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A323550 Numbers that can be expressed as (p - 1)*(q - 1) + 1, where p < q are primes.

%I A323550 #27 May 22 2025 16:36:21
%S A323550 3,5,7,9,11,13,17,19,21,23,25,29,31,33,37,41,43,45,47,49,53,57,59,61,
%T A323550 65,67,71,73,79,81,83,85,89,93,97,101,103,105,107,109,113,117,121,127,
%U A323550 131,133,137,139,141,145,149,151,157,161,163,165,167,169,173,177,179,181,185,191,193,197,199,201,205,209
%N A323550 Numbers that can be expressed as (p - 1)*(q - 1) + 1, where p < q are primes.
%C A323550 If p < q are primes and a(n) = (p - 1)*(q - 1) + 1, then x^a(n) == x (mod p*q) for every integer x.
%H A323550 Robert Israel, <a href="/A323550/b323550.txt">Table of n, a(n) for n = 1..10000</a>
%H A323550 A. Bogomolny, <a href="http://www.cut-the-knot.org/blue/Euler.shtml">Euler Function and Theorem</a>
%e A323550 181 is a term because 181 = (11 - 1)*(19 - 1) + 1. - _Bernard Schott_, Jan 19 2019
%p A323550 N:= 1000: # for terms <= N
%p A323550 S:= {}:
%p A323550 P:= select(isprime,[2,seq(i,i=3..N,2)]): nP:= nops(P):
%p A323550 for i from 1 to nP do
%p A323550   for j from i+1 to nP do
%p A323550      v:= (P[i]-1)*(P[j]-1)+1;
%p A323550      if v > N then break fi;
%p A323550      S:= S union {v}
%p A323550 od od:
%p A323550 sort(convert(S,list)); # _Robert Israel_, May 22 2025
%t A323550 nmax = 100;
%t A323550 pairs = Table[Table[(Prime[i] - 1)*(Prime[j] - 1) + 1, {i, 1, j - 1}], {j, 2,Prime[nmax]}];
%t A323550 (DeleteDuplicates@Sort@Flatten@pairs)[[1 ;; nmax]]
%o A323550 (PARI) isok(n) = {if (n % 2, forprime(p = 2, n, forprime(q = p+1, n, if (n == (p - 1)*(q - 1) + 1, return (1)););););} \\ _Michel Marcus_, Feb 25 2019
%Y A323550 Cf. A065091.
%K A323550 nonn
%O A323550 1,1
%A A323550 _Andres Cicuttin_, Jan 17 2019