET

1,2

COMMENTS

For all i, j: a(i) = a(j) => A051953(i) = A051953(j).

LINKS

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

PROG

(PARI)

up_to = 65537;

A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565

A318834(n) = { my(m=1); fordiv(n, d, if(d < n, m *= A019565(eulerphi(d)))); m; };

v318835 = rgs_transform(vector(up_to, n, A318834(n)));

A318835(n) = v318835[n];

CROSSREFS

Cf. A000010, A019565, A318831, A318834,

Cf. also A293215, A293217, A293223, A293224, A293232, A300833, A300835.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Sep 04 2018

STATUS

approved

A319709

Filter sequence combining primorial base representations of the proper divisors of n; Restricted growth sequence transform of A319708.

+10
2

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 4, 17, 18, 19, 2, 20, 2, 21, 22, 23, 18, 24, 2, 25, 26, 27, 2, 28, 2, 29, 30, 31, 2, 32, 33, 34, 12, 35, 2, 36, 37, 38, 39, 40, 2, 41, 2, 42, 43, 44, 45, 46, 2, 47, 48, 49, 2, 50, 2, 51, 52, 53, 45, 54, 2, 55, 56, 57, 2, 58, 59, 60, 61, 62, 2, 63, 64, 65, 66, 67, 68, 69, 2, 70, 71

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

OFFSET

1,2

COMMENTS

For all i, j:

a(i) = a(j) => A001065(i) = A001065(j),

a(i) = a(j) => A319713(i) = A319713(j).

LINKS

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

PROG

(PARI)

up_to = 65537;

A276086(n) = { my(i=0, m=1, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), m*=(prime(i)^((n%nextpr)/pr)); n-=(n%nextpr)); pr=nextpr); m; };

A319708(n) = { my(m=1); fordiv(n, d, if(d<n, m *= A276086(d))); (m); };

v319709 = rgs_transform(vector(up_to, n, A319708(n)));

A319709(n) = v319709[n];

CROSSREFS

Cf. A276086, A319708, A319713.

Cf. A293215, A293226, A300835 for similar constructions for other bases.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 03 2018

STATUS

approved

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