A092403 -id:A092403 - cp's OEIS frontend

A092405 a(n) = tau(n) + tau(n+1), where tau(n) = A000005(n), the number of divisors of n.

3, 4, 5, 5, 6, 6, 6, 7, 7, 6, 8, 8, 6, 8, 9, 7, 8, 8, 8, 10, 8, 6, 10, 11, 7, 8, 10, 8, 10, 10, 8, 10, 8, 8, 13, 11, 6, 8, 12, 10, 10, 10, 8, 12, 10, 6, 12, 13, 9, 10, 10, 8, 10, 12, 12, 12, 8, 6, 14, 14, 6, 10, 13, 11, 12, 10, 8, 10, 12, 10, 14, 14, 6, 10, 12, 10, 12, 10, 12, 15, 9, 6, 14, 16

Offset: 1

Jon Perry, Mar 22 2004

nonn

If a child is born to an n-year-old parent, this is the number of times the age of the parent will be a multiple of the age of the child. E.g., if n = 27, this will happen at the ages (28, 1), (29, 1),(30, 2), (30, 3), (32, 4), (35, 7), (42, 14), (36, 9), (54, 27), (56, 28). - Alexander Piperski, Sep 10 2018

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A092403, A175143, A346562.

Total /@ Partition[Array[DivisorSigma[0, #] &, 85], 2, 1] (* Michael De Vlieger, Sep 18 2018 *)

for(i=1,60,print1(","sigma(i,0)+sigma(i+1,0)))

A092405(n) = (numdiv(n)+numdiv(1+n)); \\ Antti Karttunen, Oct 07 2017

a(n) = A346562(n+1,n). - Omar E. Pol, Jul 23 2021

Extended by Ray Chandler, Mar 05 2010

A340584 Irregular triangle read by rows T(n,k) in which row n lists sigma(n) + sigma(n-1) together with the first n - 2 terms of A000203 in reverse order, with T(1,1) = 1, n >= 1.

Original entry on oeis.org

1, 4, 7, 1, 11, 3, 1, 13, 4, 3, 1, 18, 7, 4, 3, 1, 20, 6, 7, 4, 3, 1, 23, 12, 6, 7, 4, 3, 1, 28, 8, 12, 6, 7, 4, 3, 1, 31, 15, 8, 12, 6, 7, 4, 3, 1, 30, 13, 15, 8, 12, 6, 7, 4, 3, 1, 40, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 42, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 38, 28, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1

Offset: 1

Omar E. Pol, Jan 12 2021

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the base of the symmetric tower (a polycube) described in A221529 which has A000041(n-1) levels in total. The terraces of the polycube are the symmetric representation of sigma. The terraces are in the levels that are the partition numbers A000041 starting from the base. Note that for n >= 2 there are n - 1 terraces because the first terrace of the tower is formed by two symmetric representations of sigma in the same level. The volume (or the number of cubes) equals A066186(n), the sum of all parts of all partitions of n. The volume is also the sum of all divisors of all terms of the first n rows of A336811. That is due to the correspondence between divisors and partitions (cf. A336811). The growth of the volume (A066186) represents the convolution of A000203 and A000041.

			Triangle begins:
   1;
   4;
   7,  1;
  11,  3,  1;
  13,  4,  3,  1;
  18,  7,  4,  3,  1;
  20,  6,  7,  4,  3,  1;
  23, 12,  6,  7,  4,  3,  1;
  28,  8, 12,  6,  7,  4,  3,  1;
  31, 15,  8, 12,  6,  7,  4,  3,  1;
  30, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  40, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  42, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  38, 28, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the first term of row 7 is T(7,1) = 20. The other terms in row 7 are the first five terms of A000203 in reverse order, that is [6, 7, 4, 3, 1] so the 7th row of the triangle is [20, 6, 7, 4, 3, 1].
From _Omar E. Pol_, Jul 11 2021: (Start)
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due to the first two partition numbers A000041 are [1, 1]), so T(7,1) = sigma(7) + sigma(6) = 8 + 12 = 20. (End)

The length of row n is A028310(n-1).
Row sums give A024916.
Column 1 gives 1 together with A092403.
Other columns give A000203.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A346533 (mirror).
Cf. A000041, A221529, A237270, A237593, A245092, A272172 (similar), A336811, A338156, A339106, A340035.

