A351141 - OEIS

%I #20 Mar 17 2022 23:50:34

%S 37,101,577,677,15877,16901,57601,62501,33988901,34035557,113209601,

%T 113507717,121528577,121572677,345960001,346332101,635040001,

%U 635544101,7821633601,7823402501,17748634177,17749167077,24343488577,24344112677,97958984257,97962740101

%N 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 Subsequence of A002496.

%C The corresponding sequence of numbers q - p is a subsequence of A076292.

%C Conjecture: the sequence is infinite.

%C 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 The pair (257, 401) = (16^2+1, 20^2+1) is not in the sequence because 401 - 257 = 144 = 2^4*3^2.

%e The pair (577, 677) = (24^2+1, 26^2+1) is in the sequence because 577 - 677 = 100 = 2^2*5^2.

%e The pair (33988901, 34035557) = (5830^2+1, 5834^2+1) is in the sequence because 33988901 - 34035557 = 46656 = 2^6*3^6.

%p with(numtheory):

%p T:=array(1..26):nn:=350000:q:=5:j:=1:

%p for n from 4 by 2 to nn do:

%p   p:=n^2+1:

%p    if type(p, prime)=true

%p     then

%p      x:=p-q:r:=q:q:=p:

%p      u:=factorset(x):n0:=nops(u):ii:=0:d:=product(u[i],i=1..n0):

%p       for k from 2 to 20 while(ii=0) do:

%p        if d^k=x

%p         then ii=1:T[j]:=r:T[j+1]:=q:j:=j+2:

%p         else

%p        fi:

%p       od:

%p    fi:

%p od:

%p print(T):

%o (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 Cf. A002496, A076292, A096012, A193558, A216330, A339007.

%K nonn

%O 1,1

%A _Michel Lagneau_, Feb 02 2022