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.

37, 101, 577, 677, 15877, 16901, 57601, 62501, 33988901, 34035557, 113209601, 113507717, 121528577, 121572677, 345960001, 346332101, 635040001, 635544101, 7821633601, 7823402501, 17748634177, 17749167077, 24343488577, 24344112677, 97958984257, 97962740101

Offset: 1

Author

Michel Lagneau, Feb 02 2022

Keywords

• nonn

Comments

Subsequence of A002496.

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

Conjecture: the sequence is infinite.

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.

Examples

			The pair (257, 401) = (16^2+1, 20^2+1) is not in the sequence because 401 - 257 = 144 = 2^4*3^2.
The pair (577, 677) = (24^2+1, 26^2+1) is in the sequence because 577 - 677 = 100 = 2^2*5^2.
The pair (33988901, 34035557) = (5830^2+1, 5834^2+1) is in the sequence because 33988901 - 34035557 = 46656 = 2^6*3^6.
		

Crossrefs

Cf. A002496, A076292, A096012, A193558, A216330, A339007.

Programs

Maple

with(numtheory):
T:=array(1..26):nn:=350000:q:=5:j:=1:
for n from 4 by 2 to nn do:
  p:=n^2+1:
   if type(p, prime)=true
    then
     x:=p-q:r:=q:q:=p:
     u:=factorset(x):n0:=nops(u):ii:=0:d:=product(u[i],i=1..n0):
      for k from 2 to 20 while(ii=0) do:
       if d^k=x
        then ii=1:T[j]:=r:T[j+1]:=q:j:=j+2:
        else
       fi:
      od:
   fi:
od:
print(T):

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