A246637 -id:A246637 - cp's OEIS frontend

A246636 Numbers k such that C(k+2,2) divides 2^(k+1) - 1.

0, 1, 5, 17, 41, 125, 161, 377, 485, 881, 1457, 2645, 3077, 3941, 5417, 9197, 11825, 14405, 16757, 18521, 24965, 26405, 37337, 39365, 42461, 71441, 77657, 95921, 99077, 113777, 117305, 143261, 174761, 175445, 184841, 265481, 304037, 308825, 318401, 351917

Offset: 1

Clark Kimberling, Sep 01 2014

These are the numbers k such that mean of the numbers in the first k rows of Pascal' s triangle is an integer. All such k except 1 are congruent to -1 mod 6.

			The sum of the numbers in Pascal's triangle, from row 0 through row 17, is 2^18 - 1 = 262143; the number of such numbers is C(19,2) = 171, and 262143/171 = 1533; thus 17 is in this sequence, and 1533 is in A246637.

Robert Israel, Table of n, a(n) for n = 1..181

Cf. A246637, A007318.

select(k -> 2 &^(k+1) - 1 mod ((k+1)*(k+2)/2) = 0, [$0..10^6]); # Robert Israel, Nov 30 2023

z = 1000;
t = Select[Range[0, z], IntegerQ[(2^(# + 1) - 1)/Binomial[# + 2, 2]] &]

Offset corrected by Robert Israel, Nov 30 2023

A246648 Numbers k such that 2*k + 1 divides 2^(k+1) - 1.

Original entry on oeis.org

0, 1, 7, 127, 227, 647, 1351, 1907, 3239, 4607, 5219, 5975, 11447, 13159, 13919, 21527, 22049, 23759, 23939, 24839, 30959, 31283, 31583, 31967, 32767, 37223, 46091, 46511, 47267, 60479, 65663, 66527, 78539, 78599, 81727, 82799, 84311, 98405, 102671, 103967

Offset: 1

Clark Kimberling, Sep 01 2014

These are the numbers k such that mean of the k-th row of the triangle at A027926 is an integer.

Numbers k such that 2*k + 1 divides 2^k + k. - Thomas Ordowski, Jun 04 2024

			The sum of the numbers row 7 of the triangular array at A027926 is 2^8 - 1 = 255, and the number of numbers in row 7 is 15, and 255/15 = 17; thus 7 is in this sequence, and 17 is in A246649.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A246637, A246649.

filter:= k -> 2 &^ (k+1) - 1 mod (2*k+1) = 0:
select(filter, [$0..2*10^5]); # Robert Israel, Jan 10 2020

z = 140000; u = Select[Range[0, z], IntegerQ[(2^(# + 1) - 1)/(2 # + 1)] &]   (* A246648 *)
v = Table[(2^(u[[k]] + 1) - 1)/(2 u[[k]] + 1), {k, 1, 6}] (* A246649 *)

Edited and offset changed by Robert Israel, Jan 10 2020

A246649 Integers of the form (2^(k+1) - 1)/(2*k + 1).

Original entry on oeis.org

1, 1, 17, 1334440654591915542993625911497130241, 948042080603099421350928003060030968743284199473954197137709371401

Offset: 1

Clark Kimberling, Sep 01 2014

The next term has 192 digits. - Harvey P. Dale, Feb 05 2019

			The sum of the numbers row 7 of the triangular array at A027926 is 2^8 - 1 = 255, and the number of numbers in row 7 is 15, and 255/15 = 17; thus 7 is in this sequence, and 17 is in A246649.

Cf. A246637, A246648.

z = 140000; u = Select[Range[0, z], IntegerQ[(2^(# + 1) - 1)/(2 # + 1)] &]   (* A246648 *)
v = Table[(2^(u[[k]] + 1) - 1)/(2 u[[k]] + 1), {k, 1, 6}] (* A246649 *)
Select[Table[(2^(n+1)-1)/(2n+1),{n,0,250}],IntegerQ] (* Harvey P. Dale, Feb 05 2019 *)

cp's OEIS Frontend

A246636 Numbers k such that C(k+2,2) divides 2^(k+1) - 1.

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A246648 Numbers k such that 2*k + 1 divides 2^(k+1) - 1.

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Maple

Mathematica

Extensions

A246649 Integers of the form (2^(k+1) - 1)/(2*k + 1).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica