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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).
2
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
1,1
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).
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
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 sigma2(n) = sigma2(n + p), Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.
Eric Weisstein's World of Mathematics, Divisor Function.
EXAMPLE
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.
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:
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 03 2010
EXTENSIONS
Edited by Robert Israel, Aug 02 2024
STATUS
approved