A230653 Numbers k such that tau(k+1) - tau(k) = 3, where tau(k) = the number of divisors of k (A000005).

49, 99, 1023, 1681, 1935, 2499, 8649, 9603, 20449, 21903, 23715, 29583, 30975, 38024, 43263, 58563, 60515, 71824, 74528, 110223, 130321, 136899, 145924, 150543, 154449, 165649, 181475, 216224, 224675, 233288, 243049, 256035, 258063, 265225, 294849, 300303

Offset: 1

Author

Jaroslav Krizek, Oct 27 2013

Keywords

• nonn

Comments

Numbers k such that A051950(k+1) = 3.

Numbers k such that A049820(k) - A049820(k+1) = 2.

k or k+1 is a perfect square. - David A. Corneth, Feb 16 2024

Examples

			99 is in the sequence because tau(100) - tau(99) = 9 - 6 = 3.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 90 terms from Harvey P. Dale)

Crossrefs

Cf. A055927 (numbers n such that tau(n+1) - tau(n) = 1), A230115 (numbers n such that tau(n+1) - tau(n) = 2), A000005.

Programs

Mathematica

Select[ Range[ 50000], DivisorSigma[0, # ] + 3 == DivisorSigma[0, # + 1] &]
Position[Differences[DivisorSigma[0,Range[300400]]],3]//Flatten (* Harvey P. Dale, Jun 30 2022 *)

PARI

isok(n) = numdiv(n+1) - numdiv(n) == 3; \\ Michel Marcus, Oct 27 2013

Python

from sympy import divisor_count as tau
from itertools import count, islice
def agen(): # generator of terms, using comment by David A. Corneth
    for m in count(1):
        mm = m*m
        tmm = tau(mm)
        if tmm - tau(mm-1) == 3: yield mm-1
        if tau(mm+1) - tmm == 3: yield mm
print(list(islice(agen(), 36))) # Michael S. Branicky, Feb 16 2024

Extensions

More terms from Michel Marcus, Oct 27 2013