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

1,1

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.

Links

Table of n, a(n) for n=1..26.

Example

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.

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):

Prog

(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

Crossrefs

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

Sequence in context: A142941 A176973 A105019 * A090496 A005107 A139934

Adjacent sequences: A351138 A351139 A351140 * A351142 A351143 A351144

Keyword

nonn

Author

Michel Lagneau, Feb 02 2022

Status

approved