A086718 -id:A086718 - cp's OEIS frontend

A054541 Sum of first n terms equals n-th prime.

2, 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6

Offset: 1

G. L. Honaker, Jr., Apr 09 2000

nonn

Except for first term, same as A001223.
First differences of A182986. - Omar E. Pol, Oct 31 2013
A075526 is 1 together with A001223. This is 2 together with A001223. A125266 is 3 together with A001223. - Omar E. Pol, Nov 01 2013
Convolved with A024916 gives A086718. - Omar E. Pol, Dec 23 2021

Partial sums give A000040.

Cf. A001223, A075526, A086718, A024916, A125266, A182986.

Join[{2},Differences[Prime[Range[100]]]] (* Paolo Xausa, Oct 25 2023 *)

a(n) = if (n==1, 2, prime(n) - prime(n-1)); \\ Michel Marcus, Oct 31 2013

More terms from James Sellers, Apr 11 2000

A191831 a(n) = Sum_{i+j=n, i,j >= 1} tau(i)*sigma(j), where tau() = A000005(), sigma() = A000203().

Original entry on oeis.org

0, 1, 5, 12, 24, 39, 60, 87, 113, 158, 189, 249, 286, 372, 402, 516, 545, 696, 709, 886, 912, 1125, 1110, 1401, 1348, 1674, 1654, 1992, 1906, 2390, 2226, 2735, 2648, 3141, 2926, 3705, 3346, 4069, 3898, 4604, 4223, 5282, 4707, 5757, 5426, 6326, 5754, 7269, 6324, 7669, 7230, 8468, 7556, 9456, 8240, 10018, 9320, 10748, 9621, 12246

Offset: 1

N. J. A. Sloane, Jun 17 2011

This is Andrews's D_{0,1}(n).
From Omar E. Pol, Dec 08 2021: (Start)
Zero together with the convolution of A000005 and A000203.
Zero together with the convolution of A341062 and A024916.
Zero together with the convolution of the nonzero terms of A006218 and A340793.
a(n) is also the volume of a symmetric polycube which belongs to the family of symmetric polycubes that represent the convolution of A000203 with any other integer sequence, n >= 1. (End)

George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130.

Cf. A000005, A000203 A006218, A024916, A086718, A237593, A340793, A341062.

with(numtheory); D01:=n->add(tau(j)*sigma(n-j),j=1..n-1);
[seq(D01(n),n=1..60)];

Table[Sum[DivisorSigma[0, j] DivisorSigma[1, n - j], {j, n - 1}], {n, 60}] (* Michael De Vlieger, Jan 01 2017 *)

a(n)=sum(i=1,n-1,numdiv(i)*sigma(n-i)) \\ Charles R Greathouse IV, Feb 19 2013

G.f.: (Sum_{k>=1} x^k/(1 - x^k))*(Sum_{k>=1} k*x^k/(1 - x^k)). - Ilya Gutkovskiy, Jan 01 2017

A272214 Square array read by antidiagonals upwards in which T(n,k) is the product of the n-th prime and the sum of the divisors of k, n >= 1, k >= 1.

Original entry on oeis.org

2, 3, 6, 5, 9, 8, 7, 15, 12, 14, 11, 21, 20, 21, 12, 13, 33, 28, 35, 18, 24, 17, 39, 44, 49, 30, 36, 16, 19, 51, 52, 77, 42, 60, 24, 30, 23, 57, 68, 91, 66, 84, 40, 45, 26, 29, 69, 76, 119, 78, 132, 56, 75, 39, 36, 31, 87, 92, 133, 102, 156, 88, 105, 65, 54, 24, 37, 93, 116, 161, 114, 204, 104, 165, 91, 90, 36, 56

Offset: 1

Omar E. Pol, Apr 28 2016

From Omar E. Pol, Dec 21 2021: (Start)
Also triangle read by rows: T(n,j) = A000040(n-j+1)*A000203(j), 1 <= j <= n.
For a visualization of T(n,j) first consider a tower (a polycube) in which the terraces are the symmetric representation of sigma(j), for j = 1 to n, starting from the top, and the heights of the terraces are the first n prime numbers respectively starting from the base. Then T(n,j) can be represented with a set of A237271(j) right prisms of height A000040(n-j+1) since T(n,j) is also the total number of cubes that are exactly below the parts of the symmetric representation of sigma(j) in the tower.
The sum of the n-th row of triangle is A086718(n) equaling the volume of the tower whose largest side of the base is n and its total height is the n-th prime.
The tower is an member of the family of the stepped pyramids described in A245092 and of the towers described in A221529. That is an infinite family of symmetric polycubes whose volumes represent the convolution of A000203 with any other integer sequence. (End)

			The corner of the square array begins:
   2,  6,   8,  14,  12,  24,  16,  30,  26,  36, ...
   3,  9,  12,  21,  18,  36,  24,  45,  39,  54, ...
   5, 15,  20,  35,  30,  60,  40,  75,  65,  90, ...
   7, 21,  28,  49,  42,  84,  56, 105,  91, 126, ...
  11, 33,  44,  77,  66, 132,  88, 165, 143, 198, ...
  13, 39,  52,  91,  78, 156, 104, 195, 169, 234, ...
  17, 51,  68, 119, 102, 204, 136, 255, 221, 306, ...
  19, 57,  76, 133, 114, 228, 152, 285, 247, 342, ...
  23, 69,  92, 161, 138, 276, 184, 345, 299, 414, ...
  29, 87, 116, 203, 174, 348, 232, 435, 377, 522, ...
  ...
From _Omar E. Pol_, Dec 21 2021: (Start)
Written as a triangle the sequence begins:
   2;
   3,  6;
   5,  9,  8;
   7, 15, 12,  14;
  11, 21, 20,  21,  12;
  13, 33, 28,  35,  18,  24;
  17, 39, 44,  49,  30,  36, 16;
  19, 51, 52,  77,  42,  60, 24,  30;
  23, 57, 68,  91,  66,  84, 40,  45, 26;
  29, 69, 76, 119,  78, 132, 56,  75, 39, 36;
  31, 87, 92, 133, 102, 156, 88, 105, 65, 54, 24;
...
Row sums give A086718. (End)

Ivan Neretin, Table of n, a(n) for n = 1..8128

Rows 1-4 of the square array: A074400, A272027, A274535, A319527.
Columns 1-5 of the square array: A000040, A001748, A001749, A138636, A272470.
Main diagonal of the square array gives A272211.
Cf. A086718 (antidiagonal sums of the square array, row sums of the triangle).
Cf. A000203, A221529, A237270, A237271, A237593, A245092, A272173, A272400, A274824.

Table[Prime[#] DivisorSigma[1, k] &@(n - k + 1), {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Apr 28 2016 *)

T(n,k) = prime(n)*sigma(k) = A000040(n)*A000203(k), n >= 1, k >= 1.

T(n,k) = A272400(n+1,k).

cp's OEIS Frontend

A054541 Sum of first n terms equals n-th prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A191831 a(n) = Sum_{i+j=n, i,j >= 1} tau(i)*sigma(j), where tau() = A000005(), sigma() = A000203().

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A272214 Square array read by antidiagonals upwards in which T(n,k) is the product of the n-th prime and the sum of the divisors of k, n >= 1, k >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula