A319386 Semiprimes k = pq with primes p < q such that p-1 does not divide q-1.

35, 55, 77, 95, 115, 119, 143, 155, 161, 187, 203, 209, 215, 221, 235, 247, 253, 287, 295, 299, 319, 323, 329, 335, 355, 371, 377, 391, 395, 403, 407, 413, 415, 437, 473, 493, 497, 515, 517, 527, 533, 535, 551, 559, 581, 583, 589, 611, 623, 629, 635, 649, 655, 667, 689, 695, 697, 707, 713, 731

Offset

1,1

Comments

The "anti-Carmichael semiprimes" defined: semiprimes k such that lpf(k)-1 does not divide k-1; then also gpf(k)-1 does not divide k-1.

All the terms are odd and indivisible by 3.

If k is in the sequence, then gcd(k,b^k-b)=1 for some integer b.

These numbers are probably all semiprimes in A121707.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

35 = 5*7 is a term since 5-1 does not divide 7-1.
35 is a term since lpf(35)-1 = 5-1 does not divide 35-1.

Maple

N:= 1000: # for terms <= N
P:= select(isprime, {seq(i, i=5..N/5, 2)}):
S:= {}:
for p in P do
  Qs:= select(q -> q > p and q <= N/p and (q-1 mod (p-1) <> 0), P);
  S:= S union map(`*`, Qs, p);
od:
sort(convert(S, list)); # Robert Israel, Apr 14 2020

Mathematica

spndQ[n_]:=Module[{fi=FactorInteger[n][[All, 1]]}, PrimeOmega[n]==2 && Length[ fi]==2&&Mod[fi[[2]]-1, fi[[1]]-1]!=0]; Select[Range[800], spndQ] (* Harvey P. Dale, Jun 06 2021 *)

Prog

(PARI) isok(n) = {if ((bigomega(n) == 2) && (omega(n) == 2), my(p = factor(n)[1, 1], q = factor(n)[2, 1]); (q-1) % (p-1) != 0; ); }  \\ Michel Marcus, Sep 18 2018
(PARI) list(lim)=my(v=List(), s=sqrtint(lim\=1)); forprime(q=7, lim\5, forprime(p=5, min(min(q-2, s), lim\q), if((q-1)%(p-1), listput(v, p*q)))); Set(v) \\ Charles R Greathouse IV, Apr 14 2020

Crossrefs

Subsequence of A046388.

Complement of A162730 w.r.t. A006881.

Cf. A001358, A121707.

Sequence in context: A335902 A121707 A267999 * A157352 A176255 A355814

Adjacent sequences: A319383 A319384 A319385 * A319387 A319388 A319389

Keyword

nonn

Author

Thomas Ordowski, Sep 18 2018

Status

approved