A273401 - OEIS

A273401

Numbers n such that n and n + 1 have exactly the same number of odd divisors.

1, 5, 6, 10, 11, 12, 13, 19, 22, 23, 28, 37, 40, 43, 46, 47, 49, 52, 54, 58, 61, 65, 67, 69, 73, 77, 79, 82, 84, 88, 96, 103, 106, 110, 112, 114, 119, 129, 132, 136, 140, 148, 151, 154, 155, 157, 163, 166, 172, 178, 182, 185, 186, 191, 192, 193, 203, 204, 211, 215, 216, 219, 220, 221

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

If A001227(n) = A001227(n*2^m) for m >= 0 then:

1) A001227(n) is equal to number of ways to write 2n - 1 as (4*x + 2)*y + 4*x + 1 where x and y are nonnegative integers;

2) A001227(n) is equal to number of distinct values of k if k/(2n-1) + 1 divides (k/(2n - 1))^(k/(2n - 1)) + k, (k/(2n - 1))^k + k/(2n - 1) and k^(k/(2n - 1)) + k/(2n - 1).

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

5 and 6 have both two odd divisors: (1 and 5) and (1 and 3) respectively; so 5 is a term in the sequence.

MAPLE

A001227:= n -> numtheory:-tau(n)/(1+padic:-ordp(n, 2)):

R:= map(A001227, [$1..1000]):

ListTools:-SearchAll(0, A001227[2..-1]-A001227[1..-2]); # Robert Israel, May 27 2016

MATHEMATICA

Select[Range@ 221, First@ Differences@ Map[Count[Divisors@ #, _?OddQ] &, {#, # + 1}] == 0 &] (* Michael De Vlieger, Jun 26 2016 *)

SequencePosition[Table[Count[Divisors[n], _?OddQ], {n, 250}], {x_, x_}] [[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 06 2019 *)

PROG

(PARI) lista(nn) = for (n=1, nn, if (sumdiv(n, d, d%2) == sumdiv(n+1, d, d%2), print1(n, ", "))); \\ Michel Marcus, May 27 2016

(PARI) is(n)=numdiv(n>>valuation(n, 2))==numdiv((n+1)>>valuation(n+1, 2)) \\ Charles R Greathouse IV, Jul 15 2016

CROSSREFS

Cf. A001227, A206581 (primes in a(n)).

Sequence in context: A189056 A340325 A247561 * A042958 A265188 A163903

Adjacent sequences: A273398 A273399 A273400 * A273402 A273403 A273404

KEYWORD

nonn

AUTHOR

Juri-Stepan Gerasimov, May 26 2016

STATUS

approved

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