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A116990
Indices of triangular numbers whose sum of divisors is square.
4
1, 2, 11, 20, 40, 68, 92, 212, 236, 253, 266, 321, 328, 452, 582, 589, 596, 668, 695, 716, 782, 788, 836, 928, 932, 970, 991, 1012, 1065, 1076, 1173, 1264, 1300, 1336, 1388, 1436, 1490, 1549, 1796, 1854, 1927, 1995, 2159, 2228, 2252, 2468, 2545, 2588
OFFSET
1,2
COMMENTS
From Zak Seidov, Oct 19 2010: (Start)
A074285(n) = A000203(A000217(n)) = s^2.
Corresponding values of s begin: 1,2,12,24,42,72,96,216,240,192,240,288,336,456,504, 480,600,672,840,720,720,792,960,930,936,756,992,936,1008,1080,... (are most values of s multiples of 3?).
(End)
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..561 from Zak Seidov)
FORMULA
n such that A074285(n) is in A000290.
n such that sum( d | A000217(n), d ) is in A000290.
n such that A000203(A000217(n)) is in A000290.
n such that sum( d | n*(n+1)/2, d ) = k^2 for integer k.
EXAMPLE
a(1) = 1 because sigma(1*2/2) = sigma(1) = 1 = 1^2,
a(2) = 2 because sigma(2*3/2) = sigma(3) = 2^2,
a(3) = 11 because sigma(11*12/2) = sigma(66) = 144 = 12^2.
MAPLE
with(numtheory): a:=proc(n) if type(sqrt(sigma(n*(n+1)/2)), integer)=true then n else fi end: seq(a(n), n=0..3100); # Emeric Deutsch, Apr 06 2006
MATHEMATICA
Flatten@ Position[Accumulate[Range@ 2600], n_ /; IntegerQ@ Sqrt@ DivisorSigma[1, n] == True] (* Michael De Vlieger, Mar 17 2015 *)
Select[Range[2600], IntegerQ[Sqrt[DivisorSigma[1, (#(#+1))/2]]]&] (* Harvey P. Dale, Nov 19 2022 *)
PROG
(PARI) for(n=1, 1000, if(issquare(sigma(n*(n+1)/2)), print1(n", "))) \\ Zak Seidov, Mar 21 2015
CROSSREFS
See also: A000217 Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n. A074285 Sum of the divisors of n-th triangular number. A083675 Triangular number for which the sum of the proper divisors is also a triangular number. A000203 sigma(n) = sum of divisors of n. Also called sigma_1(n).
Sequence in context: A279774 A092595 A175927 * A327264 A279772 A226416
KEYWORD
easy,nonn,less
AUTHOR
Jonathan Vos Post, Apr 04 2006
EXTENSIONS
More terms from Emeric Deutsch, Apr 06 2006
Incorrect term 0 removed by Michel Marcus, Mar 17 2015
STATUS
approved