A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.

5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299, 2600, 2783, 5405, 6556, 6811, 8855, 9800, 12726, 13775, 18655, 21183, 24024, 24432, 24880, 25839, 26642, 35456, 40081, 43680, 48203, 48762, 52554, 61760, 63665, 64232, 75140, 79118, 95709, 106893, 109939

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Nelson, Penney, & Pomerance call these "Aaron numbers" because 714 is Babe Ruth's lifetime home run record, Hank Aaron's 715th home run broke this record, and 714 and 715 have the same sum of prime divisors. - David W. Wilson

Number of terms < 10^n: 1, 4, 9, 19, 40, 139, 494, 1748, 6650, ..., . - Robert G. Wilson v, Jan 23 2012

References

• John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
• P. Hoffman, The Man Who Loved Only Numbers, pp. 179-181, Hyperion, NY 1998.
• J. Roberts, Lure of Integers, pp. 250, MAA 1992.
• D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 159-160, Penguin 1986.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..6651 from Robert G. Wilson v)
• Joe K. Crump, Ruth-Aaron Pairs-an algorithm
• Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
• Brady Haran and Carl Pomerance, Aaron Numbers, Numberphile video (2017).
• 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)
• Dana Mackenzie, Homage to an Itinerant Master, Science, Vol. 275 (1997), p. 759; alternative link.
• Carol Nelson, David E. Penney, and Carl Pomerance, 714 and 715, J. Recreational Math. 7:2 (1994), pp. 87-89.
• Ivars Peterson, Playing with Ruth-Aaron pairs
• Terry Trotter, Jr., Ruth-Aaron Numbers.
• Eric Weisstein's World of Mathematics, Ruth-Aaron Pair.

Crossrefs

Cf. A006146, A039752, A039753, 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:
# Alternative:
SumPF := proc(n) option remember; add(NumberTheory:-PrimeFactors(n)) end:
seq(ifelse(SumPF(n) = SumPF(n+1), n, NULL), n = 1..3000); # Peter Luschny, Jun 11 2024

Mathematica

fQ[n_] := Plus @@ (First@# & /@ FactorInteger[n]) == Plus @@ (First@# & /@ FactorInteger[n + 1]); Select[ Range@ 100000, fQ] (* Robert G. Wilson v, Jan 22 2012 *)

PARI

sopf(n)=my(f=factor(n));sum(i=1,#f[,1],f[i,1])
is(n)=sopf(n)==sopf(n+1) \\ Charles R Greathouse IV, Jan 27 2012

Python

from sympy import factorint
def aupton(terms):
  alst, k, sopfk, sopfkp1 = [], 2, 2, 3
  while len(alst) < terms:
    if sopfkp1 == sopfk: alst.append(k)
    k, sopfk, sopfkp1 = k+1, sopfkp1, sum(p for p in factorint(k+2))
  return alst
print(aupton(42)) # Michael S. Branicky, May 24 2021