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A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.

%I A006145 #79 Mar 13 2025 12:10:40
%S A006145 5,24,49,77,104,153,369,492,714,1682,2107,2299,2600,2783,5405,6556,
%T A006145 6811,8855,9800,12726,13775,18655,21183,24024,24432,24880,25839,26642,
%U A006145 35456,40081,43680,48203,48762,52554,61760,63665,64232,75140,79118,95709,106893,109939
%N A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.
%C A006145 Nelson, Penney, & Pomerance call these "Aaron numbers" 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. - _David W. Wilson_
%C A006145 Number of terms < 10^n: 1, 4, 9, 19, 40, 139, 494, 1748, 6650, ..., . - _Robert G. Wilson v_, Jan 23 2012
%D A006145 John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
%D A006145 P. Hoffman, The Man Who Loved Only Numbers, pp. 179-181, Hyperion, NY 1998.
%D A006145 J. Roberts, Lure of Integers, pp. 250, MAA 1992.
%D A006145 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 159-160, Penguin 1986.
%H A006145 Amiram Eldar, <a href="/A006145/b006145.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..6651 from Robert G. Wilson v)
%H A006145 Joe K. Crump, <a href="http://web.archive.org/web/20070630090153/http://www.immortaltheory.com/NumberTheory/RuthAaron.htm">Ruth-Aaron Pairs-an algorithm</a>
%H A006145 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 A006145 Brady Haran and Carl Pomerance, <a href="https://www.youtube.com/watch?v=aCq04N9it8U">Aaron Numbers</a>, Numberphile video (2017).
%H A006145 G. Kreweras and Y. Poupard, <a href="/A000806/a000806.pdf">Sur les partitions en paires d'un ensemble fini totalement ordonné</a>, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74. (Annotated scanned copy)
%H A006145 Dana Mackenzie, <a href="https://doi.org/10.1126/science.275.5301.759">Homage to an Itinerant Master</a>, Science, Vol. 275 (1997), p. 759; <a href="https://www.proquest.com/openview/0b38915ee572fd265291ef3fecb3bc4d/1">alternative link</a>.
%H A006145 Carol Nelson, David E. Penney, and Carl Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/714and715.pdf">714 and 715</a>, J. Recreational Math. 7:2 (1994), pp. 87-89.
%H A006145 Ivars Peterson, <a href="https://www.sciencenews.org/article/playing-ruth-aaron-pairs">Playing with Ruth-Aaron pairs</a>
%H A006145 Terry Trotter, Jr., <a href="https://web.archive.org/web/20101130215914/https://trottermath.net/numthry/rutharon.html">Ruth-Aaron Numbers</a>.
%H A006145 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Ruth-AaronPair.html">Ruth-Aaron Pair</a>.
%p A006145 with(numtheory): for n from 1 to 10000 do t0 := 0; t1 := factorset(n);
%p A006145 for j from 1 to nops(t1) do t0 := t0+t1[ j ]; od: s[ n ] := t0; od:
%p A006145 for n from 1 to 9999 do if s[ n ] = s[ n+1 ] then lprint(n,s[ n ]); fi; od:
%p A006145 # Alternative:
%p A006145 SumPF := proc(n) option remember; add(NumberTheory:-PrimeFactors(n)) end:
%p A006145 seq(ifelse(SumPF(n) = SumPF(n+1), n, NULL), n = 1..3000); # _Peter Luschny_, Jun 11 2024
%t A006145 fQ[n_] := Plus @@ (First@# & /@ FactorInteger[n]) == Plus @@ (First@# & /@ FactorInteger[n + 1]); Select[ Range@ 100000, fQ] (* _Robert G. Wilson v_, Jan 22 2012 *)
%o A006145 (PARI) sopf(n)=my(f=factor(n));sum(i=1,#f[,1],f[i,1])
%o A006145 is(n)=sopf(n)==sopf(n+1) \\ _Charles R Greathouse IV_, Jan 27 2012
%o A006145 (Python)
%o A006145 from sympy import factorint
%o A006145 def aupton(terms):
%o A006145   alst, k, sopfk, sopfkp1 = [], 2, 2, 3
%o A006145   while len(alst) < terms:
%o A006145     if sopfkp1 == sopfk: alst.append(k)
%o A006145     k, sopfk, sopfkp1 = k+1, sopfkp1, sum(p for p in factorint(k+2))
%o A006145   return alst
%o A006145 print(aupton(42)) # _Michael S. Branicky_, May 24 2021
%Y A006145 Cf. A006146, A039752, A039753, A054378.
%K A006145 nonn
%O A006145 1,1
%A A006145 _N. J. A. Sloane_