A156061 a(n) = product of indices of distinct prime factors of n, where index(prime(k)) = k.

1, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 6, 1, 7, 2, 8, 3, 8, 5, 9, 2, 3, 6, 2, 4, 10, 6, 11, 1, 10, 7, 12, 2, 12, 8, 12, 3, 13, 8, 14, 5, 6, 9, 15, 2, 4, 3, 14, 6, 16, 2, 15, 4, 16, 10, 17, 6, 18, 11, 8, 1, 18, 10, 19, 7, 18, 12, 20, 2, 21, 12, 6, 8, 20, 12, 22, 3, 2, 13, 23, 8, 21, 14, 20, 5, 24, 6, 24, 9, 22, 15, 24, 2, 25, 4, 10, 3

Offset: 1

Author

Ctibor O. Zizka, Feb 03 2009

Keywords

• nonn
• mult

Comments

a(n) = the product of the distinct parts of the partition with Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(252)= 8; indeed, the partition having Heinz number 252 = 2*2*3*3*7 is [1,1,2,2,4] and 1*2*4 = 8. - Emeric Deutsch, Jun 03 2015

Multiplicative with a(prime(k)^e) = k. Note that in contrast to A003963, this is not fully multiplicative. a(1) = 1 as an empty product. - Antti Karttunen, Aug 13 2017

Examples

			Here primepi (A000720) gives the index of its prime argument:
n = 14 = 2 * 7, thus a(14) = primepi(2)*primepi(7) = 1*4 = 4.
n = 21 = 3 * 7, thus a(21) = primepi(3)*primepi(7) = 2*4 = 8.
n = 168 = 2^3 * 3 * 7, thus a(168)= primepi(2)*primepi(3)*primepi(7) = 1*2*4 = 8.
		

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, A007947.

Cf. also A003963, A290105, A290106, A290107.

Differs from related A290103 for the first time at n=21.

Programs

Maple

with(numtheory): a := proc(n) options operator, arrow: product(pi(factorset(n)[j]), j = 1 .. nops(factorset(n))) end proc: seq(a(n), n = 1 .. 100);  #  Emeric Deutsch, Jun 03 2015

Mathematica

Table[Apply[Times, PrimePi@ FactorInteger[n][[All, 1]]] + Boole[n == 1], {n, 100}] (* Michael De Vlieger, Aug 14 2017 *)

PARI

a(n) = {my(f=factor(n)); for (k=1, #f~, f[k,1] = primepi(f[k,1]); f[k,2] = 1); factorback(f);} \\ Michel Marcus, Aug 14 2017

Scheme

(define (A156061 n) (if (= 1 n) 1 (* (A055396 n) (A156061 (A028234 n))))) ;; Antti Karttunen, Aug 13 2017

Formula

From Antti Karttunen, Aug 13 2017: (Start)

a(1) = 1; for n > 1, a(n) = A055396(n) * a(A028234(n)).

a(n) = A003963(A007947(n)) = a(A007947(n)).

a(n) = A003963(n) / A290106(n) = A290103(n) * A290105(n).

a(A181819(n)) = A290107(n).

(End)

Extensions

a(1) = 1 prepended by Antti Karttunen, Aug 13 2017