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

1,2

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.

Mathematica

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

Prog

(PARI)
A355825(n) = factorback(apply(e->(hammingweight(e)%2), factor(n)[, 2]));
isA270428(n) = A355825(n); \\ Antti Karttunen, Jul 21 2022
(Scheme, two variants, both requiring Antti Karttunen's IntSeq-library)
(define A270428 (ZERO-POS 1 1 (COMPOSE sub1 A270419)))
(definec (chA270428 n) (cond ((= 1 n) 1) (else (* (A010060 (A067029 n)) (chA270428 (A028234 n))))))
(define A270428 (NONZERO-POS 1 1 chA270428))

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.

Sequence in context: A096432 A369938 A138302 * A183220 A187947 A242094

Adjacent sequences: A270425 A270426 A270427 * A270429 A270430 A270431

Keyword

nonn,easy

Author

Antti Karttunen, May 26 2016

Status

approved