A162730 -id:A162730 - cp's OEIS frontend

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

Thomas Ordowski, Sep 18 2018

nonn

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.

			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.

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

Subsequence of A046388.
Complement of A162730 w.r.t. A006881.
Cf. A001358, A121707.

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

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 *)

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

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

A180074 Squarefree semiprimes s=p*q, p

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 133, 134, 141, 142, 145, 146, 158, 159, 166, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 217, 218, 219, 226, 237

Offset: 1

Juri-Stepan Gerasimov, Jan 14 2011

nonn

It may seem that this is a subsequence of A162730, but it is not so, 131801 being the first counterexample. - Michel Marcus, Sep 19 2018

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A006881, A015910, A162730, A179976.

f[n_]:=With[{f=FactorInteger[n][[All,1]]},PowerMod[ 2,Times@@f,Times@@f] == 2^f[[1]]]; Select[Range[250],PrimeOmega[#]==2&&SquareFreeQ[#]&&f[#]&] (* Harvey P. Dale, Jun 06 2017 *)

isok(n) = {if ((bigomega(n) == 2) && (omega(n) == 2), my(p = factor(n)[1, 1]); lift(Mod(2, n)^n) == 2^p);} \\ Michel Marcus, Sep 19 2018

Definition and terms corrected by R. J. Mathar, Jan 14 2011

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

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A180074 Squarefree semiprimes s=p*q, p

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