A374595 Products of an odd number of distinct primes and the square of a number in A268390.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 72, 73, 78, 79, 83, 89, 97, 101, 102, 103, 105, 107, 108, 109, 110, 113, 114, 127, 130, 131, 137, 138, 139, 149, 151, 154, 157, 163, 165, 167, 170, 173, 174, 179, 180, 181, 182, 186, 190, 191, 193, 195, 197, 199, 200, 211, 222

Offset: 1

Author

Antti Karttunen and Peter Munn, Jul 13 2024

Keywords

• nonn

Comments

For any prime p: when trailing zeros are removed from the terms written in base p, every positive integer not divisible by p appears (in base p) exactly once. For example, 4 is written "11" in base 3, and "11" appears when the trailing zeros are removed from "11000", which is the term 108 written in base 3.

Numbers m such that when the exponents e_1 .. e_k in the canonical factorization m = p_1^e_1 * ... * p_k^e_k are bitwise-xored together, the result is 1.

As a set, membership is determined by prime signature. It is a coset of A268390 within the positive integers under the binary operation A059897(.,.). Using standard arithmetic operations and any given prime p, this set can be derived from A268390 by dividing by p all the terms of A268390 whose squarefree part is a multiple of p and multiplying by p all the other terms of A268390.

Examples

			30 = 2 * 3 * 5 is the product of an odd number of distinct primes times 1, and 1 is in A268390, so 30 is in the sequence.
72 = 2 * 6^2 is in the sequence since 2 is the product of 1 distinct prime and 6 is in A268390.
6 = 2 * 3 is the product of 2 (an even number) of distinct primes, so 6 is not in the sequence.
		

Links

• Index entries for sequences related to prime signature

Crossrefs

Positions of 1's in A268387, cf. A268390 (positions of 0's).

Subsequence of A000028, A026424. Differs from their intersection, A374467, by not having 128, 192, 288, etc.

Cf. A030059 (squarefree terms), A374130 (characteristic function).

See the formula section for the relationships with A006519, A008836, A059897.

Programs

PARI

isA374595 = A374130;

Formula

For prime p, {a(n) : n >= 1} = {A059897(k,p) : k in A268390}.

{a(n) : n >= 1} = {k * 2^(lambda(A006519(k))): k in A268390}, where lambda is Liouville's function, A008836, and A006519(k) is the highest power of 2 dividing k.