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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).
3
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.
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))
KEYWORD
nonn,mult
AUTHOR
Rémy Sigrist, Sep 22 2018
STATUS
approved