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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). 3
 24, 215, 280, 1079, 947519, 1362239, 2230271, 14939999, 19720007, 32509439, 45581759, 45841247, 49436927, 78436511, 82842911, 101014631, 166828031, 225622151, 225757799, 250999559, 377129087, 554998751, 619606439, 846765431, 1204092287, 1302170687, 1710035711 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The equation sigma_2(m) = sigma_2(m + k) has infinitely many solutions where k >= 2 and k is even (J.-M. De Koninck). From Amiram Eldar, Apr 19 2024: (Start) 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). The equation sigma_2(m) = sigma_2(m + 1) has only one solution: m = 6. (End) REFERENCES Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 118, entry 1079. Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B13, pp. 103-104. LINKS Amiram Eldar, Table of n, a(n) for n = 1..156 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Jean-Marie De Koninck, On the solutions of sigma_2(n) = sigma_2(n + l), Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133. Wikipedia, Schinzel's hypothesis H. EXAMPLE For m=24, sigma_2(24) = sigma_2(26) = 850. MAPLE 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: MATHEMATICA Select[Range[10^9], DivisorSigma[2, #] == DivisorSigma[2, #+2]&] PROG (PARI) is(n) = sigma(n, 2) == sigma(n + 2, 2); \\ Amiram Eldar, Apr 19 2024 (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 CROSSREFS Cf. A001157, A002961, A007373, A175199, A268043. Sequence in context: A097321 A105946 A050222 * A269496 A221434 A008655 Adjacent sequences: A169632 A169633 A169634 * A169636 A169637 A169638 KEYWORD nonn AUTHOR Michel Lagneau, Apr 04 2010 EXTENSIONS a(25)-a(27) from Donovan Johnson, Apr 14 2013 STATUS approved

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Last modified May 30 01:30 EDT 2024. Contains 372954 sequences. (Running on oeis4.)