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A175199 a(n) is the smallest integer k such that sigma_2(k) = sigma_2(k + 2n), where sigma_2(k) is the sum of squares of divisors of k (A001157).

%I A175199 #29 May 30 2025 03:16:20
%S A175199 24,430,645,860,120,864,168,1720,1935,10790,264,2580,2795,1570,16185,
%T A175199 3440,408,3870,456,21580,2355,4730,552,5160,600,5590,5805,3140,696,
%U A175199 4320,744,6880,7095,1248,840,7740,888,8170,8385,43160,984,4710,1032,9460
%N A175199 a(n) is the smallest integer k such that sigma_2(k) = sigma_2(k + 2n), where sigma_2(k) is the sum of squares of divisors of k (A001157).
%C A175199 The equation sigma_2(n) = sigma_2(n + p) has infinitely many solutions where p >= 2 and p is even (J. M. De Koninck).
%D A175199 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 827.
%D A175199 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
%H A175199 Amiram Eldar, <a href="/A175199/b175199.txt">Table of n, a(n) for n = 1..1000</a>
%H A175199 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 [alternative scanned copy].
%H A175199 J. M. De Koninck, <a href="https://ac.inf.elte.hu/Vol_021_2002/doi/127_21.pdf">On the solutions of sigma_2(n) = sigma_2(n + p)</a>, Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.
%H A175199 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function.</a>
%e A175199 For n=1, sigma_2(24) = sigma_2(26) = 850.
%e A175199 For n=2, sigma_2(430) = sigma_2(434) = 240500.
%e A175199 For n=3, sigma_2(645) = sigma_2(651) = 481000.
%p A175199 with(numtheory):for k from 2 by 2 to 200 do :indic:=0:for n from 1 to 100000 do:liste:= divisors(n) : s2 :=sum(liste[i]^2, i=1..nops(liste)):liste:=divisors(n+k):s3:=sum(liste[i]^2, i=1..nops(liste)):if s2 = s3 and indic=0 then print(k):print(n):indic:=1:else fi:od:od:
%Y A175199 Cf. A000005, A000203, A001158, A001159.
%Y A175199 Cf. A053807, A064602.
%K A175199 nonn
%O A175199 1,1
%A A175199 _Michel Lagneau_, Mar 03 2010
%E A175199 Edited by _Robert Israel_, Aug 02 2024