

A039752


RuthAaron numbers (2): sum of prime divisors of n = sum of prime divisors of n+1 (both taken with multiplicity).


16



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
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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 3109digit term determined by Jens Kruse Andersen is currently the largestknown.  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 RuthAaron "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), 120122.
S. G. Krantz, Mathematical Apocrypha, MAA, 2002, see p. 26.
Science, vol. 275, p. 759, 1997.


LINKS

P. Weisenhorn and Michael De Vlieger, Table of n, a(n) for n = 1..1121 (first 215 terms from P. Weisenhorn)
Brady Haran and Carl Pomerance, Aaron Numbers, Numberphile video (2017)
Hans Havermann, RuthAaron pairs, indexed and factored
Hans Havermann, A Large RuthAaron Pair
C. Nelson, D. E. Penney and C. Pomerance, 714 and 715, J. Recreational Math. 7 (No. 2) 1974, 8789.
Ivars Peterson, Playing with RuthAaron pairs
Carl Pomerance, RuthAaron Numbers Revisited, Paul Erdős and his Mathematics, (Budapest, 1999), Bolyai Soc. Math. Stud. 11, János Bolyai Math. Soc., Budapest, 2002, pp. 567579.
Carlos Rivera, RuthAaron Pairs Revisited
Terrel Trotter, Jr., RuthAaron Numbers [Warning: As of March 2018 this site appears to have been hacked. Proceed with great caution. The original content should be retrieved from the Wayback machine and added here.  N. J. A. Sloane, Mar 29 2018]
Terrel Trotter, Jr., 714 and 715 [Warning: As of March 2018 this site appears to have been hacked. Proceed with great caution. The original content should be retrieved from the Wayback machine and added here.  N. J. A. Sloane, Mar 29 2018]
Eric Weisstein, RuthAaron 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

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, j1, 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 *)


PROG

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


CROSSREFS

Cf. A006145, A006146, A039753, A054378, A101805, A175513.
Sequence in context: A220034 A063731 A129316 * A141536 A314561 A314562
Adjacent sequences: A039749 A039750 A039751 * A039753 A039754 A039755


KEYWORD

nonn


AUTHOR

David W. Wilson


STATUS

approved



