OFFSET
1,1
COMMENTS
So called 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, taken with multiplicity.
An infinite number of terms would follow from A175513 and the assumption of Schinzel's Hypothesis H. - Hans Havermann, Dec 15 2010
A 3109-digit term determined by Jens Kruse Andersen is currently the largest-known. - Hans Havermann, Dec 21 2010
The sum of this sequence's reciprocals is 0.42069... - Hans Havermann, Dec 21 2010
Both 417162 and 417163 are in the sequence. Hence these two numbers along with 417164 constitute a Ruth-Aaron "triple". The smallest member of the next triple is 6913943284. - Hans Havermann, Dec 01 2010, Dec 13 2010
The number of terms <= x is at most O(x (loglog x)^4 / (log x)^2) (Pomerance 1999/2002). - Tomohiro Yamada, Apr 22 2017
REFERENCES
John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
S. G. Krantz, Mathematical Apocrypha, MAA, 2002, see p. 26.
Dana Mackenzie, Homage to an itinerant master, Science 275, p. 759, 1997.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..215 from P. Weisenhorn, terms 216..1121 from Michael De Vlieger)
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)
Hans Havermann, Ruth-Aaron pairs, indexed and factored
Hans Havermann, A Large Ruth-Aaron Pair
Carol Nelson, David E. Penney and Carl Pomerance, 714 and 715, J. Recreational Math. 7 (No. 2) 1974, 87-89.
Ivars Peterson, Playing with Ruth-Aaron pairs
Carl Pomerance, Ruth-Aaron Numbers Revisited, Paul Erdős and his Mathematics, (Budapest, 1999), Bolyai Soc. Math. Stud. 11, János Bolyai Math. Soc., Budapest, 2002, pp. 567-579.
Carlos Rivera, Puzzle 358. Ruth-Aaron Pairs Revisited, The Prime Puzzles & Problems Connection.
Terrel Trotter, Jr., Ruth-Aaron Numbers (714 and 715).
Eric Weisstein, Ruth-Aaron Pair (in Wolfram MathWorld)
EXAMPLE
7129199 (7*11^2*19*443, with 7129200 = 2^4*3*5^2*13*457) is in this sequence because 7+11+11+19+443 = 2+2+2+2+3+5+5+13+457.
MAPLE
sumfactors := n -> add(map(mul, op(2, ifactors(n)))):
op(select(n->sumfactors(n)=sumfactors(n+1), [$1..100000])); # Brendan McKay, Feb 26 2026
MATHEMATICA
ppf[n_] := Plus @@ ((#[[1]] #[[2]]) & /@ FactorInteger[n]); Select[Range[50000], ppf[#] == ppf[#+1] &] (* Harvey P. Dale, Apr 27 2009 *)
PROG
(Python)
from sympy import factorint
def aupton(terms):
alst, k, sopfrk, sopfrkp1 = [], 2, 2, 3
while len(alst) < terms:
if sopfrkp1 == sopfrk: alst.append(k)
k += 1
fkp1 = factorint(k+1)
sopfrk, sopfrkp1 = sopfrkp1, sum(p*fkp1[p] for p in fkp1)
return alst
print(aupton(42)) # Michael S. Branicky, May 08 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved
