%I #50 Aug 08 2023 02:34:05
%S 1,3,7,9,13,19,21,27,29,37,39,43,49,53,57,61,63,71,79,81,87,89,91,101,
%T 107,111,113,117,129,131,133,139,147,151,159,163,169,171,173,181,183,
%U 189,193,199,203,213,223,229,237,239,243,247,251,259,261,263,267,271,273
%N All primes that divide n are of the form prime(2k), where prime(k) is k-th prime.
%C The partitions into even 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: 63 ( = 3*3*7) is in the sequence because it is the Heinz number of the partition [2, 2, 4]. - _Emeric Deutsch_, May 19 2015
%C Numbers divisible only by primes with even indices (A031215). - _Michael De Vlieger_, Dec 11 2017
%C Numbers that are not divisible by any prime with an odd index (A031368). - _Antti Karttunen_, Jul 18 2020
%C For every positive integer m there exists a unique ordered pair of positive integers (j,k) such that m = a(j)*A066208(k). - _Christopher Scussel_, Jul 01 2023
%H Antti Karttunen, <a href="/A066207/b066207.txt">Table of n, a(n) for n = 1..20000</a> (original first 1000 terms from Harry J. Smith)
%e 39 is included because 3 * 13 = prime(2) * prime(6) and 2 and 6 are both even.
%t Select[Range[273], AllTrue[PrimePi@ FactorInteger[#][[All, 1]], EvenQ] &] (* _Michael De Vlieger_, Dec 11 2017, range adjusted by _Antti Karttunen_, Jul 18 2020 *)
%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, b=0; break)); if (b, write("b066207.txt", n++, " ", m); if (n==1000, return)) ) } \\ _Harry J. Smith_, Feb 06 2010
%o (PARI) isA066207(n) = (!#select(p -> (primepi(p)%2), factor(n)[,1])); \\ _Antti Karttunen_, Jul 18 2020
%Y Cf. A031215, A031368, A066208, A215366.
%Y Cf. A297002 (a permutation).
%Y Numbers in the odd bisection of A336321.
%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(53)-a(58) from _Harry J. Smith_, Feb 06 2010
%E a(1) = 1 inserted (and the indexing of the rest of terms changed) by _Antti Karttunen_, Jul 18 2020