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

%I A374595 #23 Aug 23 2024 10:47:12
%S A374595 2,3,5,7,11,13,17,19,23,29,30,31,37,41,42,43,47,53,59,61,66,67,70,71,
%T A374595 72,73,78,79,83,89,97,101,102,103,105,107,108,109,110,113,114,127,130,
%U A374595 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
%N A374595 Products of an odd number of distinct primes and the square of a number in A268390.
%C A374595 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.
%C A374595 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.
%C A374595 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.
%H A374595 <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>
%F A374595 For prime p, {a(n) : n >= 1} = {A059897(k,p) : k in A268390}.
%F A374595 {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.
%e A374595 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.
%e A374595 72 = 2 * 6^2 is in the sequence since 2 is the product of 1 distinct prime and 6 is in A268390.
%e A374595 6 = 2 * 3 is the product of 2 (an even number) of distinct primes, so 6 is not in the sequence.
%o A374595 (PARI) isA374595 = A374130;
%Y A374595 Positions of 1's in A268387, cf. A268390 (positions of 0's).
%Y A374595 Subsequence of A000028, A026424. Differs from their intersection, A374467, by not having 128, 192, 288, etc.
%Y A374595 Cf. A030059 (squarefree terms), A374130 (characteristic function).
%Y A374595 See the formula section for the relationships with A006519, A008836, A059897.
%K A374595 nonn
%O A374595 1,1
%A A374595 _Antti Karttunen_ and _Peter Munn_, Jul 13 2024