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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).

%I A039752 #83 Mar 13 2025 12:11:38
%S A039752 5,8,15,77,125,714,948,1330,1520,1862,2491,3248,4185,4191,5405,5560,
%T A039752 5959,6867,8280,8463,10647,12351,14587,16932,17080,18490,20450,24895,
%U A039752 26642,26649,28448,28809,33019,37828,37881,41261,42624,43215,44831,44891,47544,49240
%N A039752 Ruth-Aaron numbers (2): sum of prime divisors of n = sum of prime divisors of n+1 (both taken with multiplicity).
%C A039752 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 A039752 An infinite number of terms would follow from A175513 and the assumption of Schinzel's Hypothesis H. - _Hans Havermann_, Dec 15 2010
%C A039752 A 3109-digit term determined by _Jens Kruse Andersen_ is currently the largest-known. - _Hans Havermann_, Dec 21 2010
%C A039752 The sum of this sequence's reciprocals is 0.42069... - _Hans Havermann_, Dec 21 2010
%C A039752 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 A039752 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 A039752 John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
%D A039752 S. G. Krantz, Mathematical Apocrypha, MAA, 2002, see p. 26.
%D A039752 Dana Mackenzie, Homage to an itinerant master, Science 275, p. 759, 1997.
%H A039752 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 A039752 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_4">Smith Numbers</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
%H A039752 Brady Haran and Carl Pomerance, <a href="https://www.youtube.com/watch?v=aCq04N9it8U">Aaron Numbers</a>, Numberphile video (2017)
%H A039752 Hans Havermann, <a href="http://chesswanks.com/seq/a039752.txt">Ruth-Aaron pairs, indexed and factored</a>
%H A039752 Hans Havermann, <a href="http://chesswanks.com/num/ALargeRuthAaronPair.txt">A Large Ruth-Aaron Pair</a>
%H A039752 Carol Nelson, David E. Penney and Carl Pomerance, <a href="https://math.dartmouth.edu/~carlp/714and715.pdf">714 and 715</a>, J. Recreational Math. 7 (No. 2) 1974, 87-89.
%H A039752 Ivars Peterson, <a href="https://www.sciencenews.org/article/playing-ruth-aaron-pairs">Playing with Ruth-Aaron pairs</a>
%H A039752 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 A039752 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_358.htm">Ruth-Aaron Pairs Revisited</a>
%H A039752 Terrel Trotter, Jr., <a href="https://web.archive.org/web/20101130215914/https://trottermath.net/numthry/rutharon.html">Ruth-Aaron Numbers</a>.
%H A039752 Terrel Trotter, Jr., <a href="https://web.archive.org/web/20101130215914/https://trottermath.net/numthry/rutharon.html">714 and 715</a>.
%H A039752 Eric Weisstein, <a href="https://mathworld.wolfram.com/Ruth-AaronPair.html">Ruth-Aaron Pair (in Wolfram MathWorld)</a>
%e A039752 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 A039752 anzahl:=0: n:=4: nr:=0: g:=nops(ifactors(n)[2]):
%p A039752 s[nr]:=sum(ifactors(n)[2,u][1]*ifactors(n)[2,u][2],u=1..g):
%p A039752 for j from n+1 to 1000000 do nr:=(nr+1) mod 2: g:=nops(ifactors(j)[2]):
%p A039752 s[nr]:=sum(ifactors(j)[2,u][1]*ifactors(j)[2,u][2],u=1..g):
%p A039752 if (s[0]=s[1]) then anzahl):=anzahl+1: print(anzahl,j-1,j,s[0]): end if:
%p A039752 end do:
%p A039752 # _Paul Weisenhorn_, Jul 02 2009
%t A039752 ppf[n_] := Plus @@ ((#[[1]] #[[2]]) & /@ FactorInteger[n]); Select[Range[50000], ppf[#] == ppf[#+1] &] (* _Harvey P. Dale_, Apr 27 2009 *)
%o A039752 (PARI) is_A039752(n)=A001414(n)==A001414(n+1) \\ _M. F. Hasler_, Mar 01 2014
%o A039752 (Python)
%o A039752 from sympy import factorint
%o A039752 def aupton(terms):
%o A039752   alst, k, sopfrk, sopfrkp1 = [], 2, 2, 3
%o A039752   while len(alst) < terms:
%o A039752     if sopfrkp1 == sopfrk: alst.append(k)
%o A039752     k += 1
%o A039752     fkp1 = factorint(k+1)
%o A039752     sopfrk, sopfrkp1 = sopfrkp1, sum(p*fkp1[p] for p in fkp1)
%o A039752   return alst
%o A039752 print(aupton(42)) # _Michael S. Branicky_, May 08 2021
%Y A039752 Cf. A006145, A006146, A039753, A054378, A101805, A175513.
%K A039752 nonn
%O A039752 1,1
%A A039752 _David W. Wilson_