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A351141
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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.
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0
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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):
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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