A319521 Completely multiplicative with a(prime(2*k-1)) = prime(k) and a(prime(2*k)) = 1 for any k > 0 (where prime(k) denotes the k-th prime number).

1, 2, 1, 4, 3, 2, 1, 8, 1, 6, 5, 4, 1, 2, 3, 16, 7, 2, 1, 12, 1, 10, 11, 8, 9, 2, 1, 4, 1, 6, 13, 32, 5, 14, 3, 4, 1, 2, 1, 24, 17, 2, 1, 20, 3, 22, 19, 16, 1, 18, 7, 4, 1, 2, 15, 8, 1, 2, 23, 12, 1, 26, 1, 64, 3, 10, 29, 28, 11, 6, 1, 8, 31, 2, 9, 4, 5, 2, 1

Offset: 1

Author

Rémy Sigrist, Sep 22 2018

Keywords

• nonn
• mult

Comments

See A319522 for a similar sequence.

The function n -> (a(n), A319522(n)) establishes a bijection from N to N x N (where N = A000027); see A319523 for the corresponding inverse function.

This sequence has similarities with A059905: here we keep one p-adic valuation out of two, there we keep one binary digit out of two.

Examples

			a(42) = a(prime(1)) * a(prime(2)) * a(prime(4)) = prime(1) * 1 * 1 = 2.
		

Crossrefs

Cf. A000027, A000079, A007814, A059905, A066207, A319522, A319523.

Programs

PARI

a(n) = my (f=factor(n)); prod(i=1, #f~, my (pi=primepi(f[i,1])); if (pi%2==1, prime(1+pi\2)^f[i,2], 1))

Formula

a(n) = 1 iff n = 1 or n belongs to A066207.

a(n) <= n with equality iff n is a power of 2 (A000079).

A007814(a(n)) = A007814(n).