Displaying 1-10 of 17 results found.
Numbers n such that 6n+1, 12n+1 and 18n+1 are all primes.
+10
17
1, 6, 35, 45, 51, 55, 56, 100, 121, 195, 206, 216, 255, 276, 370, 380, 426, 506, 510, 511, 710, 741, 800, 825, 871, 930, 975, 1025, 1060, 1115, 1140, 1161, 1270, 1280, 1281, 1311, 1336, 1361, 1365, 1381, 1420, 1421, 1441, 1490, 1515, 1696, 1805, 1875, 1885
COMMENTS
Main entry for this sequence is A033502. n is a Carmichael number generator giving C(n) = (6n+1)(12n+1)(18n+1).
REFERENCES
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, Wiley NY 1991, page 83, problem #20.
R. K. Guy, Unsolved Problems in Number Theory, A13.
MATHEMATICA
Select[Range[2000], And@@PrimeQ[{6, 12, 18}#+1]&] (* Harvey P. Dale, May 26 2014 *)
Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g.
+10
8
6, 10, 12, 15, 18, 20, 21, 24, 28, 33, 34, 36, 39, 40, 45, 48, 52, 57, 63, 65, 66, 68, 72, 76, 80, 85, 87, 88, 91, 93, 96, 99, 100, 105, 111, 112, 117, 120, 126, 130, 132, 133, 135, 136, 144, 145, 148, 153, 156, 160, 165, 171, 175, 176, 185, 186, 189, 190
COMMENTS
The function s_g(m) gives the sum of the base-g digits of m.
The sequence is infinite, since it contains A324460. The sequence also contains the 3-Carmichael numbers A087788 and the primary Carmichael numbers A324316. A term m must have at least 2 prime factors, and the divisor g satisfies the inequalities 1 < g < m^(1/(ord_g(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.
Note that the sequence contains the 3-Carmichael numbers, but not all Carmichael numbers. This is a nontrivial fact.
The subsequence A324460 mainly gives examples in which g is composite. See Kellner 2019.
It appears that g is usually prime: compare with A324857 (g prime) and the sparser sequence A324858 (g composite). However, g is usually composite for higher values of m. - Jonathan Sondow, Mar 17 2019
EXAMPLE
6 is a member, since 2 divides 6 and s_2(6) = 2.
MATHEMATICA
s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
f[n_] := AnyTrue[Divisors[n], s[n, #] == # &];
Select[Range[5000], f[#] &]
PROG
(PARI) isok(n) = {fordiv(n, d, if ((d>1) && (sumdigits(n, d) == d), return (1)); ); } \\ Michel Marcus, Mar 19 2019
Prime numbers n such that 2n-1 and 3n-2 are prime.
+10
5
3, 7, 37, 211, 271, 307, 331, 337, 601, 727, 1171, 1237, 1297, 1531, 1657, 2221, 2281, 2557, 3037, 3061, 3067, 4261, 4447, 4801, 4951, 5227, 5581, 5851, 6151, 6361, 6691, 6841, 6967, 7621, 7681, 7687, 7867, 8017, 8167, 8191, 8287, 8521, 8527, 8647, 8941
COMMENTS
If n, 2n-1 and 3n-2 are prime numbers, and if n >= 5, then n*(2*n-1)*(3*n-2) is a Carmichael number ( A033502). Proof: there exist numbers m such that n=6m+1 is prime (if n=6m+5, then 2n-1 = 12m+9 is composite). Let p=(6m+1)(12m+1)(18m+1) = a*b*c. Then p-1 = 6*12*18*m^3 + (6*12 + 6*18 + 12*18)*m^2 + (6 + 12 + 19)*m, so p-1 is divisible by a-1=6m, by b-1=12m, and by c-1=18m; thus p is a Carmichael number.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, A13.
EXAMPLE
For n=3, 2n-1 = 5, 3n-2 = 7.
For n=7, 2n-1 = 13, 3n-2 = 19 and 7*13*19 = 1729 (a Carmichael number).
For n=37, 2n-1 = 73, 3n-2 = 109 and 37*73*109 = 294409 (a Carmichael number).
MAPLE
with(numtheory): for n from 2 to 15000 do: if type(n, prime)=true and type(2*n-1, prime)=true and type(3*n-2, prime)=true then print (n):else fi:od:
PROG
(Magma) [ n: n in PrimesUpTo(10000) | IsPrime(2*n-1) and IsPrime(3*n-2) ];
(PARI) forprime(p=3, 10^3, isprime(2*p-1) && isprime(3*p-2) && print1(p, ", ")); \\ Joerg Arndt, Nov 29 2014
Number of 3-component Carmichael numbers C = (6M + 1)(12M + 1)(18M + 1) < 10^n.
