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A039752
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Ruth-Aaron numbers (2): sum of prime divisors of n = sum of prime divisors of n+1 (both taken with multiplicity).
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22
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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
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OFFSET
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1,1
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COMMENTS
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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
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
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REFERENCES
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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.
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LINKS
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Brady Haran and Carl Pomerance, Aaron Numbers, Numberphile video (2017)
C. Nelson, D. E. Penney and C. Pomerance, 714 and 715, J. Recreational Math. 7 (No. 2) 1974, 87-89.
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.
Terrel Trotter, Jr., Ruth-Aaron 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]
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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ppf[n_] := Plus @@ ((#[[1]] #[[2]]) & /@ FactorInteger[n]); Select[Range[50000], ppf[#] == ppf[#+1] &] (* Harvey P. Dale, Apr 27 2009 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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