A230115 Numbers n such that tau(n+1) - tau(n) = 2; where tau(n) = the number of divisors of n (A000005).

5, 7, 13, 27, 37, 51, 61, 62, 73, 74, 91, 115, 123, 146, 153, 157, 164, 187, 188, 193, 206, 235, 245, 267, 274, 277, 278, 284, 291, 313, 355, 356, 362, 369, 386, 397, 403, 411, 421, 422, 423, 425, 427, 428, 451, 457, 538, 541, 605, 613, 637, 657, 661, 667, 673

Offset: 1

Author

Jaroslav Krizek, Oct 09 2013

Keywords

• nonn

Comments

Numbers n such that A051950(n+1) = 2.

Numbers n such that A049820(n) - A049820(n+1) = 1.

Sequence of starts of first run of n (n>=2) consecutive integers m_1, m_2, ..., m_n such that tau(m_k) - tau(m_k-1) = 2, for all k=n...2: 5, 61, 421, ... (a(5) > 100000); example for n=4: tau(421) = 2, tau(422) = 4, tau(423) = 6, tau(424) = 8.

Examples

			Number 7 is in sequence because tau(8) - tau(7) = 4 - 2 = 2.
		

Links

• Jaroslav Krizek, Table of n, a(n) for n = 1..2000

Crossrefs

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

Subsequence of A162318. - Michel Marcus, Mar 26 2017

Programs

Mathematica

Select[ Range[ 50000], DivisorSigma[0, # ] + 2 == DivisorSigma[0, # + 1] &]
Flatten[Position[Partition[DivisorSigma[0,Range[700]],2,1],? (#[[2]]- #[[1]] == 2&),{1},Heads->False]] (* _Harvey P. Dale, Aug 03 2014 *)

PARI

isok(n) = (numdiv(n+1) - numdiv(n)) == 2; \\ Michel Marcus, Mar 26 2017

Python

from sympy.ntheory import divisor_count
[n for n in range(1000) if divisor_count(n + 1) - divisor_count(n) == 2] # Indranil Ghosh, Mar 26 2017