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A138046 Positive integers k such that (d(k+1) - d(k)) * (-1)^k is positive, where d(k) = the number of positive divisors of k.

%I A138046 #26 May 13 2023 17:13:18
%S A138046 45,62,74,81,105,117,134,146,164,165,188,194,206,225,254,261,273,274,
%T A138046 278,284,297,314,315,325,333,345,356,357,362,385,386,398,404,405,422,
%U A138046 428,435,441,454,458,465,477,482,494,495,513,524,525,538,554,555,561
%N A138046 Positive integers k such that (d(k+1) - d(k)) * (-1)^k is positive, where d(k) = the number of positive divisors of k.
%C A138046 The number of terms < 10^m, for m >= 1: 0, 4, 104, 1320, 15000, 162705, ..., . The smallest term which is the beginning of n consecutive terms: 45, 164, 625, 2274, 30481, 150992, 624963, 726421, ..., . - _Robert G. Wilson v_, Mar 23 2008
%H A138046 Muniru A Asiru, <a href="/A138046/b138046.txt">Table of n, a(n) for n = 1..10000</a>
%p A138046 with(numtheory): a:=proc(n) if 0<(-1)^n*(tau(n+1)-tau(n)) then n else end if end proc: seq(a(n),n=1..500); # _Emeric Deutsch_, Mar 06 2008
%p A138046 A051950 := proc(n) numtheory[tau](n)-numtheory[tau](n-1) ; end: A138046 := proc(n) option remember ; local a; if n = 1 then 45 ; else for a from A138046(n-1)+1 do if (-1)^a*A051950(a+1) > 0 then RETURN(a) ; fi ; od: fi ; end: seq(A138046(n),n=1..80) ; # _R. J. Mathar_, Mar 31 2008
%t A138046 f[n_] := (DivisorSigma[0, n + 1] - DivisorSigma[0, n])*(-1)^n; Select[ Range@ 565, f@# > 0 &] (* _Robert G. Wilson v_, Mar 23 2008 *)
%o A138046 (GAP) Filtered([1..1000],n->IsPosInt((Tau(n+1)-Tau(n))*(-1)^n)); # _Muniru A Asiru_, May 27 2018
%o A138046 (PARI) isok(n) = (numdiv(n+1) - numdiv(n))*(-1)^n > 0; \\ _Michel Marcus_, May 27 2018
%Y A138046 Cf. A138047.
%K A138046 nonn
%O A138046 1,1
%A A138046 _Leroy Quet_, Mar 02 2008
%E A138046 More terms from _Emeric Deutsch_, Mar 06 2008
%E A138046 More terms from _R. J. Mathar_ and _Robert G. Wilson v_, Mar 23 2008