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%I #26 Dec 06 2019 15:49:36
%S 5,7,13,27,37,51,61,62,73,74,91,115,123,146,153,157,164,187,188,193,
%T 206,235,245,267,274,277,278,284,291,313,355,356,362,369,386,397,403,
%U 411,421,422,423,425,427,428,451,457,538,541,605,613,637,657,661,667,673
%N Numbers n such that tau(n+1) - tau(n) = 2; where tau(n) = the number of divisors of n (A000005).
%C Numbers n such that A051950(n+1) = 2.
%C Numbers n such that A049820(n) - A049820(n+1) = 1.
%C 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.
%H Jaroslav Krizek, <a href="/A230115/b230115.txt">Table of n, a(n) for n = 1..2000</a>
%e Number 7 is in sequence because tau(8) - tau(7) = 4 - 2 = 2.
%t Select[ Range[ 50000], DivisorSigma[0, # ] + 2 == DivisorSigma[0, # + 1] &]
%t Flatten[Position[Partition[DivisorSigma[0,Range[700]],2,1],_? (#[[2]]- #[[1]] == 2&),{1},Heads->False]] (* _Harvey P. Dale_, Aug 03 2014 *)
%o (PARI) isok(n) = (numdiv(n+1) - numdiv(n)) == 2; \\ _Michel Marcus_, Mar 26 2017
%o (Python)
%o from sympy.ntheory import divisor_count
%o [n for n in range(1000) if divisor_count(n + 1) - divisor_count(n) == 2] # _Indranil Ghosh_, Mar 26 2017
%Y Cf. A000005, A055927 (numbers n such that tau(n+1) - tau(n) = 1).
%Y Subsequence of A162318. - _Michel Marcus_, Mar 26 2017
%K nonn
%O 1,1
%A _Jaroslav Krizek_, Oct 09 2013
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