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

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

Author

Jeppe Stig Nielsen, Apr 21 2021

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

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

Crossrefs

Cf. A002997, A328935.

Programs

PARI

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