A261385 Least positive integer k such that (prime(prime(k))-1)*(prime(prime(k*n))-1) = prime(p)-1 for some prime p.

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.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

Zhi-Wei Sun, Checking the conjecture for d = -1 and r = a/b (a,b = 1..60)

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

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

Cf. A000040, A257938, A261353.

Sequence in context: A172811 A247070 A140441 * A071881 A157557 A279532

Adjacent sequences: A261382 A261383 A261384 * A261386 A261387 A261388

Keyword

nonn

Author

Zhi-Wei Sun, Aug 17 2015

Status

approved