A074036 Sum of the primes from the smallest prime factor of n to the largest prime factor of n.

0, 2, 3, 2, 5, 5, 7, 2, 3, 10, 11, 5, 13, 17, 8, 2, 17, 5, 19, 10, 15, 28, 23, 5, 5, 41, 3, 17, 29, 10, 31, 2, 26, 58, 12, 5, 37, 77, 39, 10, 41, 17, 43, 28, 8, 100, 47, 5, 7, 10, 56, 41, 53, 5, 23, 17, 75, 129, 59, 10, 61, 160, 15, 2, 36, 28, 67, 58, 98, 17, 71, 5, 73, 197, 8

Offset: 1

Author

Jason Earls, Sep 15 2002

Keywords

• easy
• nonn

Comments

Obviously if n is prime then a(n) = n. However, there are composite values of n such that a(n) = n, such as 10 and 155. - Alonso del Arte, May 30 2017

Examples

			a(14) = 17 because 14 = 2 * 7 and 2 + 3 + 5 + 7 = 17.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• Erich Friedman, What's Special About This Number? Entry for 155.

Crossrefs

Cf. A055233, A169802, A169804.

Programs

Maple

f:=proc(n) local i,t1,t2,t3,t4,t5,t6; if n<=1 then RETURN(0) else
t1:=ifactors(n); t2:=t1[2]; t3:=nops(t2); t4:=0; t5:=pi(t2[1][1]); t6:=pi(t2[t3][1]);
for i from t5 to t6 do t4:=t4+ithprime(i); od; RETURN(t4); fi; end; # N. J. A. Sloane, May 24 2010
# second Maple program:
s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
a:= proc(n) option remember; uses numtheory; `if`(n<2, 0, (m->
      s(pi(max(m)))-s(pi(min(m))-1))(factorset(n)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 24 2021

Mathematica

sp[n_]:=With[{fi=FactorInteger[n][[All,1]]},Total[Prime[Range[ PrimePi[ fi[[1]]],PrimePi[fi[[-1]]]]]]]; Join[{0},Array[sp,80,2]] (* Harvey P. Dale, Dec 22 2017 *)

PARI

a(n) = if (n==1, 0, my(f = factor(n), s = 0); forprime(p=f[1,1], f[#f~,1], s += p); s); \\ Michel Marcus, May 31 2017

Formula

Given p prime and k > 0, a(p^k) = p. - Alonso del Arte, May 30 2017