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%I #79 May 08 2021 07:41:44
%S 5,8,15,77,125,714,948,1330,1520,1862,2491,3248,4185,4191,5405,5560,
%T 5959,6867,8280,8463,10647,12351,14587,16932,17080,18490,20450,24895,
%U 26642,26649,28448,28809,33019,37828,37881,41261,42624,43215,44831,44891,47544,49240
%N Ruth-Aaron numbers (2): sum of prime divisors of n = sum of prime divisors of n+1 (both taken with multiplicity).
%C 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.
%C An infinite number of terms would follow from A175513 and the assumption of Schinzel's Hypothesis H. - _Hans Havermann_, Dec 15 2010
%C A 3109-digit term determined by _Jens Kruse Andersen_ is currently the largest-known. - _Hans Havermann_, Dec 21 2010
%C The sum of this sequence's reciprocals is 0.42069... - _Hans Havermann_, Dec 21 2010
%C 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
%C The number of terms <= x is at most O(x (loglog x)^4 / (log x)^2) (Pomerance 1999/2002). - _Tomohiro Yamada_, Apr 22 2017
%D John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
%D S. G. Krantz, Mathematical Apocrypha, MAA, 2002, see p. 26.
%D Dana Mackenzie, Homage to an itinerant master, Science 275, p. 759, 1997.
%H Amiram Eldar, <a href="/A039752/b039752.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..215 from P. Weisenhorn, terms 216..1121 from Michael De Vlieger)
%H Brady Haran and Carl Pomerance, <a href="https://www.youtube.com/watch?v=aCq04N9it8U">Aaron Numbers</a>, Numberphile video (2017)
%H Hans Havermann, <a href="http://chesswanks.com/seq/a039752.txt">Ruth-Aaron pairs, indexed and factored</a>
%H Hans Havermann, <a href="http://chesswanks.com/num/ALargeRuthAaronPair.txt">A Large Ruth-Aaron Pair</a>
%H C. Nelson, D. E. Penney and C. Pomerance, <a href="https://math.dartmouth.edu/~carlp/714and715.pdf">714 and 715</a>, J. Recreational Math. 7 (No. 2) 1974, 87-89.
%H Ivars Peterson, <a href="https://www.sciencenews.org/article/playing-ruth-aaron-pairs">Playing with Ruth-Aaron pairs</a>
%H Carl Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/PDF/paper130.pdf">Ruth-Aaron Numbers Revisited</a>, Paul Erdős and his Mathematics, (Budapest, 1999), Bolyai Soc. Math. Stud. 11, János Bolyai Math. Soc., Budapest, 2002, pp. 567-579.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_358.htm">Ruth-Aaron Pairs Revisited</a>
%H Terrel Trotter, Jr., <a href="http://www.trottermath.net/numthry/rutharon.html">Ruth-Aaron Numbers</a> [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]
%H Terrel Trotter, Jr., <a href="http://www.trottermath.net/numthry/ruth714.html">714 and 715</a> [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]
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/Ruth-AaronPair.html">Ruth-Aaron Pair (in Wolfram MathWorld)</a>
%e 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.
%p anzahl:=0: n:=4: nr:=0: g:=nops(ifactors(n)[2]):
%p s[nr]:=sum(ifactors(n)[2,u][1]*ifactors(n)[2,u][2],u=1..g):
%p for j from n+1 to 1000000 do nr:=(nr+1) mod 2: g:=nops(ifactors(j)[2]):
%p s[nr]:=sum(ifactors(j)[2,u][1]*ifactors(j)[2,u][2],u=1..g):
%p if (s[0]=s[1]) then anzahl):=anzahl+1: print(anzahl,j-1,j,s[0]): end if:
%p end do:
%p # _Paul Weisenhorn_, Jul 02 2009
%t ppf[n_] := Plus @@ ((#[[1]] #[[2]]) & /@ FactorInteger[n]); Select[Range[50000], ppf[#] == ppf[#+1] &] (* _Harvey P. Dale_, Apr 27 2009 *)
%o (PARI) is_A039752(n)=A001414(n)==A001414(n+1) \\ _M. F. Hasler_, Mar 01 2014
%o (Python)
%o from sympy import factorint
%o def aupton(terms):
%o alst, k, sopfrk, sopfrkp1 = [], 2, 2, 3
%o while len(alst) < terms:
%o if sopfrkp1 == sopfrk: alst.append(k)
%o k += 1
%o fkp1 = factorint(k+1)
%o sopfrk, sopfrkp1 = sopfrkp1, sum(p*fkp1[p] for p in fkp1)
%o return alst
%o print(aupton(42)) # _Michael S. Branicky_, May 08 2021
%Y Cf. A006145, A006146, A039753, A054378, A101805, A175513.
%K nonn
%O 1,1
%A _David W. Wilson_
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