A270428 Exponentially odious numbers: 1 together with positive integers n such that all exponents in prime factorization of n are odious numbers (A000069).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Author

Antti Karttunen, May 26 2016

Keywords

• nonn
• easy

Comments

A268385 maps each term of this sequence to a unique term of A268335, and vice versa.

The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.87686263163054480657..., where f(x) = 1 - x + (1 - (1-x) * Product_{k>=0} (1-x^(2^k)))/2. - Amiram Eldar, Oct 27 2023

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), 385-395.

Crossrefs

Apart from 1, a subsequence of A270420.

Indices of ones in A270419.

Sequence A270436 sorted into ascending order.

Cf. A010060, A028234, A067029, A355825 (characteristic function).

Cf. also A262675, A268335, A268385.

Differs from its subsequence A138302 for the first time at n=113, where a(113) = 128 = 2^7, a value which does not occur in A138302.

Programs

Mathematica

odiousQ[n_] := OddQ[DigitCount[n, 2, 1]]; Select[Range[100], AllTrue[FactorInteger[#][[;;, 2]], odiousQ] &] (* Amiram Eldar, May 18 2023 *)

PARI

A355825(n) = factorback(apply(e->(hammingweight(e)%2), factor(n)[, 2]));
isA270428(n) = A355825(n); \\ Antti Karttunen, Jul 21 2022

Scheme

;; Requires Antti Karttunen's IntSeq-library.
(define A270428 (ZERO-POS 1 1 (COMPOSE sub1 A270419)))

Scheme

;; Requires Antti Karttunen's IntSeq-library.
(definec (chA270428 n) (cond ((= 1 n) 1) (else (* (A010060 (A067029 n)) (chA270428 (A028234 n))))))
(define A270428 (NONZERO-POS 1 1 chA270428))