%I M0461 N0168 #198 Oct 16 2024 20:33:37
%S 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,
%T 10,31,10,14,19,12,10,37,21,16,11,41,12,43,15,11,25,47,11,14,12,20,17,
%U 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
%N Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).
%C MacMahon calls this the potency of n.
%C Downgrades the operators in a prime decomposition. E.g., 40 factors as 2^3 * 5 and sopfr(40) = 2 * 3 + 5 = 11.
%C 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
%C a(n) <= n for all n, and a(n) = n iff n is 4 or a prime.
%C 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
%C 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
%C Except for the initial term, row sums of A027746. - _M. F. Hasler_, Feb 08 2016
%C Atanassov proves that a(n) <= A065387(n) - n. - _Charles R Greathouse IV_, Dec 06 2016
%C From _Robert G. Wilson v_, Aug 15 2022: (Start)
%C Differs from A337310 beginning with n at 64, 192, 256, 320, 448, 512, ..., .
%C The number of terms which equal k is A000607(k).
%C The first occurrence of k>1 is A056240(k).
%C The last occurrence of k>1 is A000792(k).
%C The _Amarnath Murthy_ comment of Jul 07 2001 is a result of the fundamental theorem of arithmetic.
%C (End)
%D K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 31-35.
%D Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring-2000.
%D 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 Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Daniel Forgues, <a href="/A001414/b001414.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from Franklin T. Adams-Watters)
%H Krishnaswami Alladi and Paul Erdős, <a href="https://projecteuclid.org/euclid.pjm/1102811427">On an additive arithmetic function</a>, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, <a href="https://msp.org/pjm/1977/71-2/pjm-v71-n2-p01-s.pdf">alternative link</a>.
%H Kevin S. Brown, <a href="http://www.mathpages.com/home/kmath006/kmath006.htm">The Sum of the Prime Factors of N</a>.
%H Es-said En-naoui, <a href="https://arxiv.org/abs/2301.09677">Study of the generalized Von Mangoldt function defined by L-additive function</a>, arXiv:2301.09677 [math.GM], 2023.
%H Hans Havermann, <a href="/A001414/a001414.png">Log plot of 100000 terms</a>
%H J. Iraids, K. Balodis, J. Cernenoks, M. Opmanis, R. Opmanis and K. Podnieks, <a href="http://arxiv.org/abs/1203.6462">Integer Complexity: Experimental and Analytical Results</a>, arXiv preprint arXiv:1203.6462 [math.NT], 2012.
%H Rafael Jakimczuk, <a href="http://m-hikari.com/imf/imf-2012/53-56-2012/jakimczukIMF53-56-2012-2.pdf">Sum of Prime Factors in the Prime Factorization of an Integer</a>, International Mathematical Forum, Vol. 7, No. 53 (2012), pp. 2617-2621.
%H Mohan Lal, <a href="http://dx.doi.org/10.1090/S0025-5718-1969-0242765-9">Iterates of a number-theoretic function</a>, Math. Comp., Vol. 23, No. 105 (1969), pp. 181-183.
%H P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-23.1.290">Properties of prime numbers deduced from the calculus of symmetric functions</a>, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SumofPrimeFactors.html">Sum of Prime Factors</a>.
%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Table_of_prime_factors">Table of prime factors</a>.
%H Steve Witham, <a href="/A001414/a001414_1.png">Linear-log plot</a> (The clear upper lines are n (the primes), n/2, n/3, n/4... but there is a dark band at sqrt(n).)
%H Steve Witham, <a href="/A001414/a001414_2.png">Log-log plot</a> (Differently interesting at the lower edge. Higher up, you can see sqrt(n), sqrt(n)/2, maybe sqrt(n)/3.)
%F If n = Product p_j^k_j then a(n) = Sum p_j * k_j.
%F 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
%F For n > 1: a(n) = Sum_{k=1..A001222(n)} A027746(n,k). - _Reinhard Zumkeller_, Aug 27 2011
%F 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
%e a(24) = 2+2+2+3 = 9.
%e 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.
%p A001414 := proc(n) add( op(1,i)*op(2,i),i=ifactors(n)[2]) ; end proc:
%p seq(A001414(n), n=1..100); # _Peter Luschny_, Jan 17 2011
%t a[n_] := Plus @@ Times @@@ FactorInteger@ n; a[1] = 0; Array[a, 78] (* _Ray Chandler_, Nov 12 2005 *)
%o (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 (PARI) A001414(n) = (n=factor(n))[,1]~*n[,2] \\ _M. F. Hasler_, Feb 07 2009
%o (Haskell)
%o a001414 1 = 0
%o a001414 n = sum $ a027746_row n
%o -- _Reinhard Zumkeller_, Feb 27 2012, Nov 20 2011
%o (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
%o (Python)
%o from sympy import factorint
%o def A001414(n):
%o return sum(p*e for p,e in factorint(n).items()) # _Chai Wah Wu_, Jan 08 2016
%o (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
%Y Cf. A008472 (sopf(n)), A002217, A056240, A000792, A046343, A120007, A036288.
%Y A000607(n) gives the number of values of k for which A001414(k) = n.
%Y Cf. A036349 (indices of even terms), A356163 (their char. function), A335657 (indices of odd terms), A289142 (of multiples of 3), A373371 (their char. function).
%Y For sum of squares of prime factors see A067666, for cubes see A224787.
%Y Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A056239 (with a(p)=primepi(p)), A059975 (with a(p)=p-1), A064097 (with a(p)=1+a(p-1)), A113177 (with a(p)=Fib(p)), A276085 (with a(p)=p#/p), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
%Y For other completely additive sequences see the cross-references in A104244.
%K nonn,easy,nice,changed
%O 1,2
%A _N. J. A. Sloane_