A328935 -id:A328935 - cp's OEIS frontend

A328936 The number of imprimitive Carmichael numbers (A328935) below 10^n.

4, 11, 25, 63, 134, 268, 508, 1013, 1901, 3773, 7208, 13834, 26353, 50343, 96122, 184354, 354218

Offset: 6

Amiram Eldar, Oct 31 2019

Granville and Pomerance conjectured that most Carmichael numbers are primitive, i.e. lim_{n->oo} a(n)/A055553(n) = 0.

			a(6) = 4 since there are 4 imprimitive Carmichael numbers below 10^6: 294409, 399001, 488881, 512461.

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A055553, A328935.

a(20)-a(22) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 22 2024

A328937 The number of imprimitive 3-Carmichael numbers (A087788 and A328935) below 10^n.

Original entry on oeis.org

4, 11, 25, 59, 127, 252, 471, 928, 1734, 3462, 6615, 12725, 24396, 46877, 89854, 173331, 334737, 647265, 1253176

Offset: 6

Amiram Eldar, Oct 31 2019

Granville and Pomerance conjectured that most Carmichael numbers are imprimitive, i.e. lim_{n->oo} a(n)/A132195(n) = 1.

			a(6) = 4 since there are 4 imprimitive 3-Carmichael numbers below 10^6: 294409, 399001, 488881, 512461.

J. M. Chick, Carmichael number variable relations: three-prime Carmichael numbers up to 10^24, arXiv preprint arXiv:0711.2915 [math.NT] (2007).
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.

Cf. A002997, A087788, A132195, A328935.

A328938 Least imprimitive Carmichael number (A328935) with n prime factors, or -1 if no such number exists.

Original entry on oeis.org

294409, 167979421, 1152091655881, 62411762908817281, 1516087654274358001

Offset: 3

Amiram Eldar, Oct 31 2019

From Daniel Suteu, Feb 17 2020: (Start)
a(8)  <= 42310088783100741554666880481,
a(9)  <= 21593590390253023722267234622513201,
a(10) <= 16412975107923138847512341751620644377601,
a(11) <= 325533792014488126487416882038879701391121. (End)
a(8) > 10^22. - Amiram Eldar, Apr 22 2024

Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A006931, A328935.

Escape clause added by Amiram Eldar, Apr 22 2024

A335584 Carmichael numbers (A002997) that are not minimal in their family.

Original entry on oeis.org

294409, 488881, 1152271, 3057601, 3828001, 6189121, 17098369, 19384289, 53711113, 56052361, 64377991, 82929001, 115039081, 118901521, 171454321, 172947529, 214852609, 216821881, 228842209, 279377281, 288120421, 328573477, 366652201, 492559141, 542497201

Offset: 1

Jeppe Stig Nielsen, Apr 21 2021

nonn

Let a = p_1 * p_2 *...* p_k and b = q_1 * q_2 *...* q_k be two Charmichael numbers (A002997) with the same number of factors, where p_1 < p_2 <...< p_k and q_1 < q_2 <...< q_k are primes. We say that a and b are in the same family iff the vectors [p_1 - 1, ..., p_k - 1] and [q_1 - 1, ..., q_k - 1] are parallel. In other words, the ratios (p_1-1):(p_2-1):...:(p_k-1) and (q_1-1):(q_2-1):...:(q_k-1) are equal. Sequence gives Carmichael numbers that are NOT minimal in their family.

Not a subsequence of A328935 (for example 965507554621 is primitive but not minimal).

			294409 = 37*73*109 is a Carmichael number, belonging to family 36:72:108 = 1:2:3. However, 1729 = 7*13*19 is smaller Carmichael number, and the family 6:12:18 = 1:2:3 is the same. Therefore 294409 belongs to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A328935.

is(m)=!is_A002997(m)&&return(0);f=factor(m);p=f[,1]~;r=apply(x->x-1,p);g=gcd(r);a=r/g;for(i=1,g-1,t=prod(j=1,#a,i*a[j]+1);bigomega(t)==bigomega(m)&&is_A002997(t)&&return(1));0 \\ use with suitable PROG from A002997

cp's OEIS Frontend

A328936 The number of imprimitive Carmichael numbers (A328935) below 10^n.

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A328937 The number of imprimitive 3-Carmichael numbers (A087788 and A328935) below 10^n.

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A328938 Least imprimitive Carmichael number (A328935) with n prime factors, or -1 if no such number exists.

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A335584 Carmichael numbers (A002997) that are not minimal in their family.

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