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A345331
Odd numbers k > 1 such that m^(2^v(k-1)+1) == -m (mod k) has more than one solution modulo k, where v(k) = A007814(k) is the 2-adic valuation of k.
3
15, 35, 39, 51, 55, 75, 85, 87, 91, 95, 111, 115, 119, 123, 135, 143, 153, 155, 159, 175, 183, 187, 195, 203, 205, 215, 219, 221, 235, 247, 255, 259, 267, 275, 287, 291, 295, 299, 303, 315, 319, 323, 327, 335, 339, 351, 355, 357, 365, 371, 375, 391, 395, 403
OFFSET
1,1
COMMENTS
Note that for even k, m == -1 (mod k) is a solution.
All terms are composite.
Odd composite k is a term if and only if v(p-1) > v(k-1) for some prime factors p of k. See A345330 for a proof.
This sequence and the Carmichael numbers (A002997) are disjoint: if k is a Carmichael number, then p-1 | k-1 for all prime factors p.
LINKS
EXAMPLE
51 is a term since 51 = 3 * 17 and v(17-1) = 4 > v(51-1) = 1. Also, m^(2^v(51-1)+1) == -m (mod 51) has three solutions: m == 0, 21, 30 (mod 51).
PROG
(PARI) isA345331(n) = if(!isprime(n) && n>1 && n%2, my(f=factor(n), w=omega(n)); for(i=1, w, if(valuation(f[i, 1]-1, 2) > valuation(n-1, 2), return(1))); 0, 0)
CROSSREFS
Complement of A345330 with respect to A071904.
Sequence in context: A329589 A356794 A154988 * A360110 A359163 A327934
KEYWORD
nonn
AUTHOR
Jianing Song, Jun 14 2021
STATUS
approved