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A280292 a(n) = sopfr(n) - sopf(n).

%I A280292 #26 Feb 22 2025 21:28:42
%S A280292 0,0,0,2,0,0,0,4,3,0,0,2,0,0,0,6,0,3,0,2,0,0,0,4,5,0,6,2,0,0,0,8,0,0,
%T A280292 0,5,0,0,0,4,0,0,0,2,3,0,0,6,7,5,0,2,0,6,0,4,0,0,0,2,0,0,3,10,0,0,0,2,
%U A280292 0,0,0,7,0,0,5,2,0,0,0,6,9,0,0,2,0,0,0,4,0,3,0,2,0,0,0,8,0,7,3,7,0,0,0,4,0
%N A280292 a(n) = sopfr(n) - sopf(n).
%C A280292 Alladi and Erdős (1977) proved that for all numbers m>=0, m!=1, the sequence of numbers k such that a(k) = m has a positive asymptotic density which is equal to a rational multiple of 1/zeta(2) = 6/Pi^2 (A059956). For example, when m=0, the sequence is the squarefree numbers (A005117), whose density is 6/Pi^2, and when m=2 the sequence is A081770, whose density is 1/Pi^2. - _Amiram Eldar_, Nov 02 2020
%C A280292 Sum of prime factors minus sum of distinct prime factors. Counting partitions by this statistic (sum minus sum of distinct parts) gives A364916. - _Gus Wiseman_, Feb 21 2025
%D A280292 Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See pp. 164-166.
%D A280292 Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 165.
%H A280292 Antti Karttunen, <a href="/A280292/b280292.txt">Table of n, a(n) for n = 1..20000</a>
%H A280292 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 A280292 Jean-Marie De Koninck, Paul Erdős and Aleksandar Ivić, <a href="https://doi.org/10.4153/CMB-1981-035-7">Reciprocals of certain large additive functions</a>, Canadian Mathematical Bulletin, Vol. 24, No. 2 (1981), pp. 225-231.
%H A280292 Aleksandar Ivić, <a href="https://arxiv.org/abs/math/0311505">On certain large additive functions</a>, arXiv:math/0311505 [math.NT], 2003.
%F A280292 a(n) = A001414(n) - A008472(n).
%F A280292 a(A005117(n)) = 0.
%F A280292 a(n) = A001414(A003557(n)). - _Antti Karttunen_, Oct 07 2017
%F A280292 Additive with a(1) = 0 and a(p^e) = p*(e-1) for prime p and e > 0. - _Werner Schulte_, Feb 24 2019
%F A280292 From _Amiram Eldar_, Nov 02 2020: (Start)
%F A280292 a(n) = a(A057521(n)).
%F A280292 Sum_{n<=x} a(n) ~ x*log(log(x)) + O(x) (Alladi and Erdős, 1977).
%F A280292 Sum_{n<=x, n nonsquarefree} 1/a(n) ~ c*x + O(sqrt(x)*log(x)), where c = Integral_{t=0..1} (F(t)-6/Pi^2)/t dt, and F(t) = Product_{p prime} (1-1/p)*(1-1/(t^p - p)) (De Koninck et al., 1981; Finch, 2018), or, equivalently c = Sum_{k>=2} d(k)/k = 0.1039..., where d(k) = (6/Pi^2)*A338559(k)/A338560(k) is the asymptotic density of the numbers m with a(m) = k (Alladi and Erdős, 1977; Ivić, 2003). (End)
%t A280292 Array[Total@ # - Total@ Union@ # &@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &, 105] (* _Michael De Vlieger_, Feb 25 2019 *)
%o A280292 (PARI) sopfr(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
%o A280292 sopf(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]);
%o A280292 a(n) = sopfr(n) - sopf(n);
%Y A280292 Cf. A001414 (sopfr), A008472 (sopf), A057521, A059956, A081770, A280163, A338559, A338560.
%Y A280292 A multiplicative version is A003557, firsts A064549 (sorted A001694).
%Y A280292 For length instead of sum we have A046660.
%Y A280292 For product instead of sum we have A066503, firsts A381076.
%Y A280292 Positions of first appearances are A280286 (sorted A381075).
%Y A280292 For indices instead of factors we have A380955, firsts A380956 (sorted A380957).
%Y A280292 For exponents instead of factors we have A380958, firsts A380989.
%Y A280292 A000040 lists the primes, differences A001223.
%Y A280292 A001222 counts prime factors (distinct A001221).
%Y A280292 A003963 gives product of prime indices, distinct A156061, excess A380986.
%Y A280292 A005117 lists squarefree numbers, complement A013929.
%Y A280292 A007947 gives squarefree kernel.
%Y A280292 A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
%Y A280292 A027746 lists prime factors, distinct A027748.
%Y A280292 A112798 lists prime indices (sum A056239), distinct A304038 (sum A066328).
%Y A280292 Cf. A071625, A075255, A116861, A136565, A175508, A290106, A364916, A381077.
%K A280292 nonn
%O A280292 1,4
%A A280292 _Michel Marcus_, Dec 31 2016
%E A280292 More terms from _Antti Karttunen_, Oct 07 2017