A008472 Sum of the distinct primes dividing n.

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 9, 8, 2, 17, 5, 19, 7, 10, 13, 23, 5, 5, 15, 3, 9, 29, 10, 31, 2, 14, 19, 12, 5, 37, 21, 16, 7, 41, 12, 43, 13, 8, 25, 47, 5, 7, 7, 20, 15, 53, 5, 16, 9, 22, 31, 59, 10, 61, 33, 10, 2, 18, 16, 67, 19, 26, 14, 71, 5, 73

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• nice
• easy

Comments

Sometimes called sopf(n).

Sum of primes dividing n (without repetition) (compare A001414).

Equals A051731 * A061397 = inverse Mobius transform of [0, 2, 3, 0, 5, 0, 7, ...]. - Gary W. Adamson, Feb 14 2008

Equals row sums of triangle A143535. - Gary W. Adamson, Aug 23 2008

a(n) = n if and only if n is prime. - Daniel Forgues, Mar 24 2009

a(n) = n is a new record if and only if n is prime. - Zak Seidov, Jun 27 2009

a(A001043(n)) = A191583(n);

For n > 0: a(A000079(n)) = 2, a(A000244(n)) = 3, a(A000351(n)) = 5, a(A000420(n)) = 7;

a(A006899(n)) <= 3; a(A003586(n)) = 5; a(A033846(n)) = 7; a(A033849(n)) = 8; a(A033847(n)) = 9; a(A033850(n)) = 10; a(A143207(n)) = 10. - Reinhard Zumkeller, Jun 28 2011

For n > 1: a(n) = Sum(A027748(n,k): 1 <= k <= A001221(n)). - Reinhard Zumkeller, Aug 27 2011

If n is the product of twin primes (A037074), a(n) = 2*sqrt(n+1) = sqrt(4n+4). - Wesley Ivan Hurt, Sep 07 2013

From Wilf A. Wilson, Jul 21 2017: (Start)

a(n) + 2, n > 2, is the number of maximal subsemigroups of the monoid of orientation-preserving or -reversing mappings on a set with n elements.

a(n) + 3, n > 2, is the number of maximal subsemigroups of the monoid of orientation-preserving or -reversing partial mappings on a set with n elements.

(End)

The smallest m such that a(m) = n, or 0 if no such number m exists is A064502(n). The only integers that are not in the sequence are 1, 4 and 6. - Bernard Schott, Feb 07 2022

Examples

			a(18) = 5 because 18 = 2 * 3^2 and 2 + 3 = 5.
a(19) = 19 because 19 is prime.
a(20) = 7 because 20 = 2^2 * 5 and 2 + 5 = 7.
		

Links

• Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from Franklin T. Adams-Watters)
• Johann Bartel, R. K. Bhaduri, Matthias Brack, and M. V. N. Murthy, On the asymptotic prime partitions of integers, arXiv:1609.06497 [math-ph], 2017.
• James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [_Wilf A. Wilson_, Jul 21 2017]

Crossrefs

First difference of A024924.

Cf. A001414 (sopfr), A001222, A051731, A061397, A064502, A085020, A143535.

Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), this sequence (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).

Cf. A010051.

Programs

Haskell

a008472 = sum . a027748_row  -- Reinhard Zumkeller, Mar 29 2012

Magma

[n eq 1 select 0 else &+[p[1]: p in Factorization(n)]: n in [1..100]]; // Vincenzo Librandi, Jun 24 2017

Maple

A008472 := n -> add(d, d = select(isprime, numtheory[divisors](n))):
seq(A008472(i), i = 1..40); # Peter Luschny, Jan 31 2012
A008472 := proc(n)
        add( d, d= numtheory[factorset](n)) ;
end proc: # R. J. Mathar, Jul 08 2012

Mathematica

Prepend[Array[Plus @@ First[Transpose[FactorInteger[#]]] &, 100, 2], 0]
Join[{0}, Rest[Total[Transpose[FactorInteger[#]][[1]]]&/@Range[100]]] (* Harvey P. Dale, Jun 18 2012 *)
(* Requires version 7.0+ *) Table[DivisorSum[n, # &, PrimeQ[#] &], {n, 75}] (* Alonso del Arte, Dec 13 2014 *)
Table[Sum[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)

PARI

sopf(n) = local(fac=factor(n)); sum(i=1,matsize(fac)[1],fac[i,1])

PARI

vector(100,n,vecsum(factor(n)[,1]~)) \\ Derek Orr, May 13 2015

PARI

A008472(n)=vecsum(factor(n)[,1]) \\ M. F. Hasler, Jul 18 2015

Python

from sympy import primefactors
def A008472(n): return sum(primefactors(n)) # Chai Wah Wu, Feb 03 2022

Sage

def A008472(n):
    return add(d for d in divisors(n) if is_prime(d))
print([A008472(i) for i in (1..40)]) # Peter Luschny, Jan 31 2012

Sage

[sum(prime_factors(n)) for n in range(1,74)] # Giuseppe Coppoletta, Jan 19 2015

Formula

Let n = Product_j prime(j)^k(j) where k(j) >= 1, then a(n) = Sum_j prime(j).

Additive with a(p^e) = p.

G.f.: Sum_{k >= 1} prime(k)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009

L.g.f.: -log(Product_{k>=1} (1 - x^prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017

Dirichlet g.f.: primezeta(s-1)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018

a(n) = Sum_{p|n, p prime} p. - Wesley Ivan Hurt, Feb 04 2022

From Bernard Schott, Feb 07 2022: (Start)

For n > 0: a(A001020(n)) = 11, a(A001022(n)) = 13, a(A001026(n)) = 17, a(A001029(n)) = 19, a(A009967(n)) = 23, a(A009973(n)) = 29, a(A009975(n)) = 31, a(A009981(n)) = 37, a(A009985(n)) = 41, a(A009987(n)) = 43, a(A009991(n)) = 47.

For p odd prime, a(2*p) = p+2 <==> a(A100484(n)) = A052147(n) for n > 1. (End)

a(n) = Sum_{d|n} d * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024