A253141 If n is a prime power, then a(n) = lambda(tau(n)) = A014963(A000005(n)); otherwise, a(n) = 1.

1, 2, 2, 3, 2, 1, 2, 2, 3, 1, 2, 1, 2, 1, 1, 5, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 7, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 5, 1, 2, 1, 1, 1, 1

Offset: 1

Author

Matthew Vandermast, Dec 27 2014

Keywords

• nonn
• easy

Comments

For any integer sequence a, the sequence b such that b(n) = Product_{d|n} a(d) is a divisibility sequence. Since A253139(n) = Product_{d|n} a(d), A253139 is a divisibility sequence.

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

Examples

			2 is a prime number, i.e., a prime power with 2 divisors; a(2) = A014963(2) = 2.
6 = 2*3 is not a prime power; a(6) = 1.
8 = 2^3 is a prime power with 4 divisors; a(8) = A014963(4) = 2.
32 = 2^5 is a prime power with 6 divisors; a(32) = A014963(6) = 1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Morgan Ward, A note on divisibility sequences, Bull. Amer. Math. Soc., 45 (1939), 334-336.
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000005, A014963, A253139.

Programs

Mathematica

Table[If[PrimePowerQ[n], Exp[MangoldtLambda[DivisorSigma[0, n]]], 1], {n, 1, 100}] (* Indranil Ghosh, Jul 19 2017 *)

PARI

A014963(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ This function from Charles R Greathouse IV, Jun 10 2011
A253141(n) = if(1==omega(n), A014963(numdiv(n)), 1); \\ Antti Karttunen, Jul 19 2017