

A001414


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


234



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

MacMahon calls this the potency of n.
Sometimes also called sopfr(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 iff n is prime or 4.


REFERENCES

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.
J. 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

Franklin T. AdamsWatters and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from Franklin T. AdamsWatters)
K. S. Brown, The Sum of the Prime Factors of N
J. Iraids, K. Balodis, J. Cernenoks, M. Opmanis, R. Opmanis and K. Podnieks, Integer Complexity: Experimental and Analytical Results. arXiv preprint arXiv:1203.6462, 2012.
M. Lal, Iterates of a numbertheoretic function, Math. Comp., 23 (1969), 181183.
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290316. = Coll. Papers, II, pp. 354380.
Eric Weisstein's World of Mathematics, Sum of Prime Factors
Wikipedia, Table of prime factors


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
For n > 1: a(n) = sum(k=1..A001222(n), A027746(n,k)).  Reinhard Zumkeller, Aug 27 2011


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:
seq (A001414(n), n=1..100); # Peter Luschny, Jan 17 2011


MATHEMATICA

f[n_] := Plus @@ Times @@@ FactorInteger@ n; f[1] = 0; Array[f, 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]))
(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


CROSSREFS

Cf. A008472 (sopf(n)), A002217, A056240, A000792, A046343.
A000607(n) gives the number of values of k for which A001414(k) = n.
Cf. A120007, A036288.
Sequence in context: A118503 A086295 A159303 * A134875 A134889 A181894
Adjacent sequences: A001411 A001412 A001413 * A001415 A001416 A001417


KEYWORD

nonn,easy,nice,changed


AUTHOR

N. J. A. Sloane.


STATUS

approved



