A325423 Numbers k such that sigma(2*k+1) >= sigma(2*k).

1, 7, 31, 37, 67, 73, 97, 103, 127, 157, 199, 202, 229, 241, 247, 262, 277, 283, 307, 313, 331, 337, 346, 367, 379, 382, 397, 409, 427, 457, 472, 487, 499, 517, 547, 562, 577, 607, 619, 643, 661, 667, 682, 697, 727, 757, 769, 787

Offset: 1

Author

Giuseppe Melfi, Sep 06 2019

Keywords

• nonn

Comments

The function A(x) enumerating the terms not exceeding x has the property that lim_{x->oo} A(x)/x exists (Hildebrand, 1990).

Examples

			7 is in the sequence because sigma(14) = 1+2+7+14 = 24 <= sigma(15) = 1+3+5+15 = 24;
31 is in the sequence because sigma(62) = 1+2+31+62 = 96 <= sigma(63) = 1+3+7+9+21+63 = 104.
		

References

• M. Laub, Advanced Problems: 6555. The American Mathematical Monthly, 94(8), 800 (1987). doi:10.2307/2323430.

Links

• Giuseppe Melfi, Table of n, a(n) for n = 1..2763
• Mits Kobayashi, Tim Trudgian, On integers n for which sigma(2n+1)>=sigma(2n), arXiv:1904.10064 [math.NT], 2019.
• M. Laub & L. Mattics, Problem 6555: Odd Integers with Relatively Large Divisor Sum, The American Mathematical Monthly, 97(4), 351-353 (1990). doi:10.2307/2324532.

Crossrefs

Cf. A082957.

Programs

Mathematica

Position[Partition[DivisorSigma[1,Range[2,1601]],2],?(#[[2]] >= #[[1]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Jan 10 2022 *)

PARI

isok(n) = sigma(2*n+1) >= sigma(2*n); \\ Michel Marcus, Sep 09 2019

Formula

a(n) ~ c*n with 18.2 < c < 18.6 (claimed by Kobayashi and Trudgian).