OFFSET
1,3
COMMENTS
Equals row sums of triangle A143542. - Gary W. Adamson, Aug 23 2008
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
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Harry J. Smith)
FORMULA
G.f.: Sum_{k>=1} k*x^prime(k)/(1-x^prime(k)). - Vladeta Jovovic, Aug 11 2004
Additive with a(p^e) = PrimePi(p), where PrimePi(n) = A000720(n).
a(n) = Sum_{p|n} A000720(p), where p is a prime. - Ridouane Oudra, Aug 19 2019
EXAMPLE
a(24) = 1 + 2 = 3 because 24 = 2^3 * 3 = p(1)^3 * p(2), p(k) being the k-th prime.
From Gus Wiseman, Mar 09 2019: (Start)
The distinct prime indices of 1..20 and their sums.
1: () = 0
2: (1) = 1
3: (2) = 2
4: (1) = 1
5: (3) = 3
6: (1+2) = 3
7: (4) = 4
8: (1) = 1
9: (2) = 2
10: (1+3) = 4
11: (5) = 5
12: (1+2) = 3
13: (6) = 6
14: (1+4) = 5
15: (2+3) = 5
16: (1) = 1
17: (7) = 7
18: (1+2) = 3
19: (8) = 8
20: (1+3) = 4
(End)
MAPLE
with(numtheory): seq(add(pi(d), d in factorset(n)), n=1..100); # Ridouane Oudra, Aug 19 2019
MATHEMATICA
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 *)
PROG
(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
(PARI) a(n)=my(f=factor(n)[, 1]); sum(i=1, #f, primepi(f[i])) \\ Charles R Greathouse IV, May 11 2015
(PARI) A066328(n) = vecsum(apply(primepi, (factor(n)[, 1]))); \\ Antti Karttunen, Sep 06 2018
(Python)
from sympy import primepi, primefactors
def A066328(n): return sum(map(primepi, primefactors(n))) # Chai Wah Wu, Mar 13 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Leroy Quet, Jan 01 2002
STATUS
approved