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A001414
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Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).
(Formerly M0461 N0168)
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570
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0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, 8, 17, 8, 19, 9, 10, 13, 23, 9, 10, 15, 9, 11, 29, 10, 31, 10, 14, 19, 12, 10, 37, 21, 16, 11, 41, 12, 43, 15, 11, 25, 47, 11, 14, 12, 20, 17, 53, 11, 16, 13, 22, 31, 59, 12, 61, 33, 13, 12, 18, 16, 67, 21, 26, 14, 71, 12, 73, 39, 13, 23, 18, 18
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OFFSET
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1,2
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COMMENTS
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MacMahon calls this the potency of n.
Downgrades the operators in a prime decomposition. E.g., 40 factors as 2^3 * 5 and sopfr(40) = 2 * 3 + 5 = 11.
Consider all ways of writing n as a product of zero, one, or more factors; sequence gives smallest sum of terms. - Amarnath Murthy, Jul 07 2001
a(n) <= n for all n, and a(n) = n iff n is 4 or a prime.
Look at the graph of this sequence. At the lower edge of the logarithmic scatterplot there is a set of fuzzy but unmistakable diagonal fringes, sloping toward the southeast. Their spacing gradually increases, and their slopes gradually decrease; they are more distinct toward the lower edge of the range. Is any explanation known? - Allan C. Wechsler, Oct 11 2015
For n >= 2, the glb and lub are: 3 * log(n) / log(3) <= a(n) <= n, where the lub occurs when n = 3^k, k >= 1. (Jakimczuk 2012) - Daniel Forgues, Oct 12 2015
Except for the initial term, row sums of A027746. - M. F. Hasler, Feb 08 2016
Atanassov proves that a(n) <= A065387(n) - n. - Charles R Greathouse IV, Dec 06 2016
From Robert G. Wilson v, Aug 15 2022: (Start)
Differs from A337310 beginning with n at 64, 192, 256, 320, 448, 512, ..., .
The number of terms which equal k is A000607(k).
The first occurrence of k>1 is A056240(k).
The last occurrence of k>1 is A000792(k). Also Lekraj Beedassy Apr 29 2002.
Amarnath Murthy comment of Jul 07 2001 is the result of the fundamental theorem of arithmetic.
(End)
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REFERENCES
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K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 31-35.
Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring-2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Franklin T. Adams-Watters and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from Franklin T. Adams-Watters)
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
Kevin S. Brown, The Sum of the Prime Factors of N.
Es-said En-naoui, Study of the generalized Von Mangoldt function defined by L-additive function, arXiv:2301.09677 [math.GM], 2023.
Hans Havermann, Log plot of 100000 terms
J. Iraids, K. Balodis, J. Cernenoks, M. Opmanis, R. Opmanis and K. Podnieks, Integer Complexity: Experimental and Analytical Results, arXiv preprint arXiv:1203.6462 [math.NT], 2012.
Rafael Jakimczuk, Sum of Prime Factors in the Prime Factorization of an Integer, International Mathematical Forum, Vol. 7, No. 53 (2012), pp. 2617-2621.
Mohan Lal, Iterates of a number-theoretic function, Math. Comp., Vol. 23, No. 105 (1969), pp. 181-183.
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380.
Eric Weisstein's World of Mathematics, Sum of Prime Factors.
Wikipedia, Table of prime factors.
Steve Witham, Linear-log plot (The clear upper lines are n (the primes), n/2, n/3, n/4... but there is a dark band at sqrt(n).)
Steve Witham, Log-log plot (Differently interesting at the lower edge. Higher up, you can see sqrt(n), sqrt(n)/2, maybe sqrt(n)/3.)
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FORMULA
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If n = Product p_j^k_j then a(n) = Sum p_j * k_j.
Dirichlet g.f. f(s)*zeta(s), where f(s) = Sum_{p prime} p/(p^s-1) = Sum_{k>0} primezeta(k*s-1) is the Dirichlet g.f. for A120007. Totally additive with a(p^e) = p*e. - Franklin T. Adams-Watters, Jun 02 2006
For n > 1: a(n) = Sum_{k=1..A001222(n)} A027746(n,k). - Reinhard Zumkeller, Aug 27 2011
Sum_{n>=1} (-1)^a(n)/n^s = ((2^s + 1)/(2^s - 1))*zeta(2*s)/zeta(s), if Re(s)>1 and 0 if s=1 (Alladi and Erdős, 1977). - Amiram Eldar, Nov 02 2020
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EXAMPLE
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a(24) = 2+2+2+3 = 9.
a(30) = 10: 30 can be written as 30, 15*2, 10*3, 6*5, 5*3*2. The corresponding sums are 30, 17, 13, 11, 10. Among these 10 is the least.
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MAPLE
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A001414 := proc(n) local e, j; e := ifactors(n)[2]: add(e[j][1]*e[j][2], j=1..nops(e)) end:
seq(A001414(n), n=1..100); # Peter Luschny, Jan 17 2011
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MATHEMATICA
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a[n_] := Plus @@ Times @@@ FactorInteger@ n; a[1] = 0; Array[a, 78] (* Ray Chandler, Nov 12 2005 *)
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PROG
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(PARI) a(n)=local(f); if(n<1, 0, f=factor(n); sum(k=1, matsize(f)[1], f[k, 1]*f[k, 2]))
(PARI) A001414(n) = (n=factor(n))[, 1]~*n[, 2] \\ M. F. Hasler, Feb 07 2009
(Haskell)
a001414 1 = 0
a001414 n = sum $ a027746_row n
-- Reinhard Zumkeller, Feb 27 2012, Nov 20 2011
(Sage) [sum(factor(n)[j][0]*factor(n)[j][1] for j in range(0, len(factor(n)))) for n in range(1, 79)] # Giuseppe Coppoletta, Jan 19 2015
(Python)
from sympy import factorint
def A001414(n):
return sum(p*e for p, e in factorint(n).items()) # Chai Wah Wu, Jan 08 2016
(Magma) [n eq 1 select 0 else (&+[j[1]*j[2]: j in Factorization(n)]): n in [1..100]]; // G. C. Greubel, Jan 10 2019
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CROSSREFS
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Cf. A008472 (sopf(n)), A002217, A056240, A000792, A046343, A120007, A036288.
A000607(n) gives the number of values of k for which A001414(k) = n.
For sum of squares of prime factors see A067666, for cubes see A224787.
Other completely additive sequences with primes p mapped to a function of p include p-1: A059975, p#/p: A276085, T(p): A341885.
For other completely additive sequences see the cross-references in A104244.
Sequence in context: A159303 A262049 A337310 * A134875 A134889 A303702
Adjacent sequences: A001411 A001412 A001413 * A001415 A001416 A001417
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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