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

1,2

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.

Links

Table of n, a(n) for n=1..79.

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).

Example

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

Prog

(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))

Crossrefs

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

Sequence in context: A303781 A216377 A253515 * A281589 A302436 A283167

Adjacent sequences: A319518 A319519 A319520 * A319522 A319523 A319524

Keyword

nonn,mult

Author

Rémy Sigrist, Sep 22 2018

Status

approved