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A169635 Integers m such that sigma_2(m) = sigma_2(m + 2) where sigma_2(m) is the sum of squares of divisors of m (A001157).

%I A169635 #36 May 30 2025 03:15:21
%S A169635 24,215,280,1079,947519,1362239,2230271,14939999,19720007,32509439,
%T A169635 45581759,45841247,49436927,78436511,82842911,101014631,166828031,
%U A169635 225622151,225757799,250999559,377129087,554998751,619606439,846765431,1204092287,1302170687,1710035711
%N A169635 Integers m such that sigma_2(m) = sigma_2(m + 2) where sigma_2(m) is the sum of squares of divisors of m (A001157).
%C A169635 The equation sigma_2(m) = sigma_2(m + k) has infinitely many solutions where k >= 2 and k is even (J.-M. De Koninck).
%C A169635 From _Amiram Eldar_, Apr 19 2024: (Start)
%C A169635 De Koninck's proof is based on the assumption of Schinzel's hypothesis H. If q, r = q + 2, s = q^2 + q + 1, and p = q^2 + 3*q + 3 are all primes, then p*q is a term (the values of q+1 are the terms of A268043).
%C A169635 The equation sigma_2(m) = sigma_2(m + 1) has only one solution: m = 6. (End)
%D A169635 Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 118, entry 1079.
%D A169635 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B13, pp. 103-104.
%H A169635 Amiram Eldar, <a href="/A169635/b169635.txt">Table of n, a(n) for n = 1..156</a>
%H A169635 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 A169635 Jean-Marie 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 + l)</a>, Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.
%H A169635 Wikipedia, <a href="https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H">Schinzel's hypothesis H</a>.
%e A169635 For m=24, sigma_2(24) = sigma_2(26) = 850.
%p A169635 with(numtheory):for n from 1 to 500000000 do:liste:= divisors(n) : s2 :=sum(liste[i]^2, i=1..nops(liste)):liste:=divisors(n+2):s3:=sum(liste[i]^2, i=1..nops(liste)):if s2 = s3 then print(n):else fi:od:
%t A169635 Select[Range[10^9], DivisorSigma[2,#] == DivisorSigma[2,#+2]&]
%o A169635 (PARI) is(n) = sigma(n, 2) == sigma(n + 2, 2); \\ _Amiram Eldar_, Apr 19 2024
%o A169635 (PARI) lista(mmax) = {my(s1 = sigma(1, 2), s2 = sigma(2, 2), s3, s4); forstep(m = 3, mmax, 2, s3 = sigma(m, 2); s4 = sigma(m+1, 2); if(s1 == s3, print1(m - 2, ", ")); if(s2 == s4, print1(m - 1, ", ")); s1 = s3; s2 = s4);} \\ _Amiram Eldar_, Apr 19 2024
%Y A169635 Cf. A001157, A002961, A007373, A175199, A268043.
%K A169635 nonn
%O A169635 1,1
%A A169635 _Michel Lagneau_, Apr 04 2010
%E A169635 a(25)-a(27) from _Donovan Johnson_, Apr 14 2013