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

%I A156061 #38 Feb 14 2020 16:34:37
%S A156061 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,
%T A156061 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,
%U A156061 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
%N A156061 a(n) = product of indices of distinct prime factors of n, where index(prime(k)) = k.
%C A156061 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
%C A156061 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
%H A156061 Antti Karttunen, <a href="/A156061/b156061.txt">Table of n, a(n) for n = 1..10000</a>
%H A156061 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A156061 From _Antti Karttunen_, Aug 13 2017: (Start)
%F A156061 a(1) = 1; for n > 1, a(n) = A055396(n) * a(A028234(n)).
%F A156061 a(n) = A003963(A007947(n)) = a(A007947(n)).
%F A156061 a(n) = A003963(n) / A290106(n) = A290103(n) * A290105(n).
%F A156061 a(A181819(n)) = A290107(n).
%F A156061 (End)
%e A156061 Here primepi (A000720) gives the index of its prime argument:
%e A156061 n = 14 = 2 * 7, thus a(14) = primepi(2)*primepi(7) = 1*4 = 4.
%e A156061 n = 21 = 3 * 7, thus a(21) = primepi(3)*primepi(7) = 2*4 = 8.
%e A156061 n = 168 = 2^3 * 3 * 7, thus a(168)= primepi(2)*primepi(3)*primepi(7) = 1*2*4 = 8.
%p A156061 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
%t A156061 Table[Apply[Times, PrimePi@ FactorInteger[n][[All, 1]]] + Boole[n == 1], {n, 100}] (* _Michael De Vlieger_, Aug 14 2017 *)
%o A156061 (Scheme) (define (A156061 n) (if (= 1 n) 1 (* (A055396 n) (A156061 (A028234 n))))) ;; _Antti Karttunen_, Aug 13 2017
%o A156061 (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
%Y A156061 Cf. A000040, A000720, A007947.
%Y A156061 Cf. also A003963, A290105, A290106, A290107.
%Y A156061 Differs from related A290103 for the first time at n=21.
%K A156061 nonn,mult
%O A156061 1,3
%A A156061 _Ctibor O. Zizka_, Feb 03 2009
%E A156061 a(1) = 1 prepended by _Antti Karttunen_, Aug 13 2017