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Sums of prime divisors of Ruth-Aaron numbers (A006145).
4

%I #34 May 31 2024 05:54:25

%S 5,5,7,18,15,20,44,46,29,31,50,30,20,34,75,162,146,46,14,113,53,66,

%T 333,36,514,318,43,193,279,418,30,121,55,485,200,136,77,37,211,587,

%U 147,269,477,108,136,235,185,290,333,309,493,177,199,223,641,531,182,368

%N Sums of prime divisors of Ruth-Aaron numbers (A006145).

%D John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.

%D Dana Mackenzie, Homage to an itinerant master, Science, vol. 275, p. 759, 1997.

%D Carol Nelson, David E. Penney, and Carl Pomerance, 714 and 715. Journal of Recreational Mathematics 7(2):87-89, 1974.

%H Amiram Eldar, <a href="/A006146/b006146.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 Ivars Peterson, <a href="https://www.sciencenews.org/article/playing-ruth-aaron-pairs">Playing with Ruth-Aaron pairs</a>

%H Ivars Peterson's MathTrek, <a href="https://web.archive.org/web/20130702062101/http://www.maa.org/mathland/mathland_6_30.html">Playing with Ruth-Aaron Pairs</a> [In the internet archive]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Ruth-AaronPair.html">Ruth-Aaron Pair</a>

%F a(n) = A008472(A006145(n)) = A008472(A006145(n) + 1). - _Amiram Eldar_, Nov 24 2019

%p with(numtheory): for n from 1 to 10000 do t0 := 0; t1 := factorset(n);

%p for j from 1 to nops(t1) do t0 := t0+t1[ j ]; od: s[ n ] := t0; od:

%p for n from 1 to 9999 do if s[ n ] = s[ n+1 ] then lprint(n,s[ n ]); fi; od:

%t Cases[Partition[(Plus@@(First@#&/@FactorInteger@#)&/@Range@100000),2,1],{a_,a_}:>a] (* _Hans Rudolf Widmer_, May 31 2024 *)

%o (Python)

%o from sympy import primefactors

%o def aupton(terms):

%o alst, k, sopfk, sopfkp1 = [], 0, 0, 1

%o while len(alst) < terms:

%o k, sopfk, sopfkp1 = k+1, sopfkp1, sum(p for p in primefactors(k+1))

%o if sopfkp1 == sopfk: alst.append(sopfk)

%o return alst

%o print(aupton(58)) # _Michael S. Branicky_, May 05 2021

%Y Cf. A006145, A008472, A039752, A039753, A039703, A054378.

%K nonn

%O 1,1

%A _N. J. A. Sloane_