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 A039752 Ruth-Aaron numbers (2): sum of prime divisors of n = sum of prime divisors of n+1 (both taken with multiplicity). 15
 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 (list; graph; refs; listen; history; text; internal format)
 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 "triplet". The smallest member of the next triplet 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. 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 (2017) Hans Havermann, Ruth-Aaron pairs, indexed and factored Hans Havermann, A Large Ruth-Aaron Pair C. Nelson, D. E. Penney and C. 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) 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 From Paul Weisenhorn, Jul 02 2009: (Start) 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: (End) 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 A065905 A286056 Adjacent sequences:  A039749 A039750 A039751 * A039753 A039754 A039755 KEYWORD nonn AUTHOR STATUS approved

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