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A273401 Numbers n such that n and n + 1 has exactly the same number of odd divisors. 3
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 *)

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: A191472 A189056 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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Last modified February 22 18:27 EST 2018. Contains 299469 sequences. (Running on oeis4.)