A196421 -id:A196421 - cp's OEIS frontend

A077320 Triangle in which n-th row contains n smallest multiples of the n-th prime.

2, 3, 6, 5, 10, 15, 7, 14, 21, 28, 11, 22, 33, 44, 55, 13, 26, 39, 52, 65, 78, 17, 34, 51, 68, 85, 102, 119, 19, 38, 57, 76, 95, 114, 133, 152, 23, 46, 69, 92, 115, 138, 161, 184, 207, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290

Offset: 1

Amarnath Murthy, Nov 04 2002

A000040 (primes) gives initial terms of rows.
A033286 contains the final terms of rows.
Sum of the n-th row = prime(n)*A000217(n), by definition.
a(A000217(n) + 1) = prime(n+1), by definition.

			From _Bruno Berselli_, Sep 05 2017: (Start)
Triangle begins:
   2;
   3, 6;
   5, 10,  15;
   7, 14,  21,  28;
  11, 22,  33,  44,  55;
  13, 26,  39,  52,  65,  78;
  17, 34,  51,  68,  85, 102, 119;
  19, 38,  57,  76,  95, 114, 133, 152;
  23, 46,  69,  92, 115, 138, 161, 184, 207;
  29, 58,  87, 116, 145, 174, 203, 232, 261, 290;
  31, 62,  93, 124, 155, 186, 217, 248, 279, 310, 341;
  37, 74, 111, 148, 185, 222, 259, 296, 333, 370, 407, 444;
  41, 82, 123, 164, 205, 246, 287, 328, 369, 410, 451, 492, 533;
  43, 86, 129, 172, 215, 258, 301, 344, 387, 430, 473, 516, 559, 602, etc.
(End)

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

Cf. A000040, A000217, A033286.

Row sums give A196421. - Omar E. Pol, Mar 12 2012

Table[Prime[n]*Range[n], {n, 10}] // Flatten (* Ivan Neretin, May 02 2019 *)

T(n,k) = k*prime(n) with 1 <= k <= n. - Bruno Berselli, Sep 05 2017

A085783 Product of a prime and a triangular number.

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 20, 21, 23, 29, 30, 31, 33, 37, 39, 41, 42, 43, 45, 47, 50, 51, 53, 56, 57, 59, 61, 63, 66, 67, 69, 70, 71, 72, 73, 75, 78, 79, 83, 84, 87, 89, 90, 93, 97, 101, 102, 103, 105, 107, 108, 109, 110, 111

Offset: 1

Jon Perry, Jul 23 2003

nonn

			15 = 5*t(2) is a term.

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

Cf. A000040, A000217.

A196421 is a subsequence. - Michel Marcus, May 15 2018

N:= 200: # for all terms <= N
T:= [seq(i*(i+1)/2,i=1..floor((sqrt(4*N+1)-1)/2))]:
P:= select(isprime,[2,seq(i,i=3..N,2)]):
sort(select(`<=`,convert({seq(seq(p*t,p=P),t=T)},list),N)); # Robert Israel, May 15 2018

107 and 109 inserted by Robert Israel, May 15 2018

A196303 Primes p such that (p-1)*2^p+1 is also prime.

Original entry on oeis.org

2, 3, 7, 1471, 1483, 61627, 88651

Offset: 1

Juri-Stepan Gerasimov, Oct 02 2011

nonn

			a(1) = 2 because 2 and (2-1)*2^2 + 1 = 5 are both prime.
a(2) = 3 because 3 and (3-1)*2^3 + 1 = 17 are both prime.
a(3) = 7 because 7 and (7-1)*2^7 + 1 = 769 are both prime.

Prime terms of A128001.

Cf. A029544, A196421, A196446.

Select[Prime[Range[9000]],PrimeQ[(#-1)2^#+1]&] (* Harvey P. Dale, Jan 19 2012 *)

forprime(n=1,1e4,if(ispseudoprime((n-1)<Charles R Greathouse IV, Oct 09 2011

a(7) corrected by Michael S. Branicky, May 14 2025

A378493 Dot product of the first n primes and the first n triangular numbers.

Original entry on oeis.org

2, 11, 41, 111, 276, 549, 1025, 1709, 2744, 4339, 6385, 9271, 13002, 17517, 23157, 30365, 39392, 49823, 62553, 77463, 94326, 114313, 137221, 163921, 195446, 230897, 269831, 313273, 360688, 413233, 476225, 545393, 622250, 704955, 798825, 899391, 1009762

Offset: 1

Harvey P. Dale, Nov 28 2024

nonn

			a(3) = dot product of {2,3,5} and {1,3,6} = 2*1+3*3+5*6 = 41.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Partial sums of A196421.

Cf. A000040, A000217, A166486.

a:= proc(n) option remember; `if`(n<1, 0,
      a(n-1)+ithprime(n)*n*(n+1)/2)
    end:
seq(a(n), n=1..37);  # Alois P. Heinz, Nov 28 2024

Table[Prime[Range[n]].Accumulate[Range[n]],{n,50}]

a(n) mod 2 = A166486(n-1).

A356868 a(n) = n^2 * prime(n).

Original entry on oeis.org

2, 12, 45, 112, 275, 468, 833, 1216, 1863, 2900, 3751, 5328, 6929, 8428, 10575, 13568, 17051, 19764, 24187, 28400, 32193, 38236, 43907, 51264, 60625, 68276, 75087, 83888, 91669, 101700, 122047, 134144, 149193, 160684, 182525, 195696, 214933, 235372, 254007, 276800, 300899

Offset: 1

Alex Ratushnyak, Sep 01 2022

Cf. A000040, A000290, A004232, A033286, A196421.

a[n_] := n^2 * Prime[n]; Array[a, 40] (* Amiram Eldar, Sep 02 2022 *)

from sympy import prime
def a(n): return n**2 * prime(n)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Sep 01 2022

cp's OEIS Frontend

A077320 Triangle in which n-th row contains n smallest multiples of the n-th prime.

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A085783 Product of a prime and a triangular number.

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A196303 Primes p such that (p-1)*2^p+1 is also prime.

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Extensions

A378493 Dot product of the first n primes and the first n triangular numbers.

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Formula

A356868 a(n) = n^2 * prime(n).

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