A374459 Nonsquarefree exponentially odd numbers.

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536

Offset

1,1

Comments

First differs from A301517 at n = 1213. A301517(1213) = 12500 = 2^2 * 5^5 is not an exponentially odd number.

Numbers whose exponents in their prime factorization are all odd and at least one of them is larger than 1.

All the squarefree numbers (A005117) are exponentially odd. Therefore, the sequence of exponentially odd numbers (A268335) is a disjoint union of the squarefree numbers and this sequence.

The asymptotic density of this sequence is A065463 - A059956 = 0.096515099145... .

Links

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

Formula

a(n) = A268335(A374460(n)).

Sum_{n>=1} 1/a(n)^s = zeta(2*s) * (Product_{p prime} (1 + 1/p^s - 1/p^(2*s))) - zeta(s)/zeta(2*s) for s > 1.

Example

8 = 2^3 is a term since 3 is odd and larger than 1.

Mathematica

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, OddQ] && ! AllTrue[e, # == 1 &]]; Select[Range[1000], q]

Prog

(PARI) is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] %2), return(0))); for(i = 1, #e, if(e[i] >1, return(1))); 0; }

Crossrefs

Intersection of A013929 (or A046099) and A268335.

Subsequence of A301517.

Subsequences: A062838 \ {1}, A065036, A102838, A113850, A113852, A179671, A190011, A335988 \ {1}.

Cf. A005117, A374460.

Sequence in context: A377844 A240111 A301517 * A381312 A195086 A366761

Adjacent sequences: A374456 A374457 A374458 * A374460 A374461 A374462

Keyword

nonn,easy

Author

Amiram Eldar, Jul 09 2024

Status

approved