Table[If[n <= 2, {Total@ #}, Prepend[#2, Total@ #1] & @@ TakeDrop[#, 2]] &@ DivisorSigma[1, Range[n, 1, -1]], {n, 14}] // Flatten (* Michael De Vlieger, Jan 13 2021 *)

A346533 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.

Original entry on oeis.org

1, 4, 1, 7, 1, 3, 11, 1, 3, 4, 13, 1, 3, 4, 7, 18, 1, 3, 4, 7, 6, 20, 1, 3, 4, 7, 6, 12, 23, 1, 3, 4, 7, 6, 12, 8, 28, 1, 3, 4, 7, 6, 12, 8, 15, 31, 1, 3, 4, 7, 6, 12, 8, 15, 13, 30, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 40, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 42, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 38

Offset: 1

Omar E. Pol, Jul 22 2021

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the top of the symmetric tower (a polycube) described in A221529.
The height of the tower equals A000041(n-1).
The terraces of the tower are the symmetric representation of sigma.
The terraces are in the levels that are the partition numbers A000041 starting from the base.
Note that for n >= 2 there are n - 1 terraces because the lower terrace of the tower is formed by two symmetric representations of sigma in the same level.

			Triangle begins:
  1;
  4;
  1, 7;
  1, 3, 11;
  1, 3,  4, 13;
  1, 3,  4,  7, 18;
  1, 3,  4,  7,  6, 20;
  1, 3,  4,  7,  6, 12, 23;
  1, 3,  4,  7,  6, 12,  8, 28;
  1, 3,  4,  7,  6, 12,  8, 15, 31;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 30;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 40;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 42;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 28, 38;
  ...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the last term of row 7 is T(7,6) = 20. The other terms in row 7 are the first five terms of A000203, so the 7th row of the triangle is [1, 3, 4, 7, 6, 20].
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due the first two partition numbers A000041 are [1, 1]), so T(7,6) = sigma(7) + sigma(6) = 8 + 12 = 20.
.
Illustration of initial terms:
   Row 1    Row 2      Row 3      Row 4        Row 5          Row 6
.
    1        4         1 7        1 3 11       1 3 4 13       1 3 4 7 18
.   _        _ _       _ _ _      _ _ _ _      _ _ _ _ _      _ _ _ _ _ _
   |_|      |   |     |_|   |    |_| |   |    |_| | |   |    |_| | | |   |
            |_ _|     |    _|    |_ _|   |    |_ _|_|   |    |_ _|_| |   |
                      |_ _|      |      _|    |_ _|  _ _|    |_ _|  _|   |
                                 |_ _ _|      |     |        |_ _ _|    _|
                                              |_ _ _|        |        _|
                                                             |_ _ _ _|
.

Paolo Xausa, Table of n, a(n) for n = 1..11176 (rows 1..150 of the triangle, flattened)
Omar E. Pol, Illustration of the partitions of 10, prism and tower

Mirror of A340584.
The length of row n is A028310(n-1).
Row sums give A024916.
Leading diagonal gives A092403.
Other diagonals give A000203.
Companion of A346562.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A000041, A221529, A237270, A237593, A245092, A245093 (similar), A336811, A336812, A338156, A339106, A340035.

A346533row[n_]:=If[n==1,{1},Join[DivisorSigma[1,Range[n-2]],{Total[DivisorSigma[1,{n-1,n}]]}]];Array[A346533row,15] (* Paolo Xausa, Oct 23 2023 *)

A227306 Numbers k that divide sigma(k) + sigma(k-1).

