A006146 Sums of prime divisors of Ruth-Aaron numbers (A006145).

5, 5, 7, 18, 15, 20, 44, 46, 29, 31, 50, 30, 20, 34, 75, 162, 146, 46, 14, 113, 53, 66, 333, 36, 514, 318, 43, 193, 279, 418, 30, 121, 55, 485, 200, 136, 77, 37, 211, 587, 147, 269, 477, 108, 136, 235, 185, 290, 333, 309, 493, 177, 199, 223, 641, 531, 182, 368

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
• Dana Mackenzie, Homage to an itinerant master, Science, vol. 275, p. 759, 1997.
• Carol Nelson, David E. Penney, and Carl Pomerance, 714 and 715. Journal of Recreational Mathematics 7(2):87-89, 1974.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74. (Annotated scanned copy)
• Ivars Peterson, Playing with Ruth-Aaron pairs
• Ivars Peterson's MathTrek, Playing with Ruth-Aaron Pairs [In the internet archive]
• Eric Weisstein's World of Mathematics, Ruth-Aaron Pair

Crossrefs

Cf. A006145, A008472, A039752, A039753, A039703, A054378.

Programs

Maple

with(numtheory): for n from 1 to 10000 do t0 := 0; t1 := factorset(n);
for j from 1 to nops(t1) do t0 := t0+t1[ j ]; od: s[ n ] := t0; od:
for n from 1 to 9999 do if s[ n ] = s[ n+1 ] then lprint(n,s[ n ]); fi; od:

Mathematica

Cases[Partition[(Plus@@(First@#&/@FactorInteger@#)&/@Range@100000),2,1],{a_,a_}:>a] (* Hans Rudolf Widmer, May 31 2024 *)

Python

from sympy import primefactors
def aupton(terms):
  alst, k, sopfk, sopfkp1 = [], 0, 0, 1
  while len(alst) < terms:
    k, sopfk, sopfkp1 = k+1, sopfkp1, sum(p for p in primefactors(k+1))
    if sopfkp1 == sopfk: alst.append(sopfk)
  return alst
print(aupton(58)) # Michael S. Branicky, May 05 2021

Formula

a(n) = A008472(A006145(n)) = A008472(A006145(n) + 1). - Amiram Eldar, Nov 24 2019