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

1,3

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from indices in prime factorization

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).

Example

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.

Mathematica

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

Prog

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

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.

Sequence in context: A232890 A300441 A375378 * A103484 A016444 A280831

Adjacent sequences: A295662 A295663 A295664 * A295666 A295667 A295668

Keyword

nonn,mult

Author

Antti Karttunen, Nov 26 2017

Status

approved