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%I #29 Feb 16 2024 17:25:42
%S 49,99,1023,1681,1935,2499,8649,9603,20449,21903,23715,29583,30975,
%T 38024,43263,58563,60515,71824,74528,110223,130321,136899,145924,
%U 150543,154449,165649,181475,216224,224675,233288,243049,256035,258063,265225,294849,300303
%N Numbers k such that tau(k+1) - tau(k) = 3, where tau(k) = the number of divisors of k (A000005).
%C Numbers k such that A051950(k+1) = 3.
%C Numbers k such that A049820(k) - A049820(k+1) = 2.
%C k or k+1 is a perfect square. - _David A. Corneth_, Feb 16 2024
%H David A. Corneth, <a href="/A230653/b230653.txt">Table of n, a(n) for n = 1..10000</a> (first 90 terms from Harvey P. Dale)
%e 99 is in the sequence because tau(100) - tau(99) = 9 - 6 = 3.
%t Select[ Range[ 50000], DivisorSigma[0, # ] + 3 == DivisorSigma[0, # + 1] &]
%t Position[Differences[DivisorSigma[0,Range[300400]]],3]//Flatten (* _Harvey P. Dale_, Jun 30 2022 *)
%o (PARI) isok(n) = numdiv(n+1) - numdiv(n) == 3; \\ _Michel Marcus_, Oct 27 2013
%o (Python)
%o from sympy import divisor_count as tau
%o from itertools import count, islice
%o def agen(): # generator of terms, using comment by _David A. Corneth_
%o for m in count(1):
%o mm = m*m
%o tmm = tau(mm)
%o if tmm - tau(mm-1) == 3: yield mm-1
%o if tau(mm+1) - tmm == 3: yield mm
%o print(list(islice(agen(), 36))) # _Michael S. Branicky_, Feb 16 2024
%Y Cf. A055927 (numbers n such that tau(n+1) - tau(n) = 1), A230115 (numbers n such that tau(n+1) - tau(n) = 2), A000005.
%K nonn
%O 1,1
%A _Jaroslav Krizek_, Oct 27 2013
%E More terms from _Michel Marcus_, Oct 27 2013
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