A066207 All primes that divide n are of the form prime(2k), where prime(k) is k-th prime.

1, 3, 7, 9, 13, 19, 21, 27, 29, 37, 39, 43, 49, 53, 57, 61, 63, 71, 79, 81, 87, 89, 91, 101, 107, 111, 113, 117, 129, 131, 133, 139, 147, 151, 159, 163, 169, 171, 173, 181, 183, 189, 193, 199, 203, 213, 223, 229, 237, 239, 243, 247, 251, 259, 261, 263, 267, 271, 273

Offset

1,2

Comments

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

Numbers divisible only by primes with even indices (A031215). - Michael De Vlieger, Dec 11 2017

Numbers that are not divisible by any prime with an odd index (A031368). - Antti Karttunen, Jul 18 2020

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000 (original first 1000 terms from Harry J. Smith)

Example

39 is included because 3 * 13 = prime(2) * prime(6) and 2 and 6 are both even.

Mathematica

Select[Range[273], AllTrue[PrimePi@ FactorInteger[#][[All, 1]], EvenQ] &] (* Michael De Vlieger, Dec 11 2017, range adjusted by Antti Karttunen, Jul 18 2020 *)

Prog

(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
(PARI) isA066207(n) = (!#select(p -> (primepi(p)%2), factor(n)[, 1])); \\ Antti Karttunen, Jul 18 2020

Crossrefs

Cf. A031215, A031368, A066208, A215366.

Cf. A297002 (a permutation).

Numbers in the odd bisection of A336321.

Sequence in context: A225495 A111223 A266836 * A141544 A172407 A088649

Adjacent sequences: A066204 A066205 A066206 * A066208 A066209 A066210

Keyword

nonn

Author

Leroy Quet, Dec 16 2001

Extensions

Offset changed from 0 to 1 by Harry J. Smith, Feb 06 2010

a(53)-a(58) from Harry J. Smith, Feb 06 2010

a(1) = 1 inserted (and the indexing of the rest of terms changed) by Antti Karttunen, Jul 18 2020

Status

approved