cp's OEIS Frontend

A332820 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.

%I A332820 #49 Jan 17 2023 16:32:13
%S A332820 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 A332820 84,86,90,92,93,95,106,110,112,117,119,120,122,123,124,125,141,142,
%U A332820 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 A332820 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 A332820 The positive integers are partitioned between this sequence, A332821 and A332822, which list the integers in respective cosets of the subgroup.
%C A332820 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 A332820 It follows that for any n in this sequence, all powers n^k are present (k >= 0), as are all cubes.
%C A332820 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 A332820 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 A332820 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 A332820 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 A332820 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 A332820 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 A332820 {a(n) : n >= 1} = {1} U {2 * A332822(k) : k >= 1} U {A003961(a(k)) : k >= 1}.
%F A332820 {a(n) : n >= 1} = {1} U {a(k)^2 : k >= 1} U {A331590(2, A332822(k)) : k >= 1}.
%F A332820 From _Peter Munn_, Mar 17 2021: (Start)
%F A332820 {a(n) : n >= 1} = {k : k >= 1, 3|A048675(k)}.
%F A332820 {a(n) : n >= 1} = {k : k >= 1, 3|A195017(k)}.
%F A332820 {a(n) : n >= 1} = {A332821(k)/2 : k >= 1, 2|A332821(k)}.
%F A332820 {a(n) : n >= 1} = {A332822(k)/3 : k >= 1, 3|A332822(k)}.
%F A332820 (End)
%t A332820 Select[Range@ 225, Or[Mod[Total@ #, 3] == 0 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]], # == 1] &] (* _Michael De Vlieger_, Mar 15 2020 *)
%o A332820 (PARI) isA332820(n) =  { my(f = factor(n)); !((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); };
%Y A332820 Positions of zeros in A332823; equivalently, numbers in row 3k of A277905 for some k >= 0.
%Y A332820 Cf. A048675, A195017, A332821, A332822, A353350 (characteristic function), A353348 (its Dirichlet inverse), A359830 (complement).
%Y A332820 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 A332820 Subsequences: A000578\{0}, A006094, A090090, A099788, A245630 (A191002 in ascending order), A244726\{0}, A325698, A338471, A338556, A338907.
%Y A332820 Subsequence of {1} U A268388.
%K A332820 nonn
%O A332820 1,2
%A A332820 _Antti Karttunen_ and _Peter Munn_, Feb 25 2020
%E A332820 New name from _Peter Munn_, Mar 08 2021