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 A066328 a(n) = sum of indices of distinct prime factors of n; here, index(i-th prime) = i. 20
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 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) = A056239(A007947(n)). - Antti Karttunen, Sep 06 2018 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}] & /@ 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), 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 CROSSREFS Cf. A143542. - Gary W. Adamson, Aug 23 2008 Cf. A000720, A056239, A136565. Cf. A001221, A046660, A112798, A114638, A116861, A304360. Sequence in context: A304740 A326619 A326567 * A330417 A330415 A319225 Adjacent sequences: A066325 A066326 A066327 * A066329 A066330 A066331 KEYWORD nonn AUTHOR Leroy Quet, Jan 01 2002 STATUS approved

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Last modified February 2 05:15 EST 2023. Contains 359997 sequences. (Running on oeis4.)