A066208 All primes that divide n are of the form prime(2k-1), where prime(k) is k-th prime.

1, 2, 4, 5, 8, 10, 11, 16, 17, 20, 22, 23, 25, 31, 32, 34, 40, 41, 44, 46, 47, 50, 55, 59, 62, 64, 67, 68, 73, 80, 82, 83, 85, 88, 92, 94, 97, 100, 103, 109, 110, 115, 118, 121, 124, 125, 127, 128, 134, 136, 137, 146, 149, 155, 157, 160, 164, 166, 167, 170, 176, 179, 184

Offset: 1

Author

Leroy Quet, Dec 16 2001

Keywords

• nonn

Comments

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

From Peter Munn, Aug 11 2022: (Start)

Closed under multiplication.

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.

(End)

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

Examples

			20 is included because 20 = 2^2 * 5 = p(1)^2 * p(3) and 1 and 3 are both odd.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1001

Crossrefs

Cf. A066207.

See comments for the relationship to A206284, A215366, A297845.

Programs

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

Extensions

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

a(61) and a(62) from Harry J. Smith, Feb 06 2010

1 prepended by Peter Munn, Aug 11 2022