A378252 Least prime power > 2^n.

2, 3, 5, 9, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659, 4294967311, 8589934609

Offset

0,1

Comments

Prime powers are listed by A246655.

Conjecture: All terms except 9 are prime. Hence this is the same as A014210 after 9. Confirmed up to n = 1000.

From Lechoslaw Ratajczak, Oct 13 2025: (Start)

Let b(i,j,t) (i >= 1, 1 <= j <= i, t >= 1) be the recursive sequence defined by b(i,j,1) = i mod j, b(i,j,t) = (i + Sum_{k=1..t-1} b(i,j,k)) mod j, and s(m) counts numbers u (1 <= u <= m) for which there is such r that b(m,u,r) attains value of 0 (the initial terms of s(m) (m >= 1) are: 1, 2, 3, 3, 4, 5, 4, 4, 7, 6,...).

Conjecture: if i and j are coprime then b(i,j,t) attains value of 0 iff j is power of 2. If this conjecture and the Wiseman conjecture are true, the consecutive indices of records of the sequence c(d) = totient(d) - d + s(d) (d >= 2) are consecutive terms of (a(n)) (the initial terms of c(d) (d >= 2) are: 1, 2, 1, 3, 1, 3, 0, 4, 0, 4,...). (End)

Links

Table of n, a(n) for n=0..33.

Mathematica

Table[NestWhile[#+1&, 2^n+1, !PrimePowerQ[#]&], {n, 0, 20}]

Prog

(Python)
from itertools import count
from sympy import primefactors
def A378252(n): return next(i for i in count(1+(1<<n)) if len(primefactors(i))==1) # Chai Wah Wu, Dec 02 2024
(PARI) a(n) = my(x=2^n+1); while (!isprimepower(x), x++); x; \\ Michel Marcus, Dec 03 2024

Crossrefs

Subtracting 2^n appears to give A013597 except at term 3.

For prime we have A014210.

For previous we have A014234.

For perfect power we have A357751.

For squarefree we have A372683.

A000015 gives the least prime power >= n.

A000040 lists the primes, differences A001223.

A000961 and A246655 list the prime powers, differences A057820.

A024619 and A361102 list the non prime powers, diffs A375708 and A375735.

A031218 gives the greatest prime power <= n.

A244508 counts prime powers between powers of 2.

Prime powers between primes are counted by A080101 and A366833.

Cf. A007918, A037035, A053707, A059305, A065514, A069584, A151800, A304521, A377051, A377282, A377283, A377289, A377432, A378249.

Sequence in context: A341960 A137155 A014227 * A334816 A064769 A320641

Adjacent sequences: A378249 A378250 A378251 * A378253 A378254 A378255

Keyword

nonn

Author

Gus Wiseman, Nov 30 2024

Status

approved