A137510 -id:A137510 - cp's OEIS frontend

A264440 Row lengths of the irregular triangle A137510 (number of divisors d of n with 1 < d < n, or 0 if no such d exists).

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 1, 2, 2, 4, 1, 6, 1, 4, 2, 2, 2, 7, 1, 2, 2, 6, 1, 6, 1, 4, 4, 2, 1, 8, 1, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 10, 1, 2, 4, 5, 2, 6, 1, 4, 2, 6, 1, 10, 1, 2, 4, 4, 2, 6, 1, 8, 3, 2, 1, 10, 2, 2

Offset: 1

Wolfdieter Lang, Jan 16 2016

See A032741 for the number of divisors d of n with 1 <= d < n, n >= 1.

See A070824 for the number of the divisors d of n with 1 < d < n, n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000040, A032741, A070824, A137510.

seq(max(1,numtheory:-tau(n)-2), n=1..100); # Robert Israel, Jan 20 2016

Array[DivisorSigma[0, #] - 2 &, {80}] /. n_ /; n < 2 -> 1 (* Michael De Vlieger, Jan 16 2016 *)

A264440(n) = max(1,numdiv(n)-2); \\ After Robert Israel's formula. - Antti Karttunen, May 25 2017

a(1) = 1; a(n) = 1 if n is prime, otherwise a(n) = A070824(n).
a(1) = 1; a(n) = 1 if n is prime, otherwise a(n) = A032741(n) - 1.
a(n) = max(1, A000005(n)-2). - Robert Israel, Jan 20 2016

A233773 Triangle read by rows in which row n lists the proper divisors of n together with -n.

Original entry on oeis.org

-1, 1, -2, 1, -3, 1, 2, -4, 1, -5, 1, 2, 3, -6, 1, -7, 1, 2, 4, -8, 1, 3, -9, 1, 2, 5, -10, 1, -11, 1, 2, 3, 4, 6, -12, 1, -13, 1, 2, 7, -14, 1, 3, 5, -15, 1, 2, 4, 8, -16, 1, -17, 1, 2, 3, 6, 9, -18, 1, -19, 1, 2, 4, 5, 10, -20, 1, 3, 7, -21, 1, 2, 11, -22, 1, -23

Offset: 1

Omar E. Pol, Dec 31 2013

The same as A027750 but with the last term of every row multiplied by -1.

The sum of row n gives the abundance of n.

			Written as an irregular triangle in which row n has length A000005(n) the sequence begins:
-1;
1, -2;
1, -3;
1, 2, -4;
1, -5;
1, 2, 3, -6;
1, -7;
1, 2, 4, -8;
1, 3, -9;
1, 2, 5, -10;
1, -11;
1, 2, 3, 4, 6, -12;
...

Eric Weisstein's World of Mathematics, Abundance
Eric Weisstein's World of Mathematics, Quasiperfect Number

Row sums give A033880.
Cf. A000005, A000203, A000396, A001065, A005100, A005101, A027750, A027751, A033879.
Cf. A137510, A163870.

Flatten[Table[Join[{Most[Divisors[n]]},{-n}],{n,30}]] (* Harvey P. Dale, Feb 20 2016 *)

A348554 Irregular triangle read by rows: row n gives the divisors d of 2n with 1 < d < 2n, for n >= 2.

Original entry on oeis.org

2, 2, 3, 2, 4, 2, 5, 2, 3, 4, 6, 2, 7, 2, 4, 8, 2, 3, 6, 9, 2, 4, 5, 10, 2, 11, 2, 3, 4, 6, 8, 12, 2, 13, 2, 4, 7, 14, 2, 3, 5, 6, 10, 15, 2, 4, 8, 16, 2, 17, 2, 3, 4, 6, 9, 12, 18, 2, 19, 2, 4, 5, 8, 10, 20, 2, 3, 6, 7, 14, 21, 2, 4, 11, 22, 2, 23, 2, 3, 4, 6, 8, 12, 16, 24, 2, 5, 10, 25

Offset: 2

Wolfdieter Lang, Oct 22 2021

This gives the rows 2*n of A137510, for n >= 2.
The length of row n is A069930(n) = tau(2*n) - 2 = A099777(n) - 2.
The sum of row n is A346880(n) = A062731(n) - (2*n + 1).

			The irregular triangle T(n, k) begins:
n, 2*n / k 1  2  3  4  5  6  7 ...
----------------------------------
2,   4:    2
3,   6:    2  3
4,   8:    2  4
5,  10:    2  5
6   12:    2  3  4 6
7,  14:    2  7
8,  16:    2  4  8
9,  18:    2  3  6  9
10, 20:    2  4  5 10
11, 22:    2 11
12, 24:    2  3  4  6  8 12
13, 26:    2 13
14, 28:    2  4  7 14
15, 30:    2  3  5  6 10 15
16, 32:    2  4  8  1
17, 34:    2 17
18, 36:    2  3  4  6  9 12 18
19, 38:    2 19
20, 40:    2  4  5  8 10 20
...

Cf. A069930, A099777, A137510, A346880.

Flatten@Table[Select[Divisors[2n],1<#<2n&],{n,2,25}] (* Giorgos Kalogeropoulos, Oct 22 2021 *)

row(n) = select(x->((x>1) && (x<2*n)), divisors(2*n)); \\ Michel Marcus, Oct 23 2021

T(n, k) = A137510(2*n, k), for n >= 2 and k = 1, 2, ..., A069930(n).

cp's OEIS Frontend

A264440 Row lengths of the irregular triangle A137510 (number of divisors d of n with 1 < d < n, or 0 if no such d exists).

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A233773 Triangle read by rows in which row n lists the proper divisors of n together with -n.

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A348554 Irregular triangle read by rows: row n gives the divisors d of 2n with 1 < d < 2n, for n >= 2.

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cp's OEIS Frontend

A264440 Row lengths of the irregular triangle A137510 (number of divisors d of n with 1 < d < n, or 0 if no such d exists).

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Maple

Mathematica

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A233773 Triangle read by rows in which row n lists the proper divisors of n together with -n.

Views

Author

Keywords

Comments

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Programs

Mathematica

A348554 Irregular triangle read by rows: row n gives the divisors d of 2*n with 1 < d < 2*n, for n >= 2.

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Author

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Mathematica

PARI

Formula

A348554 Irregular triangle read by rows: row n gives the divisors d of 2n with 1 < d < 2n, for n >= 2.