A066328 - OEIS

%I #43 Mar 13 2024 10:43:49

%S 0,1,2,1,3,3,4,1,2,4,5,3,6,5,5,1,7,3,8,4,6,6,9,3,3,7,2,5,10,6,11,1,7,

%T 8,7,3,12,9,8,4,13,7,14,6,5,10,15,3,4,4,9,7,16,3,8,5,10,11,17,6,18,12,

%U 6,1,9,8,19,8,11,8,20,3,21,13,5,9,9,9,22,4,2,14,23,7,10,15,12,6,24,6,10

%N a(n) = sum of indices of distinct prime factors of n; here, index(i-th prime) = i.

%C Equals row sums of triangle A143542. - _Gary W. Adamson_, Aug 23 2008

%C a(n) = the sum 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_{j=1..r} (p_j-th prime) (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(75) = 5; indeed, the partition having Heinz number 75 = 3*5*5 is [2,3,3] and 2 + 3 = 5. - _Emeric Deutsch_, Jun 04 2015

%H Antti Karttunen, <a href="/A066328/b066328.txt">Table of n, a(n) for n = 1..65537</a> (terms 1..1000 from Harry J. Smith)

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>

%F G.f.: Sum_{k>=1} k*x^prime(k)/(1-x^prime(k)). - _Vladeta Jovovic_, Aug 11 2004

%F Additive with a(p^e) = PrimePi(p), where PrimePi(n) = A000720(n).

%F a(n) = A056239(A007947(n)). - _Antti Karttunen_, Sep 06 2018

%F a(n) = Sum_{p|n} A000720(p), where p is a prime. - _Ridouane Oudra_, Aug 19 2019

%e a(24) = 1 + 2 = 3 because 24 = 2^3 * 3 = p(1)^3 * p(2), p(k) being the k-th prime.

%e From _Gus Wiseman_, Mar 09 2019: (Start)

%e The distinct prime indices of 1..20 and their sums.

%e    1: () = 0

%e    2: (1) = 1

%e    3: (2) = 2

%e    4: (1) = 1

%e    5: (3) = 3

%e    6: (1+2) = 3

%e    7: (4) = 4

%e    8: (1) = 1

%e    9: (2) = 2

%e   10: (1+3) = 4

%e   11: (5) = 5

%e   12: (1+2) = 3

%e   13: (6) = 6

%e   14: (1+4) = 5

%e   15: (2+3) = 5

%e   16: (1) = 1

%e   17: (7) = 7

%e   18: (1+2) = 3

%e   19: (8) = 8

%e   20: (1+3) = 4

%e (End)

%p with(numtheory): seq(add(pi(d), d in factorset(n)), n=1..100); # _Ridouane Oudra_, Aug 19 2019

%t PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_] := (Plus @@ PrimePi[ PrimeFactors[n]]); Table[ f[n], {n, 91}] (* _Robert G. Wilson v_, May 04 2004 *)

%o (PARI) { for (n=1, 1000, f=factor(n); a=0; for (i=1, matsize(f)[1], a+=primepi(f[i, 1])); write("b066328.txt", n, " ", a) ) } \\ _Harry J. Smith_, Feb 10 2010

%o (PARI) a(n)=my(f=factor(n)[,1]); sum(i=1,#f,primepi(f[i])) \\ _Charles R Greathouse IV_, May 11 2015

%o (PARI) A066328(n) = vecsum(apply(primepi,(factor(n)[,1]))); \\ _Antti Karttunen_, Sep 06 2018

%o (Python)

%o from sympy import primepi, primefactors

%o def A066328(n): return sum(map(primepi,primefactors(n))) # _Chai Wah Wu_, Mar 13 2024

%Y Cf. A143542. - _Gary W. Adamson_, Aug 23 2008

%Y Cf. A000720, A056239, A136565.

%Y Cf. A001221, A046660, A112798, A114638, A116861, A304360.

%K nonn

%O 1,3

%A _Leroy Quet_, Jan 01 2002