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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A152117 a(n) = n*(n-th prime) + (n+1)*((n+1)-th prime).

Original entry on oeis.org

8, 21, 43, 83, 133, 197, 271, 359, 497, 631, 785, 977, 1135, 1307, 1553, 1851, 2101, 2371, 2693, 2953, 3271, 3647, 4045, 4561, 5051, 5407, 5777, 6157, 6551, 7327, 8129, 8713, 9247, 9941, 10651, 11245, 12003, 12707, 13433, 14259, 14941, 15815, 16705
Offset: 1

Views

Author

Klaus Brockhaus, Dec 10 2008

Keywords

Comments

a(n) = A033286(n) + A033286(n+1).

Examples

			5*(fifth prime) + 6*(sixth prime) = 5*11 + 6*13 = 55 + 78 = 133.

Links

Crossrefs

Cf. A000040 (prime numbers), A033286 (n*(n-th prime)), A033287 (first differences of A033286), A119487 (primes in this sequence).

Programs

  • Magma
    [ n*NthPrime(n)+(n+1)*NthPrime(n+1): n in [1..43] ];
  • Mathematica
    Total/@Partition[Times@@@Table[{n,Prime[n]},{n,50}],2,1] (* Harvey P. Dale, Aug 13 2019 *)
  • PARI
    a(n) = n*prime(n) + (n+1)*prime(n+1); \\ Michel Marcus, Feb 05 2016

A090942 n-th arithmetic mean = prime(n).

Original entry on oeis.org

2, 4, 9, 13, 27, 23, 41, 33, 55, 83, 51, 103, 89, 69, 103, 143, 155, 95, 175, 147, 113, 205, 171, 227, 289, 201, 155, 215, 165, 229, 547, 255, 329, 205, 489, 221, 373, 385, 319, 407, 419, 263, 611, 279, 373, 289, 763, 787, 419, 327, 433, 545, 345, 781, 581, 593
Offset: 1

Views

Author

Amarnath Murthy, Dec 29 2003

Keywords

Comments

Partial sums give A033286. - Omar E. Pol, Apr 20 2015
In other words, this is the unique sequence such that for all n >= 1, (1/n) * Sum_{k=1..n} a(k) = prime(n). - Antti Karttunen, Apr 30 2015

Examples

			From _Omar E. Pol_, Apr 20 2015: (Start)
Illustration of initial terms:
Consider a diagram in the first quadrant of the square grid in which the length of the n-th horizontal line segment is equal to the n-th prime, as shown below:
.   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  51|
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _          83|   |
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _      55|           |   |
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  33|       |           |   |
.  |_ _ _ _ _ _ _ _ _ _ _ _ _      41|   |       |           |   |
.  |_ _ _ _ _ _ _ _ _ _ _  23|       |   |       |           |   |
.  |_ _ _ _ _ _ _      27|   |       |   |       |           |   |
.  |_ _ _ _ _  13|       |   |       |   |       |           |   |
.  |_ _ _   9|   |       |   |       |   |       |           |   |
.  |_ _ 4|   |   |       |   |       |   |       |           |   |
.  |_2_|_|_ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _ _ _ _ _|_ _|
.
a(n) is also the area (or the number of cells) in the n-th region of the diagram. For example: a(4) = 7 + 6 = 13.
(End)

Links

Crossrefs

Cf. A000040, A007504, A033286, A033287, A141042, A152535.

Programs

  • Magma
    [n le 1 select 2 else n*NthPrime(n) - (n-1)*NthPrime(n-1): n in [1..60]]; // G. C. Greubel, Feb 04 2019
  • Mathematica
    Table[If[n==1, 2, nPrime@n -(n-1)Prime[n-1]], {n, 60}] (* Michael De Vlieger, Apr 20 2015 *)
  • PARI
    vector(60, n, if(n==1,2, n*prime(n) -(n-1)*prime(n-1))) \\ G. C. Greubel, Feb 04 2019
  • Sage
    [2] + [n*nth_prime(n) - (n-1)*nth_prime(n-1) for n in (2..60)] # G. C. Greubel, Feb 04 2019

Formula

a(n) = n*prime(n) - (n-1)*prime(n-1).
a(n) = A033287(n-2), n>1. - R. J. Mathar, Sep 08 2008
a(n) = A000040(n) + A141042(n-1), n >=2. - Omar E. Pol, Apr 20 2015

Extensions

Corrected and extended by Ray Chandler, Dec 31 2003
Showing 1-2 of 2 results.

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