A068208 Numbers k such that tau(k) - tau(k+1) = 1.

4, 8, 81, 441, 625, 1089, 2024, 2401, 3025, 3968, 5624, 5929, 6561, 6723, 7569, 8281, 8463, 12321, 13225, 13688, 14161, 14641, 14884, 15375, 16641, 20164, 21608, 24025, 25921, 26895, 28561, 34225, 46225, 55225, 55695, 61009, 62001, 67081

Offset: 1

Author

Benoit Cloitre, Mar 23 2002

Keywords

• nonn
• easy

Comments

Numbers k such that A000005(k) - A000005(k+1) = 1.

a(n) cannot be a prime.

a(n) or a(n) + 1 is a perfect square as its number of divisors must be odd. - David A. Corneth, Dec 26 2020

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Enrique Pérez Herrero)

Crossrefs

Cf. A000005.

Programs

Mathematica

Select[Range[10^5], DivisorSigma[0, #] - DivisorSigma[0, # + 1] == 1 &] (* Michael De Vlieger, Dec 02 2015 *)
Position[Differences[DivisorSigma[0,Range[70000]]],-1]//Flatten (* Harvey P. Dale, May 22 2020 *)

PARI

isok(n) = numdiv(n) - numdiv(n+1) == 1; \\ Michel Marcus, Dec 02 2015

PARI

list(lim) = {my(d, e, k2); for(k = 2, lim, e = factor(k)[, 2]; d = vecprod(apply(x -> 2*x + 1, e)); k2 = k^2; if(numdiv(k2-1) == d+1, print1(k2-1, ", ")); if(numdiv(k2+1) == d-1, print1(k2, ", ")));} \\ Amiram Eldar, Apr 20 2025