A342188 Numbers k such that both k and k+1 are not exponentially squarefree numbers.

80, 624, 2511, 5264, 6399, 7695, 7856, 10287, 13040, 14640, 15471, 15632, 18063, 19375, 20624, 20816, 23247, 23408, 25839, 27135, 28560, 28592, 31023, 31184, 33615, 35072, 36015, 36368, 38799, 38960, 39375, 40816, 41391, 44144, 46250, 46575, 46736, 49167, 51920

Offset

1,1

Comments

The numbers of terms not exceeding 10^k for k = 2, 3, ..., are 1, 2, 7, 72, 719, 7226, 72238, 722565, 7225651, ... Apparently this sequence has an asymptotic density 0.00007225...

The asymptotic density of this sequence is 1 + Product_{p prime} (1 - 2*(p-1)*Sum_{k>=4, k nonsquarefree} 1/p^(k+1)) - 2*A262276 = 0.000722566657736771344086... (Srichan et al., 2026, a consequence of Theorem 1.1). - Amiram Eldar, Jul 22 2026

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Teerapat Srichan, Nithi Rungtanapirom, and Saeree Wananiyakul, On the distribution of consecutive M-free integers, RAIRO - Theoretical Informatics and Applications, Vol. 60 (2026), Article 22.

Example

80 is a term since 80 = 2^4 * 5 and 81 = 3^4 both have a nonsquarefree exponent in their prime factorization.

Mathematica

expSqFQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], SquareFreeQ]; Select[Range[5*10^4], !expSqFQ[#] && !expSqFQ[# + 1] &]

Prog

(PARI) isA130897(k) = #select(x -> !issquarefree(x), factor(k)[, 2]) > 0;
isok(k) = isA130897(k) && isA130897(k+1); \\ Amiram Eldar, Jul 22 2026

Crossrefs

Cf. A013929, A130897, A209061.

Similar sequences: A068140, A068781, A342187, A342189.

Sequence in context: A388001 A389955 A068782 * A255478 A204342 A235090

Adjacent sequences: A342185 A342186 A342187 * A342189 A342190 A342191

Keyword

nonn,easy,changed

Author

Amiram Eldar, Mar 04 2021

Status

approved