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Search: id:A001414
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| A001414 |
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Integer log of n: sum of primes dividing n (with repetition).
(Formerly M0461 N0168)
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+0
172
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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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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; sequence gives smallest sum of terms. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 07 2001
a(n)=n iff n is prime or 4.
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REFERENCES
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M. Lal, Iterates of a number-theoretic function, Math. Comp., 23 (1969), 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.
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.
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).
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LINKS
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Daniel Forgues, Table of n, a(n) for n=1..100000
K. S. Brown, The Sum of the Prime Factors of N
M. L. Perez et al., eds., Smarandache Notions Journal
Eric Weisstein's World of Mathematics, Sum of Prime Factors
Wikipedia, Table of prime factors
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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(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
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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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MATHEMATICA
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Prepend[ Array[ Plus @@ Map[ Times @@ #1&, FactorInteger[ # ] ]&, 100, 2 ], 0 ]
Table[Plus @@ Times @@@ FactorInteger[n], {n, 100}] - Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 12 2005
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PROGRAM
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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] [From M. F. Hasler (MHasler(AT)univ-ag.fr), Feb 07 2009]
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CROSSREFS
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Cf. A008472 (sopf(n)), A002217, A056240, A000792, A046343.
A000607(n) gives the number of values of k for which A001414(k) = n.
Cf. A120007.
Sequence in context: A118503 A086295 A159303 this_sequence A134875 A134889 A094802
Adjacent sequences: A001411 A001412 A001413 this_sequence 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 (njas(AT)research.att.com).
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