I find that the sum is near an integers for 7,11,13,17, and 29, but not for 163. The integer parts appear in A089034 a(n)={p^4 - 1}/240, where p=prime(n+3). terms: 10, 61, 119, 348, 543, 1166, Pari code: ----------------- default(realprecision,1000); default(format,"g.50"); { for (k=1, 1000, s = sum(n=1,100, n^3/(exp(2*Pi*n/k)-1)); if ( abs(s-round(s))<0.001, print(k,": ---- "); print(s); c = contfrac(s); print(vector(20,n,c[n])); ); ); } Output: ----------------- 7: ---- 10.000000000000000190161767888662674830486095865010 [10, 5258680601799450, 4, 1, 10, 1, 1, 1, 2, 1, 4, 2, 1, 1, 1, 3, 1, 3, 1, 1] 11: ---- 60.999999999999999999783017186376408591503435429769 [60, 1, 4608659936241490513, 5, 1, 6, 1, 7, 10, 1, 127, 1, 1, 3, 2, 1, 3, 9, 1, 1] 13: ---- 118.99999999999999822060853662753902153501763247951 [118, 1, 561989882824721, 38, 1, 1, 27, 1, 9, 1, 1, 1, 3, 1, 1, 2, 5, 2, 1, 2] 17: ---- 347.99999999978099522542236078594550239032616790505 [347, 1, 4566110495, 1, 1, 2, 1, 1, 1, 3, 1, 5, 1, 1, 1, 5, 1, 1, 2, 3] 19: ---- 542.99999998765415727636451319177064725216471202466 [542, 1, 80998924, 1, 9, 2, 7, 3, 4, 1, 4, 101, 2, 3, 1, 1, 4, 1, 2, 1] 23: ---- 1165.9999950595915588813284161938468112041252275357 [1165, 1, 202411, 2, 2, 2, 2, 2, 7, 1, 2, 3, 1, 3, 1, 1, 22, 2, 1, 1] ----------------- Remove the if() statement to observe nearness to rationals for many k.