A039752 Ruth-Aaron numbers (2): sum of prime divisors of n = sum of prime divisors of n+1 (both taken with multiplicity).

5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248, 4185, 4191, 5405, 5560, 5959, 6867, 8280, 8463, 10647, 12351, 14587, 16932, 17080, 18490, 20450, 24895, 26642, 26649, 28448, 28809, 33019, 37828, 37881, 41261, 42624, 43215, 44831, 44891, 47544, 49240

Offset: 1

Author

David W. Wilson

Keywords

• nonn

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

Examples

			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.
		

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, Ruth-Aaron Pairs Revisited
• Terrel Trotter, Jr., Ruth-Aaron Numbers.
• Terrel Trotter, Jr., 714 and 715.
• Eric Weisstein, Ruth-Aaron Pair (in Wolfram MathWorld)

Crossrefs

Cf. A006145, A006146, A039753, A054378, A101805, A175513.

Programs

Maple

anzahl:=0: n:=4: nr:=0: g:=nops(ifactors(n)[2]):
s[nr]:=sum(ifactors(n)[2,u][1]*ifactors(n)[2,u][2],u=1..g):
for j from n+1 to 1000000 do nr:=(nr+1) mod 2: g:=nops(ifactors(j)[2]):
s[nr]:=sum(ifactors(j)[2,u][1]*ifactors(j)[2,u][2],u=1..g):
if (s[0]=s[1]) then anzahl):=anzahl+1: print(anzahl,j-1,j,s[0]): end if:
end do:
# Paul Weisenhorn, Jul 02 2009

Mathematica

ppf[n_] := Plus @@ ((#[[1]] #[[2]]) & /@ FactorInteger[n]); Select[Range[50000], ppf[#] == ppf[#+1] &] (* Harvey P. Dale, Apr 27 2009 *)

PARI

is_A039752(n)=A001414(n)==A001414(n+1) \\ M. F. Hasler, Mar 01 2014

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