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A268375
Numbers k for which A001222(k) = A267116(k).
16
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, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 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
OFFSET
1,2
COMMENTS
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.
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
A225546 maps the set of terms 1:1 onto A138302. - Peter Munn, Jan 26 2020
Equivalently, numbers k for which A064547(k) = A331591(k). - Amiram Eldar, Dec 23 2023
LINKS
EXAMPLE
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.
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.
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.
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.
MATHEMATICA
{1}~Join~Select[Range@ 160, PrimeOmega@ # == BitOr @@ Map[Last, FactorInteger@ #] &] (* Michael De Vlieger, Feb 04 2016 *)
PROG
(Scheme, with Antti Karttunen's IntSeq-library)
(define A268375 (ZERO-POS 1 1 A268374))
CROSSREFS
Indices of zeros in A268374, also in A289618.
Cf. A091862 (characteristic function), A268376 (complement).
Cf. A000961, A054753, A191555 (subsequences).
Related to A138302 via A225546.
Cf. also A318363 (a permutation).
Sequence in context: A347454 A119848 A265640 * A048683 A231876 A334298
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Feb 03 2016
STATUS
approved