A115170 - OEIS

A115170

Non-cubefree numbers k such that 2k+1 is also non-cubefree (A046099).

40, 256, 312, 472, 688, 904, 1120, 1200, 1312, 1336, 1552, 1768, 1984, 2187, 2200, 2312, 2416, 2456, 2632, 2848, 2875, 3064, 3280, 3312, 3429, 3496, 3712, 3928, 3944, 4144, 4312, 4360, 4576, 4792, 5008, 5224, 5312, 5440, 5562, 5656, 5872, 6088, 6250, 6304, 6312, 6345, 6520, 6655, 6688, 6736, 6952

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OFFSET

1,1

COMMENTS

The probability that a random integer is cubefree is 1/zeta(3) (see A088453).

The asymptotic density of this sequence is 1 - 15/(7*zeta(3)) + (7/8)*Product_{p odd prime} (1-2/p^3) = 1 - (15/7)*A088453 + (7/6)*A340153 = 0.00704953764810822631953... . - Amiram Eldar, Aug 20 2026

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..700 from R. J. Mathar)

EXAMPLE

312 is a term because 2^3 divides 312 and 5^3 divides 312*2+1.

MATHEMATICA

noncfQ[n_] := AnyTrue[FactorInteger[n][[;; , 2]], # > 2 &]; Select[Range[7000], noncfQ[#] && noncfQ[2*# + 1] &] (* Amiram Eldar, May 25 2025 *)

PROG

(PARI) iscubefree(k) = if(k == 1, 1, vecmax(factor(k)[, 2]) < 3);

isok(k) = !iscubefree(k) && !iscubefree(2*k+1); \\ Amiram Eldar, Aug 20 2026

CROSSREFS

Cf. A005384 (n and 2n+1 prime), A046099, A111153 (n and 2n+1 semiprime), A117204 (n and 2n+1 squarefree), A115228.

Cf. A088453, A340153.