A268375 - OEIS

%I #39 Dec 23 2023 14:31:57

%S 1,2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,25,27,28,29,31,32,37,41,

%T 43,44,45,47,48,49,50,52,53,59,61,63,64,67,68,71,73,75,76,79,80,81,83,

%U 89,92,97,98,99,101,103,107,109,112,113,116,117,121,124,125,127,128,131,137,139,144,147,148,149,151,153

%N Numbers k for which A001222(k) = A267116(k).

%C Numbers k whose prime factorization k = p_1^e_1 * ... * p_m^e_m contains no pair of exponents e_i and e_j (i and j distinct) whose base-2 representations have at least one shared digit-position in which both exponents have a 1-bit.

%C Equivalently, numbers k such that the factors in the (unique) factorization of k into powers of squarefree numbers with distinct exponents that are powers of two, are prime powers. For example, this factorization of 90 is 10^1 * 3^2, so 90 is not included, as 10 is not prime; whereas this factorization of 320 is 5^1 * 2^2 * 2^4, so 320 is included as 5 and 2 are both prime. - _Peter Munn_, Jan 16 2020

%C A225546 maps the set of terms 1:1 onto A138302. - _Peter Munn_, Jan 26 2020

%C Equivalently, numbers k for which A064547(k) = A331591(k). - _Amiram Eldar_, Dec 23 2023

%H Antti Karttunen, <a href="/A268375/b268375.txt">Table of n, a(n) for n = 1..10000</a>

%e 12 = 2^2 * 3^1 is included in the sequence as the exponents 2 ("10" in binary) and 1 ("01" in binary) have no 1-bits in the same position, and 18 = 2^1 * 3^2 is included for the same reason.

%e On the other hand, 24 = 2^3 * 3^1 is NOT included in the sequence as the exponents 3 ("11" in binary) and 1 ("01" in binary) have 1-bit in the same position 0.

%e 720 = 2^4 * 3^2 * 5^1 is included as the exponents 1, 2 and 4 ("001", "010" and "100" in binary) have no 1-bits in shared positions.

%e Likewise, 10! = 3628800 = 2^8 * 3^4 * 5^2 * 7^1 is included as the exponents 1, 2, 4 and 8 ("0001", "0010", "0100" and "1000" in binary) have no 1-bits in shared positions. And similarly for any term of A191555.

%t {1}~Join~Select[Range@ 160, PrimeOmega@ # == BitOr @@ Map[Last, FactorInteger@ #] &] (* _Michael De Vlieger_, Feb 04 2016 *)

%o (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define A268375 (ZERO-POS 1 1 A268374))

%Y Indices of zeros in A268374, also in A289618.

%Y Cf. A001222, A064547, A267116, A046645, A289617, A331591.

%Y Cf. A091862 (characteristic function), A268376 (complement).

%Y Cf. A000961, A054753, A191555 (subsequences).

%Y Related to A138302 via A225546.

%Y Cf. also A318363 (a permutation).

%K nonn,base

%O 1,2

%A _Antti Karttunen_, Feb 03 2016