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A329979 Prime numbers which can be represented as p^i * q^j - (p + q) where p and q are distinct odd primes and i,j > 0.

%I A329979 #27 Aug 26 2025 14:23:14
%S A329979 7,11,19,23,31,37,43,47,53,59,67,71,79,83,101,103,107,127,131,137,139,
%T A329979 149,163,167,179,181,191,199,211,223,229,233,239,251,263,271,283,293,
%U A329979 307,311,331,347,349,359,367,373,379,383,397,419,421,431,439,443,463,467,479,491,499
%N A329979 Prime numbers which can be represented as p^i * q^j - (p + q) where p and q are distinct odd primes and i,j > 0.
%C A329979 Numbers of this form are an attempt to generalize Mersenne numbers (see link).
%H A329979 Robert Israel, <a href="/A329979/b329979.txt">Table of n, a(n) for n = 1..10000</a>
%H A329979 Craig J. Beisel, <a href="https://math.stackexchange.com/questions/3451929/can-it-be-shown-that-numbers-of-a-certain-form-produce-primes-more-often-than-ex">Can it be shown that numbers of a certain form produce primes more often than expected?</a>, Math StackExchange, November 2019.
%p A329979 N:= 1000: # for terms <= N
%p A329979 P:= select(isprime, [seq(i,i=3..(N+3)/2,2)]):
%p A329979 S:= {}:
%p A329979 for ip from 1 to nops(P) do
%p A329979   p:= P[ip];
%p A329979   for i from 1 while p^i*3 - (p+3) <= N do
%p A329979     for iq from 1 to ip-1 do
%p A329979        q:= P[iq];
%p A329979        if p^i*q - (p+q) > N then break fi;
%p A329979        for j from 1 do
%p A329979          x:= p^i * q^j - (p+q);
%p A329979          if x > N then break fi;
%p A329979          if isprime(x) then S:= S union {x} fi;
%p A329979 od od od od:
%p A329979 sort(convert(S,list));  # _Robert Israel_, Aug 25 2025
%o A329979 (PARI) z=[];forprime(a=3,1000, forprime(b=a+2,1000, for(i=1,10, for(j=1,10, y=a+b; x=a^i*b^j-y; if(x<500 && isprime(x) && setsearch(z,x)==0,z=setunion(z,[x])) )))); print(z)
%K A329979 nonn,changed
%O A329979 1,1
%A A329979 _Craig J. Beisel_, Nov 26 2019
%E A329979 Definition clarified by _Robert Israel_, Aug 25 2025