+10
4
0, 1, 1, 2, 2, 3, 7, 10, 16, 25, 50, 86, 150, 256, 436, 783, 1435, 2631, 4765, 8766, 16320, 30601, 57719, 109504, 208822, 400643, 771735, 1494772, 2903761, 5658670, 11059937, 21696205, 42670184, 84144873, 66369603, 329733896, 655014986, 1303918824, 2601139051
COMMENTS
Note that this is different from the count of 3-Carmichael numbers, A132195. The numbers counted here are neither those listed in A087788 (3 arbitrary prime factors) nor those listed in A033502 (where 6m + 1, 12m + 1 and 18m + 1 are all prime). - M. F. Hasler, Apr 14 2015
REFERENCES
Posting by Harvey Dubner (harvey(AT)dubner.com) to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Nov 23 1998.
EXTENSIONS
Terms updated (from Dubner's paper) by Amiram Eldar, Aug 11 2017
The least Chernick's "universal form" Carmichael number with n prime factors.
+10
4
1729, 63973, 26641259752490421121, 1457836374916028334162241, 24541683183872873851606952966798288052977151461406721, 53487697914261966820654105730041031613370337776541835775672321, 58571442634534443082821160508299574798027946748324125518533225605795841
COMMENTS
Chernick proved that U(k, m) = (6m + 1)*(12m + 1)*Product_{i = 1..k-2} (9*(2^i)m + 1), for k >= 3 and m >= 1 is a Carmichael number, if all the factors are primes and, for k >= 4, 2^(k-4) divides m. He called U(k, m) "universal forms". This sequence gives a(k) = U(k, m) with the least value of m. The least values of m for k = 3, 4, ... are 1, 1, 380, 380, 780320, 950560, 950560, 3208386195840, 31023586121600, ...
LINKS
Samuel S. Wagstaff, Jr., Large Carmichael numbers, Mathematical Journal of Okayama University, Vol. 22, (1980), pp. 33-41.
EXAMPLE
For k=3, m = 1, a(3) = U(3, 1) = (6*1 + 1)*(12*1 + 1)*(18*1 + 1) = 1729.
For k=4, m = 1, a(4) = U(4, 1) = (6*1 + 1)*(12*1 + 1)*(18*1 + 1)*(36*1 + 1) = 63973.
For k=5, m = 380, a(5) = U(5, 1) = (6*380 + 1)*(12*380 + 1)*(18*380 + 1)*(36*380 + 1)*(72*380 + 1) = 26641259752490421121.
MATHEMATICA
fc[k_] := If[k < 4, 1, 2^(k - 4)]; a={}; Do[v = Join[{6, 12}, 2^Range[k-2]*9];
w = fc[k]; x = v*w; m = 1; While[! AllTrue[x*m + 1, PrimeQ], m++]; c=Times @@ (x*m + 1); AppendTo[a, c], {k, 3, 9}]; a
Carmichael numbers of the form (6*k + 1)*(12*k + 1)*(18*k + 1), where only two of the three numbers 6*k + 1, 12*k + 1, 18*k + 1 are prime.
+10
3
172081, 1773289, 4463641, 47006785, 295643089, 798770161, 1150270849, 1420379065, 1976295241, 18390744505, 122160500281, 134642101321, 215741809801, 228944441089, 263610459505, 321140603665, 374464040689, 444722065201, 676328168881, 1009514855521
PROG
(Magma) lst:=[]; for k in [1..920] do t:={n: n in [1..3] | IsPrime(6*k*n+1)}; if #Set(t) eq 2 then c:=&*[6*k*n+1: n in [1..3]]; if IsOne(c mod CarmichaelLambda(c)) then lst:=Append(lst, c); end if; end if; end for; lst;
Carmichael numbers of the form (6*k + 1)*(12*k + 1)*(18*k + 1), where only one of the three numbers 6*k + 1, 12*k + 1, 18*k + 1 is prime.
+10
3
13992265, 1504651681, 14782305601, 292869912385, 2387706608354185, 4484354372982001, 70895950685971489, 807481759780458361, 1659264916949696161, 118023300545190612481, 140695625117781970705, 11710597360056333492601, 19818019625768288167321
PROG
(Magma) lst:=[]; for k in [1..2482095] do t:={n: n in [1..3] | IsPrime(6*k*n+1)}; if #Set(t) eq 1 then c:=&*[6*k*n+1: n in [1..3]]; if IsOne(c mod CarmichaelLambda(c)) then lst:=Append(lst, c); end if; end if; end for; lst;
Carmichael numbers of the form C = (30n-p)*(60n-(2p+1))*(90n-(3p+2)), where n is a natural number and p, 2p+1, 3p+2 are all three prime numbers.
