cp's OEIS Frontend

A351141 Pairs of primes (p,q) = (A002496(m), A002496(m+1)) such that q-p is a power r of the product of its prime factors for some m.

%I A351141 #20 Mar 17 2022 23:50:34
%S A351141 37,101,577,677,15877,16901,57601,62501,33988901,34035557,113209601,
%T A351141 113507717,121528577,121572677,345960001,346332101,635040001,
%U A351141 635544101,7821633601,7823402501,17748634177,17749167077,24343488577,24344112677,97958984257,97962740101
%N A351141 Pairs of primes (p,q) = (A002496(m), A002496(m+1)) such that q-p is a power r of the product of its prime factors for some m.
%C A351141 Subsequence of A002496.
%C A351141 The corresponding sequence of numbers q - p is a subsequence of A076292.
%C A351141 Conjecture: the sequence is infinite.
%C A351141 The corresponding powers r are given by the sequence b(n) = 6, 2, 10, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... It seems that b(n) = 2 for n > 5.
%e A351141 The pair (257, 401) = (16^2+1, 20^2+1) is not in the sequence because 401 - 257 = 144 = 2^4*3^2.
%e A351141 The pair (577, 677) = (24^2+1, 26^2+1) is in the sequence because 577 - 677 = 100 = 2^2*5^2.
%e A351141 The pair (33988901, 34035557) = (5830^2+1, 5834^2+1) is in the sequence because 33988901 - 34035557 = 46656 = 2^6*3^6.
%p A351141 with(numtheory):
%p A351141 T:=array(1..26):nn:=350000:q:=5:j:=1:
%p A351141 for n from 4 by 2 to nn do:
%p A351141   p:=n^2+1:
%p A351141    if type(p, prime)=true
%p A351141     then
%p A351141      x:=p-q:r:=q:q:=p:
%p A351141      u:=factorset(x):n0:=nops(u):ii:=0:d:=product(u[i],i=1..n0):
%p A351141       for k from 2 to 20 while(ii=0) do:
%p A351141        if d^k=x
%p A351141         then ii=1:T[j]:=r:T[j+1]:=q:j:=j+2:
%p A351141         else
%p A351141        fi:
%p A351141       od:
%p A351141    fi:
%p A351141 od:
%p A351141 print(T):
%o A351141 (PARI) lista(nn) = my(lastp=2); forprime(p=nextprime(lastp+1), nn, if (issquare(p-1), if (ispowerful(p-lastp), my(f=factor(p-lastp)[,2]); if (vecmin(f) == vecmax(f), print1(lastp, ", ", p, ", "));); lastp = p;);); \\ _Michel Marcus_, Feb 03 2022
%Y A351141 Cf. A002496, A076292, A096012, A193558, A216330, A339007.
%K A351141 nonn
%O A351141 1,1
%A A351141 _Michel Lagneau_, Feb 02 2022