Original entry on oeis.org

2, 6, 34, 50, 216, 236, 262, 386, 898, 924, 945, 1456, 2380, 5356, 6468, 6624, 8362, 14100, 23496, 26938, 46594, 80876, 196344, 212796, 1661136, 4070200, 4160920, 4626700, 5244548, 5462384, 17062316, 60464628, 217408416, 248621604, 262792908, 265371336, 323987588

Offset: 1

Alex Ratushnyak, Jul 05 2013

nonn

Is 945 the only odd term? - Zak Seidov, Jul 06 2013
945 and 19910536425 are the only odd terms below 2^36. - Alex Ratushnyak, Jul 08 2013
The third odd term is a(58) = 841488503841. - Giovanni Resta, Apr 04 2014

Giovanni Resta, Table of n, a(n) for n = 1..65 (terms < 10^13)

Cf. A000203, A092403, A007691, A068078, A186103, A072188.

With[{nn=324*10^6},Select[Thread[{Total/@Partition[DivisorSigma[ 1,Range[ nn]],2,1],Range[ 2,nn]}],Divisible[#[[1]],#[[2]]]&][[All,2]]] (* Harvey P. Dale, May 29 2020 *)

A067282 Numbers k such that phi(k) + phi(k+1) divides sigma(k) + sigma(k+1).

Original entry on oeis.org

1, 5, 52, 55, 185, 506, 551, 590, 644, 667, 707, 2285, 2587, 2758, 7551, 10366, 11336, 11564, 11798, 12750, 16616, 16703, 16764, 17383, 18239, 24350, 24415, 26586, 33263, 35541, 40382, 63248, 76247, 76622, 92379, 95069, 97341, 106312, 111388

Offset: 1

Benoit Cloitre, Feb 23 2002

nonn

Presumably the ratio (sigma(n)+sigma(n+1))/(phi(n)+phi(n+1)) can be arbitrarily large. - Labos Elemer, Sep 17 2004

The first term for which the ratio is k for k = 2, 3, ... is 1, 5, 644, 6513584, ... - Amiram Eldar, Mar 02 2020

Amiram Eldar, Table of n, a(n) for n = 1..200

Cf. A000010, A000203, A092403, A092404.

Select[Range[120000], Divisible[DivisorSigma[1, #] + DivisorSigma[1, # + 1], EulerPhi[#] + EulerPhi[# + 1]] &] (* Amiram Eldar, Mar 02 2020 *)
Select[Partition[Table[{n,EulerPhi[n],DivisorSigma[1,n]},{n,111400}],2,1], Divisible[ #[[1,3]]+#[[2,3]],#[[1,2]]+#[[2,2]]]&][[All,1,1]] (* Harvey P. Dale, Apr 25 2020 *)

More terms from Labos Elemer, Sep 17 2004

A252922 a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3), with a(1)=0, a(2)=1, a(3)=4.

Original entry on oeis.org

0, 1, 4, 8, 14, 17, 25, 26, 35, 36, 46, 43, 58, 54, 66, 62, 79, 73, 88, 77, 101, 94, 110, 92, 120, 115, 133, 113, 138, 126, 158, 134, 167, 143, 165, 150, 193, 177, 189, 154, 206, 188, 228, 182, 224, 206, 234, 198, 244, 229, 274, 222, 263, 224, 272, 246, 312, 272, 290, 230, 318, 290, 326, 262, 327, 315, 355, 296

Offset: 1

Omar E. Pol, Dec 24 2014

This is also a rectangular array read by rows, with four columns, in which T(j,k) is the number of cells (also the area) of the j-th gap between the arms in the k-th quadrant of the spiral of the symmetric representation of sigma described in A239660, with j >= 1 and 1 <= k <= 4 and starting with T(1,1) = 0, see example.

