

A001414


Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).
(Formerly M0461 N0168)


607



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

1,2


COMMENTS

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
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).
The Amarnath Murthy comment of Jul 07 2001 is a result of the fundamental theorem of arithmetic.
(End)


REFERENCES

K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 3135.
Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 123, Spring2000.
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).


LINKS

Steve Witham, Linearlog 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, Loglog plot (Differently interesting at the lower edge. Higher up, you can see sqrt(n), sqrt(n)/2, maybe sqrt(n)/3.)


FORMULA

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^s1) = Sum_{k>0} primezeta(k*s1) is the Dirichlet g.f. for A120007. Totally additive with a(p^e) = p*e.  Franklin T. AdamsWatters, Jun 02 2006
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


EXAMPLE

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.


MAPLE

A001414 := proc(n) local e, j; e := ifactors(n)[2]: add(e[j][1]*e[j][2], j=1..nops(e)) end:


MATHEMATICA

a[n_] := Plus @@ Times @@@ FactorInteger@ n; a[1] = 0; Array[a, 78] (* Ray Chandler, Nov 12 2005 *)


PROG

(PARI) a(n)=local(f); if(n<1, 0, f=factor(n); sum(k=1, matsize(f)[1], f[k, 1]*f[k, 2]))
(Haskell)
a001414 1 = 0
a001414 n = sum $ a027746_row n
(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
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


CROSSREFS

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 p1: A059975, p#/p: A276085, T(p): A341885.
For other completely additive sequences see the crossreferences in A104244.


KEYWORD

nonn,easy,nice


AUTHOR



STATUS

approved



