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%I #39 Dec 16 2018 13:01:11
%S 57,85,213,224,354,476,568,594,812,1218,1235,1316,1484,2103,2470,2492,
%T 2643,2840,2996,3836,3978,4026,4544,4810,4844,5012,6125,6356,6524,
%U 7364,7532,7648,8876,9272,9328,10098,11107,11797,12572,12594,13412,13640
%N Numbers m such that m and m+22 have the same sum of divisors.
%C If 3*k-1 and 14*k-1 are both prime with k>1, then n = 28*(3*k-1) belongs to this sequence. The number of such integers n <= x would be asymptotically cx/(log x)^2 for some constant c > 0 from the Hardy-Littlewood conjecture D in Partitio Numerorum. - _Tomohiro Yamada_, Oct 03 2018
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 62, p. 22, Ellipses, Paris 2008.
%D W. Sierpinski, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 110.
%D Tomohiro Yamada, On equations sigma(n) = sigma(n+k) and phi(n) = phi(n+k), J. Comb. Number Theory 9 (2017), 15-21.
%H Tomohiro Yamada, <a href="/A172333/b172333.txt">Table of n, a(n) for n = 1..46702</a> (All terms < 2^28, first 2000 terms from Muniru A Asiru)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes</a>, Acta Math. 44 (1923), 1-70.
%H Tomohiro Yamada, <a href="https://arxiv.org/abs/1001.2511">On equations sigma(n) = sigma(n+k) and phi(n) = phi(n+k)<</a>, arXiv:1001.2511 [math.NT], 2010.
%p with(numtheory):for n from 1 to 20000 do;if sigma(n) = sigma(n+22) then print(n); else fi ; od;
%o (PARI) isok(k) = sigma(k)==sigma(k+22); \\ _Altug Alkan_, Oct 03 2018
%o (GAP) Filtered([1..13700],k->Sigma(k)=Sigma(k+22)); # _Muniru A Asiru_, Oct 20 2018
%Y Cf. A000203, A015861, A002961, A015865, A015867, A015858, A015859, A015860.
%K nonn
%O 1,1
%A _Michel Lagneau_, Feb 01 2010