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A261385
Least positive integer k such that (prime(prime(k))-1)*(prime(prime(k*n))-1) = prime(p)-1 for some prime p.
3
1, 3, 221, 15, 13, 137, 63, 103, 44, 2, 31, 3, 45, 3, 4, 104, 38, 237, 61, 19, 56, 183, 22, 11, 15, 374, 9, 5, 42, 97, 2, 47, 4, 19, 23, 399, 3, 103, 29, 10, 2, 109, 51, 1, 52, 80, 23, 64, 76, 2, 218, 3, 7, 98, 4, 145, 10, 12, 213, 87, 36, 181, 28, 169, 71, 25, 72, 71, 54, 50
OFFSET
1,2
COMMENTS
Conjecture: Let d be any nonzero integer. Then each positive rational number r can be written as m/n, where m and n are positive integers with (prime(prime(m))+d)*(prime(prime(n))+d) = prime(p)+d for some prime p.
This conjecture implies that for any nonzero integer d the equation x*y = z with x,y,z in the set {prime(p)+d: p is prime} has infinitely many solutions.
EXAMPLE
a(3) = 221 since (prime(prime(221))-1)*(prime(prime(221*3)-1) = (prime(1381)-1)*(prime(4957)-1) = 11446*48130 = 550895980 = prime(28890079)-1 with 28890079 prime.
MATHEMATICA
f[n_]:=Prime[Prime[n]]-1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0; Label[bb]; k=k+1; If[PQ[f[k]*f[k*n]+1], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 70}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 17 2015
STATUS
approved