A060831 -id:A060831 - cp's OEIS frontend

A262535 Triangle read by rows T(n,k) in which column k lists the partial sums of the k-th column of triangle A261699.

1, 2, 3, 3, 4, 3, 5, 8, 6, 8, 3, 7, 15, 3, 8, 15, 3, 9, 24, 6, 10, 24, 6, 5, 11, 35, 6, 5, 12, 35, 9, 5, 13, 48, 9, 5, 14, 48, 9, 12, 15, 63, 12, 12, 5, 16, 63, 12, 12, 5, 17, 80, 12, 12, 5, 18, 80, 15, 21, 5, 19, 99, 15, 21, 5, 20, 99, 15, 21, 10, 21, 120, 18, 21, 10, 7, 22, 120, 18, 32, 10, 7, 23, 143, 18, 32, 10, 7, 24, 143, 21, 32, 10, 7, 25, 168, 21, 32, 15, 7, 26, 168, 21, 45, 15, 7, 27, 195, 24, 45, 15, 16

Offset: 1

Omar E. Pol, Sep 24 2015

Conjecture: the sum of row n gives A078471(n), the sum of all odd divisors of all positive integers <= n.
Row n has length A003056(n) hence column k starts in row A000217(k).
Column 1 gives A000027.

			Triangle begins:
1;
2;
3,    3;
4,    3;
5,    8;
6,    8,  3;
7,   15,  3;
8,   15,  3;
9,   24,  6;
10,  24,  6,  5;
11,  35,  6,  5;
12,  35,  9,  5;
13,  48,  9,  5;
14,  48,  9, 12;
15,  63, 12, 12,  5;
16,  63, 12, 12,  5;
17,  80, 12, 12,  5;
18,  80, 15, 21,  5;
19,  99, 15, 21,  5;
20,  99, 15, 21, 10;
21, 120, 18, 21, 10,  7;
22, 120, 18, 32, 10,  7;
23, 143, 18, 32, 10,  7;
24, 143, 21, 32, 10,  7;
25, 168, 21, 32, 15,  7;
26, 168, 21, 45, 15,  7;
27, 195, 24, 45, 15, 16;
...
For n = 6 the sum of all odd divisors of all positive integers <= 6 is (1) + (1) + (1 + 3) + (1) + (1 + 5) + (1 + 3) = 17. On the other hand the sum of the 6th row of triangle is 6 + 8 + 3 = 17 equaling the sum of all odd divisors of all positive integers <= 6.

Cf. A000027, A000217, A001227, A003056, A060831, A078471, A236104, A237593, A261699.

A285999 Total number of odd divisors of all positive integers <= n, minus the total number of middle divisors of all positive integers <= n.

Original entry on oeis.org

0, 0, 2, 2, 4, 4, 6, 6, 8, 10, 12, 12, 14, 16, 18, 18, 20, 22, 24, 24, 28, 30, 32, 32, 34, 36, 40, 40, 42, 44, 46, 46, 50, 52, 54, 56, 58, 60, 64, 64, 66, 68, 70, 72, 76, 78, 80, 80, 82, 84, 88, 90, 92, 94, 98, 98, 102, 104, 106, 108, 110, 112, 116, 116, 120, 122, 124, 126, 130, 132, 134, 134, 136, 138, 144, 146, 148, 152

Offset: 1

Omar E. Pol, May 14 2017

nonn

Conjecture 1: a(n) is also twice the total number of partitions of all positive integers <= n into an even number of consecutive parts.
Conjecture 2: a(n) is also the total number of equidistant subparts of the symmetric representations of sigma of all positive integers <= n. Thus a(n) is also the total number of equidistant subparts in the terraces of the stepped pyramid with n levels described in A245092.
For more information about the "subparts" of the symmetric representation of sigma see A279387 and A237593.

Partial sums of A281009.

Cf. A060831, A196020, A235791, A236104, A237048, A237591, A237593, A240542, A245092, A279387, A285901, A285902.

Accumulate@ Table[DivisorSum[n, 1 &, OddQ] - DivisorSum[n, 1 &, Sqrt[n/2] <= # < Sqrt[2 n] &], {n, 78}] (* Michael De Vlieger, May 18 2017 *)

Conjecture: a(n) = A060831(n) - A240542(n).

Conjecture: a(n) = 2*A285902(n).

A338733 Partial sums of A054843.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 13, 15, 16, 19, 22, 24, 26, 28, 30, 35, 36, 38, 41, 43, 45, 50, 52, 54, 56, 59, 61, 65, 68, 70, 74, 76, 77, 81, 83, 87, 91, 93, 95, 99, 101, 103, 107, 109, 111, 118, 120, 122, 124, 127, 130, 134, 136, 138, 142, 147, 149, 153, 155, 157, 161, 163, 165, 171, 172

Offset: 0

N. J. A. Sloane, Dec 06 2020

nonn

Suggested by the important sequence A060831, which gives partial sums of A001227.

Cf. A001227, A054843, A060831.

A339009 Numbers k such that the average number of odd divisors of {1..k} is an integer.

Original entry on oeis.org

1, 2, 165, 170, 1274, 9437, 69720, 69732, 69734, 69736, 515230, 515236, 515246, 28132043, 28132063, 28132079

Offset: 1

Ilya Gutkovskiy, Nov 18 2020

Numbers k that divide A060831(k) where A060831(k) = Sum_{j=1..k} A001227(j).

The sequence also includes: 83860580242, 4578632504347, 4578632504465, 4578632504515. - Daniel Suteu, Nov 24 2020

			165 is in the sequence because the average number of odd divisors of {1..165} is an integer: A060831(165) / 165 = 495 / 165 = 3.

Eric Weisstein's World of Mathematics, Odd Divisor Function.

Cf. A001227, A048290, A050226, A060831, A063971, A064610, A064612.

s[n_] := Module[{c = 0, k = 1, sum = 0, seq = {}}, While[c < n, sum += DivisorSigma[0, k/2^IntegerExponent[k, 2]]; If[Divisible[sum, k], c++; AppendTo[seq, k]]; k++]; seq]; s[13] (* Amiram Eldar, Nov 18 2020 *)

f(n) = my(n2=n\2); sum(k=1, sqrtint(n), n\k)*2-sqrtint(n)^2-sum(k=1, sqrtint(n2), n2\k)*2+sqrtint(n2)^2; \\ A060831
isok(k) = (f(k) % k) == 0; \\ Michel Marcus, Nov 25 2020

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A262535 Triangle read by rows T(n,k) in which column k lists the partial sums of the k-th column of triangle A261699.

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A285999 Total number of odd divisors of all positive integers <= n, minus the total number of middle divisors of all positive integers <= n.

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A338733 Partial sums of A054843.

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A339009 Numbers k such that the average number of odd divisors of {1..k} is an integer.

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Mathematica

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