A373210 Least k such that the smallest prime >= 2^k is 2^k + 2*n + 1, or -1 if no such k exists.

0, 3, 5, 10, 9, 29, 64, 22, 13, 162, 19, 39, 34, 14, 17, 36, 60, 25, 74, 87, 121, 24, 151, 209, 170, 111, 35, 50, 188, 45, 96, 247, 193, 124, 49, 115, 258, 83, 173, 254, 56, 167, 136, 138, 279, 148, 314, 153, 158, 106, 199, 434, 93, 161, 6954, 104, 719, 240, 164

Offset: 0

Author

Jianing Song, May 28 2024

Keywords

• sign

Comments

a(n) = -1 if 2*n + 1 is a Sierpiński number (for example when 2*n + 1 = 78557); cf. A076336. See also A067760.

Conjecture: a(n) != -1 if 2*n + 1 is not a Sierpiński number. In other words, if 2*n + 1 is not a Sierpiński number, then there exists some k >= 1 such that 2^k + 1, 2^k + 3, ..., 2^k + 2*n - 1 are all composite while 2^k + 2*n + 1 is prime.

a(54), a(75), a(83), a(128), a(159), a(176), ... > 5000 (if not equal to -1), which means that 109, 151, 167, 257, 319, 353, ... do not present among the first 5000 terms of A092131.

a(75) = 5880, a(83) = 5513. - Michael S. Branicky, May 28 2024

a(128) > 7000. - Michael S. Branicky, May 30 2024

Examples

			a(6) = 64, because the smallest prime >= 2^k is not 2^k + 13 for 0 <= k <= 63, while the smallest prime >= 2^64 is 2^64 + 13.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 0..127

Crossrefs

Cf. A067760, A076336, A092131.

Programs

PARI

A373210_first_N_terms(N) = my(v = vector(N+1, i, -1), dist); v[1] = 0; for(i=2, oo, dist = nextprime(2^i) - 2^i; if(dist <= 2*N+1 && v[(dist+1)/2] == -1, v[(dist+1)/2] = i); if(vecmin(v) > -1, return(v))) \\ Warning: ignoring Sierpinski numbers

Extensions

a(54) and beyond from Michael S. Branicky, May 29 2024