We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

			a(5) = sigma(4) + sigma(3) + sigma(2) = 7 + 4 + 3 = 14. On the other hand a(5) = A024916(4) - A024916(1) = 15 - 1 = 14.
...
Also, if written as a rectangular array T(j,k) with four columns the sequence begins:
    0,   1,   4,   8;
   14,  17,  25,  26;
   35,  36,  46,  43;
   58,  54,  66,  62;
   79,  73,  88,  77;
  101,  94, 110,  92;
  120, 115, 133, 113;
  138, 126, 158, 134;
  167, 143, 165, 150;
  193, 177, 189, 154;
  206, 188, 228, 182;
  224, 206, 234, 198;
  244, 229, 274, 222;
  263, 224, 272, 246;
  312, 272, 290, 230;
  318, 290, 326, 262;
  ...
In this case T(2,1) = a(5) = 14.

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

Cf. A000203, A010883, A024916, A092403, A112610, A193553, A196020, A236104, A237270, A237271, A237593, A239052, A239053, A239931-A239934, A239660, A240020, A244050, A245092, A262626.

L:= [0,0,0,seq(numtheory:-sigma(n), n=1..100)]:
L[1..101]+L[2..102]+L[3..103]; # Robert Israel, Dec 07 2016

a252922[n_] := Block[{f}, f[1] = 0; f[2] = 1; f[3] = 4;
  f[x_] := DivisorSigma[1, x - 1] + DivisorSigma[1, x - 2] +
DivisorSigma[1, x - 3]; Table[f[i], {i, n}]]; a252922[68] (* Michael De Vlieger, Dec 27 2014 *)

v=concat([0,1,4],vector(100,n,sigma(n)+sigma(n+1)+sigma(n+2))) \\ Derek Orr, Dec 30 2014

a(1) = 0, a(2) = sigma(1) = 1, a(3) = sigma(2) + sigma(1) = 4; for n >= 4, a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3).

a(n) = A024916(n-1) - A024916(n-4) for n >= 5.

A348335 a(n) = smallest k such that the sum of the divisors of the n numbers from k to k+n-1 equals sigma(k+n), or -1 if no such k exists.

Original entry on oeis.org

14, 1, 591357

Offset: 1

Metin Sariyar, Oct 13 2021

a(4) > 10^9, if it exists. - Amiram Eldar, Oct 13 2021

			a(1) = 14 because sigma(14) = sigma(15) = 24; a(1) = A002961(1).
a(2) = 1 because sigma(1) + sigma(2) = 1 + 3 = 4, the same as sigma(3) = 4; a(2) = A104149(1).
a(3) = 591357 because sigma(591357) + sigma(591358) + sigma(591359) = 866880 + 890352 + 599760 = 2356992, the same as sigma(591360) = 2356992.

Cf. A000203, A002961, A065900, A092403, A104149, A252922.

a[n_] := Module[{sig = DivisorSigma[1, Range[n]], k = n + 1}, While[(s = DivisorSigma[1, k]) != Plus @@ sig, sig = Join[Drop[sig, 1], {s}]; k++]; k - n]; Array[a, 3] (* Amiram Eldar, Oct 29 2021 *)

isok(m, nb) = sum(i=1, nb, sigma(m+i-1)) == sigma(m+nb);
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Oct 28 2021

cp's OEIS Frontend

A092405 a(n) = tau(n) + tau(n+1), where tau(n) = A000005(n), the number of divisors of n.

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A340584 Irregular triangle read by rows T(n,k) in which row n lists sigma(n) + sigma(n-1) together with the first n - 2 terms of A000203 in reverse order, with T(1,1) = 1, n >= 1.

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A346533 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.

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A227306 Numbers k that divide sigma(k) + sigma(k-1).

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A067282 Numbers k such that phi(k) + phi(k+1) divides sigma(k) + sigma(k+1).

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A252922 a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3), with a(1)=0, a(2)=1, a(3)=4.

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A348335 a(n) = smallest k such that the sum of the divisors of the n numbers from k to k+n-1 equals sigma(k+n), or -1 if no such k exists.

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