A055927 Numbers k such that k + 1 has one more divisor than k.

1, 3, 9, 15, 25, 63, 121, 195, 255, 361, 483, 729, 841, 1443, 3363, 3481, 3721, 5041, 6241, 10201, 15625, 17161, 18224, 19321, 24963, 31683, 32761, 39601, 58564, 59049, 65535, 73441, 88208, 110889, 121801, 143641, 145923, 149769, 167281

Offset: 1

Author

Labos Elemer, Jul 21 2000

Keywords

• nonn

Comments

Numbers k such that d(k+1) - d(k) = 1, where d(k) is A000005(k), the number of divisors.

Numbers k such that A049820(k) = A049820(k+1). - Jaroslav Krizek, Feb 10 2014

Numbers k such that A051950(k+1) = 1. - Danny Rorabaugh, Oct 05 2017

Examples

			a(4) = 15, as 15 has 4 and 16 has 5 divisors. a(6) = 63, as 63 and 64 have 6 and 7 divisors respectively.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Donovan Johnson)

Crossrefs

Numbers where repetition occurs in A049820.

Cf. A000005, A006073, A045983, A049820, A075044.

Programs

Mathematica

Select[ Range[ 200000], DivisorSigma[0, # ] + 1 == DivisorSigma[0, # + 1] &]
Position[Differences[DivisorSigma[0,Range[170000]]],1]//Flatten (* Harvey P. Dale, Jul 06 2025 *)

PARI

for(n=1,1000,if(numdiv(n+1)-numdiv(n)==1,print1(n,", "))); /* Joerg Arndt, Apr 09 2011 */

Extensions

More terms from David W. Wilson, Sep 06 2000, who remarks that every element is of form n^2 or n^2 - 1.