A295665 Fully multiplicative with a(prime(m)) = prime(k) when m = prime(k), and a(prime(m)) = 1 when m is not a prime.

1, 1, 2, 1, 3, 2, 1, 1, 4, 3, 5, 2, 1, 1, 6, 1, 7, 4, 1, 3, 2, 5, 1, 2, 9, 1, 8, 1, 1, 6, 11, 1, 10, 7, 3, 4, 1, 1, 2, 3, 13, 2, 1, 5, 12, 1, 1, 2, 1, 9, 14, 1, 1, 8, 15, 1, 2, 1, 17, 6, 1, 11, 4, 1, 3, 10, 19, 7, 2, 3, 1, 4, 1, 1, 18, 1, 5, 2, 1, 3, 16, 13, 23, 2, 21, 1, 2, 5, 1, 12, 1, 1, 22, 1, 3, 2, 1, 1, 20, 9, 1, 14, 1, 1, 6

Offset: 1

Author

Antti Karttunen, Nov 26 2017

Keywords

• nonn
• mult

Comments

The number of applications to reach 1 is A322027(n). Positions of first appearances are A076610. - Gus Wiseman, Jan 17 2020

Examples

			For n = 360 = 2^3 * 3^2 * 5 = prime(1)^3 * prime(2)^2 * prime(3), 1 is not a prime, but 2 and 3 are, thus a(360) = 2^2 * 3 = 12.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000040, A000720, A010051, A295666.

Cf. also A003963, A257538.

Positions of 1's are A320628.

Positions of terms > 1 are A331386.

Primes of prime index are A006450.

Primes of nonprime index are A007821.

Products of primes of prime index are A076610.

Products of primes of nonprime index are A320628.

The number of prime prime indices is A257994.

The number of nonprime prime indices is A330944.

Numbers whose prime indices are not all prime are A330945.

Cf. A001222, A007097, A018252, A056239, A112798, A320633, A322027, A330947, A330948.

Programs

Mathematica

Table[Times@@Cases[FactorInteger[n],{p_?(PrimeQ[PrimePi[#]]&),k_}:>PrimePi[p]^k],{n,40}] (* Gus Wiseman, Jan 17 2020 *)

Scheme

(definec (A295665 n) (if (= 1 n) 1 (let ((k (A055396 n))) (* (if (zero? (A010051 k)) 1 k) (A295665 (A032742 n))))))

Formula

Multiplicative with a(p^e) = A000720(p)^(e*A010051(A000720(p))).

a(1) = 1; for n > 1, if A055396(n) is a prime, then a(n) = A055396(n) * a(A032742(n)), otherwise a(n) = a(A032742(n)).

Other identities. For all n >= 1:

a(A006450(n)) = A000040(n).

a(A007097(n)) = A007097(n-1).

a(A294876(n)) = A295666(n).