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
KEYWORD
nonn
AUTHOR
Thomas Ordowski, Sep 18 2018
STATUS
approved