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A335584
Carmichael numbers (A002997) that are not minimal in their family.
2
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
1,1
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).
EXAMPLE
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.
PROG
(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
CROSSREFS
Sequence in context: A224973 A328664 A328935 * A182206 A178997 A328938
KEYWORD
nonn
AUTHOR
Jeppe Stig Nielsen, Apr 21 2021
STATUS
approved