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%I #49 Jan 17 2023 16:32:13
%S 1,6,8,14,15,20,26,27,33,35,36,38,44,48,50,51,58,63,64,65,68,69,74,77,
%T 84,86,90,92,93,95,106,110,112,117,119,120,122,123,124,125,141,142,
%U 143,145,147,156,158,160,161,162,164,170,171,177,178,185,188,196,198,201,202,208,209,210,214,215,216,217,219,221,225
%N Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.
%C The positive integers are partitioned between this sequence, A332821 and A332822, which list the integers in respective cosets of the subgroup.
%C As the sequence lists the integers in a multiplicative subgroup of the positive rationals, the sequence is closed under multiplication and, provided the result is an integer, under division.
%C It follows that for any n in this sequence, all powers n^k are present (k >= 0), as are all cubes.
%C If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting numbers are a permutation of the full sequence; and if we take the square root of each square term we get the full sequence.
%C There are no primes in the sequence, therefore if k is present and p is a prime, k*p and k/p are absent (noting that k/p might not be an integer). This property extends from primes to all terms of A050376 (often called Fermi-Dirac primes), therefore to squares of primes, 4th powers of primes etc.
%C The terms are the even numbers in A332821 halved. The terms are also the numbers m such that 5m is in A332821, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332822, and so on for alternate primes: 7, 13, 19 etc.
%C The numbers that are half of the even terms of this sequence are in A332822, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332821, which consists exactly of those numbers. These properties extend in a pattern of alternating primes as described in the previous paragraph.
%C If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.
%C If m and n are in this sequence then so is m*n (the definition of "multiplicative semigroup"), while if n is in this sequence, and x is in the complement A359830, then n*x is in A359830. This essentially follows from the fact that A048675 is totally additive sequence. Compare to A329609. - _Antti Karttunen_, Jan 17 2023
%F {a(n) : n >= 1} = {1} U {2 * A332822(k) : k >= 1} U {A003961(a(k)) : k >= 1}.
%F {a(n) : n >= 1} = {1} U {a(k)^2 : k >= 1} U {A331590(2, A332822(k)) : k >= 1}.
%F From _Peter Munn_, Mar 17 2021: (Start)
%F {a(n) : n >= 1} = {k : k >= 1, 3|A048675(k)}.
%F {a(n) : n >= 1} = {k : k >= 1, 3|A195017(k)}.
%F {a(n) : n >= 1} = {A332821(k)/2 : k >= 1, 2|A332821(k)}.
%F {a(n) : n >= 1} = {A332822(k)/3 : k >= 1, 3|A332822(k)}.
%F (End)
%t Select[Range@ 225, Or[Mod[Total@ #, 3] == 0 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]], # == 1] &] (* _Michael De Vlieger_, Mar 15 2020 *)
%o (PARI) isA332820(n) = { my(f = factor(n)); !((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); };
%Y Positions of zeros in A332823; equivalently, numbers in row 3k of A277905 for some k >= 0.
%Y Cf. A048675, A195017, A332821, A332822, A353350 (characteristic function), A353348 (its Dirichlet inverse), A359830 (complement).
%Y Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
%Y Subsequences: A000578\{0}, A006094, A090090, A099788, A245630 (A191002 in ascending order), A244726\{0}, A325698, A338471, A338556, A338907.
%Y Subsequence of {1} U A268388.
%K nonn
%O 1,2
%A _Antti Karttunen_ and _Peter Munn_, Feb 25 2020
%E New name from _Peter Munn_, Mar 08 2021
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