%I A001414 M0461 N0168
%S A001414 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,
%T A001414 31,10,14,19,12,10,37,21,16,11,41,12,43,15,11,25,47,11,14,12,20,17,53,
               11,
%U A001414 16,13,22,31,59,12,61,33,13,12,18,16,67,21,26,14,71,12,73,39,13,23,18,
               18
%N A001414 Integer log of n: sum of primes dividing n (with repetition).
%C A001414 MacMahon calls this the potency of n.
%C A001414 Sometimes also called sopfr(n).
%C A001414 Downgrades the operators in a prime decomposition. E.g. 40 factors as 
               2^3 * 5 and sopfr(40) = 2 * 3 + 5 = 11.
%C A001414 Consider all ways of writing n as a product; sequence gives smallest 
               sum of terms. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 
               07 2001
%C A001414 a(n)=n iff n is prime or 4.
%D A001414 M. Lal, Iterates of a number-theoretic function, Math. Comp., 23 (1969), 
               181-183.
%D A001414 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.
%D A001414 Amarnath Murthy, Generalization of Partition function and introducing 
               Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 
               1-2-3, Spring-2000.
%D A001414 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.
%D A001414 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.
%D A001414 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001414 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001414 Daniel Forgues, <a href="b001414.txt">Table of n, a(n) for n=1..100000</
               a>
%H A001414 K. S. Brown, <a href="http://www.mathpages.com/home/kmath006/kmath006.htm">
               The Sum of the Prime Factors of N</a>
%H A001414 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/
               ">Smarandache Notions Journal</a>
%H A001414 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SumofPrimeFactors.html">Sum of Prime Factors</a>
%H A001414 Wikipedia, <a href="http://www.wikipedia.org/wiki/Table_of_prime_factors">
               Table of prime factors</a>
%F A001414 If n = Product (p_j^k_j) then a(n) = Sum (p_j * k_j).
%F A001414 Dirichlet g.f. f(s)*zeta(s), where f(s) = sum_{p prime} p/(p^s-1) = sum_{k>
               0} primezeta(ks-1) is the Dirichlet g.f. for A120007. Totally additive 
               with a(p^e) = p*e. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jun 02 2006
%e A001414 a(24)=2+2+2+3=9.
%e A001414 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.
%t A001414 Prepend[ Array[ Plus @@ Map[ Times @@ #1&, FactorInteger[ # ] ]&, 100, 
               2 ], 0 ]
%t A001414 Table[Plus @@ Times @@@ FactorInteger[n], {n, 100}] - Ray Chandler (rayjchandler(AT)sbcglobal.net), 
               Nov 12 2005
%o A001414 (PARI) a(n)=local(f); if(n<1,0,f=factor(n); sum(k=1,matsize(f)[1],f[k,
               1]*f[k,2]))
%o A001414 (PARI) A001414(n) = (n=factor(n))[,1]~*n[,2] [From M. F. Hasler (MHasler(AT)univ-ag.fr), 
               Feb 07 2009]
%Y A001414 Cf. A008472 (sopf(n)), A002217, A056240, A000792, A046343.
%Y A001414 A000607(n) gives the number of values of k for which A001414(k) = n.
%Y A001414 Cf. A120007.
%Y A001414 Sequence in context: A118503 A086295 A159303 this_sequence A134875 A134889 
               A094802
%Y A001414 Adjacent sequences: A001411 A001412 A001413 this_sequence A001415 A001416 
               A001417
%K A001414 nonn,easy,nice
%O A001414 1,2
%A A001414 N. J. A. Sloane (njas(AT)research.att.com).