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).

24, 430, 645, 860, 120, 864, 168, 1720, 1935, 10790, 264, 2580, 2795, 1570, 16185, 3440, 408, 3870, 456, 21580, 2355, 4730, 552, 5160, 600, 5590, 5805, 3140, 696, 4320, 744, 6880, 7095, 1248, 840, 7740, 888, 8170, 8385, 43160, 984, 4710, 1032, 9460

Offset: 1

Author

Michel Lagneau, Mar 03 2010

Keywords

• nonn

Comments

The equation sigma_2(n) = sigma_2(n + p) has infinitely many solutions where p >= 2 and p is even (J. M. De Koninck).

Examples

			For n=1, sigma_2(24) = sigma_2(26) = 850.
For n=2, sigma_2(430) = sigma_2(434) = 240500.
For n=3, sigma_2(645) = sigma_2(651) = 481000.
		

References

• 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.
• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• 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].
• J. M. De Koninck, On the solutions of sigma_2(n) = sigma_2(n + p), Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.
• Eric Weisstein's World of Mathematics, Divisor Function.

Crossrefs

Cf. A000005, A000203, A001158, A001159.

Cf. A053807, A064602.

Programs

Maple

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:

Extensions

Edited by Robert Israel, Aug 02 2024