A078563 a(n) is the least positive integer k such that g(k) = n*g(k-1), where g(k) = prime(k+1) - prime(k).

3, 2, 11, 1022, 34, 46, 4714, 295, 99, 14372, 297, 263, 40026, 429, 985, 161441, 1457, 3087, 413695, 2344, 1879, 278832, 14939, 8423, 887313, 2810, 4260, 3589373, 7810, 13820, 12007816, 51037, 45507, 13186859, 15783, 30765, 6957876, 57765, 24554, 50613572, 23283

Offset

1,1

Comments

Conjecture: The equation g(k) = n*g(k-1), for a fixed positive integer n, is always solvable for k.

Links

Table of n, a(n) for n=1..41.

Example

k = 2 is the least positive integer such that g(k) = 5-3 = 2*(3-2) = 2*g(k-1), so a(2) = 2.

Mathematica

pg[n_] := Module[{r = 0, i = 2, a, b, c, p = False}, While[ ! p, a = Prime[i - 1]; b = Prime[i]; c = Prime[i + 1]; If[c - b == n (b - a), r = i; p = True]; i = i + 1]; r]; Table[pg[i], {i, 1, 30}]

Prog

(PARI) a(n) = {my(g=1, p=3, q); for(k=2, oo, q=p; p=nextprime(p+1); if(g*n == g=p-q, return(k))); } \\ Jinyuan Wang, Jul 30 2020

Crossrefs

Cf. A001223.

Sequence in context: A362995 A178384 A372803 * A016560 A122407 A346122

Adjacent sequences: A078560 A078561 A078562 * A078564 A078565 A078566

Keyword

nonn

Author

Joseph L. Pe, Jan 07 2003

Extensions

More terms from Jinyuan Wang, Jul 30 2020

Status

approved