+10
2
1729, 172081, 294409, 1773289, 4463641, 56052361, 118901521, 172947529, 216821881, 228842209, 295643089, 798770161, 1150270849, 1299963601, 1504651681, 1976295241, 2301745249, 9624742921, 11346205609, 13079177569
COMMENTS
These numbers can be reduced to only two possible forms: C =(30n-23)*(60n-47)*(90n-71) or C = (30n-29)*(60n-59)*(90n-89). In the first form, for the particular case when 30n-23,60n-47 and 90n-71 are all three prime numbers, we obtain the Chernick numbers of the form 10m+1 (for k = 5n-4 we have C = (6k+1)*(12k+1)*(18k+1)). In the second form, for the particular case when 30n-29,60n-59 and 90n-89 are all three prime numbers, we obtain the Chernick numbers of the form 10m+9 (for k = 5n-5 we have C = (6k+1)*(12k+1)*(18k+1)).
So the Chernick numbers can be divided into two categories: Chernick numbers of the form (30n+7)*(60n+13)*(90n+19) and Chernick numbers of the form (30n+1)*(60n+1)*(90n+1).
PROG
(PARI) list(lim)={
my(v=List(), f);
for(k=1, round(solve(x=(lim/162000)^(1/3), lim^(1/3), (30*x-23)*(60*x-47)*(90*x-71)-lim)),
n=(30*k-23)*(60*k-47)*(90*k-71)-1;
f=factor(30*k-23);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(60*k-47);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(90*k-71);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
listput(v, n+1)
);
for(k=2, round(solve(x=(lim/162000)^(1/3), lim^(1/3), (30*x-29)*(60*x-59)*(90*x-89)-lim)),
n=(30*k-29)*(60*k-59)*(90*k-89)-1;
f=factor(30*k-29);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(60*k-59);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(90*k-89);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
listput(v, n+1)
);
vecsort(Vec(v))
Carmichael numbers of the form (6*k+1)*(12*k+1)*(18*k+1) which are the product of four prime numbers.
+10
2
172081, 1773289, 4463641, 295643089, 798770161, 1976295241, 122160500281, 374464040689, 444722065201, 676328168881, 1009514855521, 2382986541601, 3022286597929, 9031805532361, 33648448111489, 155773422536761, 206932492972801, 366715617643441, 708083570971801
MAPLE
with(numtheory); P:=proc(q)local a, b, k, ok, n;
for n from 0 to q do a:=(6*n+1)*(12*n+1)*(18*n+1); b:=ifactors(a)[2];
if issqrfree(a) and nops(b)=4 then ok:=1;
for k from 1 to 4 do if not type((a-1)/(b[k][1]-1), integer) then ok:=0;
break; fi; od; if ok=1 then print(a); fi;
MATHEMATICA
g[n_] := (6*n+1)*(12*n+1)*(18*n+1); testQ[n_] := Block[{p, e}, {p, e} = Transpose@ FactorInteger@ n; e == {1, 1, 1, 1} && Max[Mod[n-1, p-1]] == 0]; Select[g /@ Range[10^4], testQ] (* Giovanni Resta, May 21 2013 *)
PROG
(Magma) [c: n in [1..10^4] | #PrimeDivisors(c) eq 4 and IsOne(c mod CarmichaelLambda(c)) where c is (6*n+1)*(12*n+1)*(18*n+1)];
Numbers k such that (6*k+1)*(12*k+1)*(18*k+1) is a Carmichael number which is the product of four prime numbers.
+10
2
5, 11, 15, 61, 85, 115, 455, 661, 700, 805, 920, 1225, 1326, 1910, 2961, 4935, 5425, 6565, 8175, 10885, 11375, 12155, 13230, 18315, 37800, 39325, 45325, 59726, 69440, 99645, 113120, 121365, 129850, 144685, 211945, 353465, 378940, 389896, 392625
MAPLE
with(numtheory); P:=proc(q)local a, b, k, ok, n;
for n from 0 to q do a:=(6*n+1)*(12*n+1)*(18*n+1); b:=ifactors(a)[2];
if issqrfree(a) and nops(b)=4 then ok:=1;
for k from 1 to 4 do if not type((a-1)/(b[k][1]-1), integer) then ok:=0;
break; fi; od; if ok=1 then print(n); fi;
MATHEMATICA
IsCarmichaelQ[n_] := Module[{f}, If[EvenQ[n] || PrimeQ[n], False, f = Transpose[FactorInteger[n]][[1]]; Union[Mod[n-1, f-1]] == {0}]]; n = 0; t = {}; While[Length[t] < 39, n++; c = (6*n + 1)*(12*n + 1)*(18*n + 1); If[SquareFreeQ[c] && Length[FactorInteger[c]] == 4 && IsCarmichaelQ[c], AppendTo[t, n]]]; t (* T. D. Noe, Jan 23 2013 *)
PROG
(Magma) [n: n in [1..4*10^5] | #PrimeDivisors(c) eq 4 and IsOne(c mod CarmichaelLambda(c)) where c is (6*n+1)*(12*n+1)*(18*n+1)];
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