%I #44 Sep 09 2023 11:27:09
%S 1,2,4,5,8,10,11,16,17,20,22,23,25,31,32,34,40,41,44,46,47,50,55,59,
%T 62,64,67,68,73,80,82,83,85,88,92,94,97,100,103,109,110,115,118,121,
%U 124,125,127,128,134,136,137,146,149,155,157,160,164,166,167,170,176,179,184
%N All primes that divide n are of the form prime(2k-1), where prime(k) is k-th prime.
%C The partitions into odd parts, encoded by their Heinz numbers. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by _Alois P. Heinz_ in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: 50 ( = 2*5*5) is in the sequence because it is the Heinz number of the partition [1, 3, 3]. - _Emeric Deutsch_, May 19 2015
%C From _Peter Munn_, Aug 11 2022: (Start)
%C Closed under multiplication.
%C Encodings, as defined in A206284, of even polynomials with nonnegative integer coefficients; so closed under application of A297845(.,.), which represents the multiplication of polynomials encoded this way.
%C (End)
%C For every positive integer m there exists a unique ordered pair of positive integers (j,k) such that m = a(j)*A066207(k). - _Christopher Scussel_, Aug 08 2023
%H Harry J. Smith, <a href="/A066208/b066208.txt">Table of n, a(n) for n = 1..1001</a>
%e 20 is included because 20 = 2^2 * 5 = p(1)^2 * p(3) and 1 and 3 are both odd.
%o (PARI) { n=0; for (m=2, 10^9, f=factor(m); b=1; for(i=1, matsize(f)[1], if (primepi(f[i, 1])%2 == 0, b=0; break)); if (b, write("b066208.txt", n++, " ", m); if (n==1000, return)) ) } \\ _Harry J. Smith_, Feb 06 2010
%Y Cf. A066207.
%Y See comments for the relationship to A206284, A215366, A297845.
%K nonn
%O 1,2
%A _Leroy Quet_, Dec 16 2001
%E Offset changed from 0 to 1 by _Harry J. Smith_, Feb 06 2010
%E a(61) and a(62) from _Harry J. Smith_, Feb 06 2010
%E 1 prepended by _Peter Munn_, Aug 11 2022