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Numbers k such that k + 1 has one more divisor than k.
14

%I #41 Feb 16 2024 10:12:41

%S 1,3,9,15,25,63,121,195,255,361,483,729,841,1443,3363,3481,3721,5041,

%T 6241,10201,15625,17161,18224,19321,24963,31683,32761,39601,58564,

%U 59049,65535,73441,88208,110889,121801,143641,145923,149769,167281

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

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

%C Numbers k such that A049820(k) = A049820(k+1). - _Jaroslav Krizek_, Feb 10 2014

%C Numbers k such that A051950(k+1) = 1. - _Danny Rorabaugh_, Oct 05 2017

%H Giovanni Resta, <a href="/A055927/b055927.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Donovan Johnson)

%e 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.

%t Select[ Range[ 200000], DivisorSigma[0, # ] + 1 == DivisorSigma[0, # + 1] &]

%o (PARI) for(n=1,1000,if(numdiv(n+1)-numdiv(n)==1,print1(n,", "))); /* _Joerg Arndt_, Apr 09 2011 */

%Y Numbers where repetition occurs in A049820.

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

%K nonn

%O 1,2

%A _Labos Elemer_, Jul 21 2000

%E More terms from _David W. Wilson_, Sep 06 2000, who remarks that every element is of form n^2 or n^2 - 1.