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A275418 Numbers n such that n - 1 has exactly as many odd divisors as n + 1.

%I A275418 #59 Sep 08 2022 08:46:17
%S A275418 3,4,6,11,12,13,18,21,23,25,27,30,34,39,42,45,47,56,57,60,72,75,81,86,
%T A275418 87,92,93,94,95,99,102,105,108,109,117,123,124,131,135,138,139,142,
%U A275418 144,147,150,155,159,160,165,169,177,180,184,186,192,193,198,202,204,207,213,214,216
%N A275418 Numbers n such that n - 1 has exactly as many odd divisors as n + 1.
%C A275418 Numbers n > 1 such that d(2n - 2) + d(n + 1) = d(2n + 2) + d(n - 1) where d = A000005.
%C A275418 Conjectures:
%C A275418 (1) There are only finitely many terms n such that A001227(n - 1) = A001227(n + 1) is odd: 3, 99, 577, 3363, ... (see A276188).
%C A275418 (2) There are only finitely many terms n such that A001227(n - 1) = A001227(n) = A001227(n + 1) = 2: 6, 11, 12, 13, 23, 47, 192, 193, 383, 786432, ... (see also A181490-A181493, A276136).
%C A275418 (3) There are only finitely many prime terms p such that A001227(p - 1) = A001227(p + 1) is prime: 11, 13, 23, 47, 193, 383, 577, ... (see also A275598).
%C A275418 I don't find any more for conjecture #3 up to 10^10. - _Charles R Greathouse IV_,  Aug 22 2016
%H A275418 Charles R Greathouse IV, <a href="/A275418/b275418.txt">Table of n, a(n) for n = 1..10000</a>
%e A275418 3 is in this sequence because 2 and 4 both have only one odd divisor, 1.
%e A275418 4 is in this sequence because 3 and 5 both have exactly two odd divisors each (1 and 3 for the former, 1 and 5 for the latter).
%p A275418 N:= 1000: # to get all terms < N
%p A275418 nod:= proc(n) numtheory:-tau(n/2^padic:-ordp(n,2)) end proc:
%p A275418 X:= map(nod,[$1..N]):
%p A275418 select(t -> X[t+1]=X[t-1], [$2..N-1]); # _Robert Israel_, Aug 04 2016
%t A275418 f[n_] := Count[Divisors@ n, k_ /; OddQ@ k]; Select[Range[2, 240], f[# - 1] == f[# + 1] &] (* _Michael De Vlieger_, Jul 28 2016 *)
%t A275418 Flatten[Position[Partition[Table[Count[Divisors[n],_?OddQ],{n,300}],3,1],_?(#[[1]]==#[[3]]&),{1},Heads->False]]+1 (* _Harvey P. Dale_, Nov 02 2016 *)
%o A275418 (Magma) [n: n in [2..216] | NumberOfDivisors(2*(n-1))+ NumberOfDivisors(n+1) eq NumberOfDivisors(2*(n+1))+ NumberOfDivisors(n-1)];
%o A275418 (PARI) a001227(n) = sumdiv(n, d, d%2);
%o A275418 is(n) = a001227(n-1)==a001227(n+1) \\ _Felix Fröhlich_, Jul 27 2016
%o A275418 (PARI) is(n)=numdiv((n-1)>>valuation(n-1,2)) == numdiv((n+1)>>valuation(n+1,2)) \\ _Charles R Greathouse IV_, Jul 29 2016
%Y A275418 Supersequence of A014574.
%Y A275418 Cf. A000005, A001227, A067888, A181490-A181493, A273401, A275598, A276136, A276188.
%K A275418 nonn,easy
%O A275418 1,1
%A A275418 _Juri-Stepan Gerasimov_, Jul 27 2016
%E A275418 Name edited by _Alonso del Arte_, Aug 23 2016