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

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, 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, 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

Offset: 1

Author

Michel Marcus, Dec 31 2016

Keywords

• nonn

Comments

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

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

References

• 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.
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 165.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
• Jean-Marie De Koninck, Paul Erdős and Aleksandar Ivić, Reciprocals of certain large additive functions, Canadian Mathematical Bulletin, Vol. 24, No. 2 (1981), pp. 225-231.
• Aleksandar Ivić, On certain large additive functions, arXiv:math/0311505 [math.NT], 2003.

Crossrefs

Cf. A001414 (sopfr), A008472 (sopf), A057521, A059956, A081770, A280163, A338559, A338560.

A multiplicative version is A003557, firsts A064549 (sorted A001694).

For length instead of sum we have A046660.

For product instead of sum we have A066503, firsts A381076.

Positions of first appearances are A280286 (sorted A381075).

For indices instead of factors we have A380955, firsts A380956 (sorted A380957).

For exponents instead of factors we have A380958, firsts A380989.

A000040 lists the primes, differences A001223.

A001222 counts prime factors (distinct A001221).

A003963 gives product of prime indices, distinct A156061, excess A380986.

A005117 lists squarefree numbers, complement A013929.

A007947 gives squarefree kernel.

A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).

A027746 lists prime factors, distinct A027748.

A112798 lists prime indices (sum A056239), distinct A304038 (sum A066328).

Cf. A071625, A075255, A116861, A136565, A175508, A290106, A364916, A381077.

Programs

Mathematica

Array[Total@ # - Total@ Union@ # &@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &, 105] (* Michael De Vlieger, Feb 25 2019 *)

PARI

sopfr(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
sopf(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]);
a(n) = sopfr(n) - sopf(n);

Formula

a(n) = A001414(n) - A008472(n).

a(A005117(n)) = 0.

a(n) = A001414(A003557(n)). - Antti Karttunen, Oct 07 2017

Additive with a(1) = 0 and a(p^e) = p*(e-1) for prime p and e > 0. - Werner Schulte, Feb 24 2019

From Amiram Eldar, Nov 02 2020: (Start)

a(n) = a(A057521(n)).

Sum_{n<=x} a(n) ~ x*log(log(x)) + O(x) (Alladi and Erdős, 1977).

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)

Extensions

More terms from Antti Karttunen, Oct 07 2017