A065342 -id:A065342 - cp's OEIS frontend

A100484 The primes doubled; Even semiprimes.

4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 1

Reinhard Zumkeller, Nov 22 2004

Essentially the same as A001747.
Right edge of the triangle in A065342. - Reinhard Zumkeller, Jan 30 2012
A253046(a(n)) > a(n). - Reinhard Zumkeller, Dec 26 2014
Apart from first term, these are the tau2-primes as defined in [Anderson, Frazier] and [Lanterman]. - Michel Marcus, May 15 2019
For every positive integer b and each m in this sequence b^(m-1) == b (mod m). - Florian Baur, Nov 26 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. D. Anderson and Andrea M. Frazier, On a general theory of factorization in integral domains, Rocky Mountain J. Math., Volume 41, Number 3 (2011), 663-705. See pp. 698, 699, 702.
James Lanterman, Irreducibles in the Integers modulo n, arXiv:1210.2991 [math.NT], 2012.
Eric Weisstein's World of Mathematics, Semiprime

Subsequence of A091376. After the initial 4 also a subsequence of A039956.
Cf. A046315, A152099, A179740.
Cf. A001748, A253046, A353478 (characteristic function).
Row 3 of A286625, column 3 of A286623.

2*Filtered([1..300],IsPrime); # Muniru A Asiru, Oct 05 2018

List([1..70], n-> 2*Primes[n]); # G. C. Greubel, May 18 2019

a100484 n = a100484_list !! (n-1)
a100484_list = map (* 2) a000040_list
-- Reinhard Zumkeller, Jan 31 2012

[2*p: p in PrimesUpTo(350)]; // Vincenzo Librandi, Mar 27 2014

A100484:=n->2*ithprime(n); seq(A100484(n), n=1..70); # Wesley Ivan Hurt, Mar 27 2014

2*Prime[Range[70]] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

2*primes(70) \\ Charles R Greathouse IV, Aug 21 2011

[2*nth_prime(n) for n in (1..70)] # G. C. Greubel, May 18 2019

a(n) = 2 * A000040(n).
a(n) = A001747(n+1).
n>1: A000005(a(n)) = 4; A000203(a(n)) = 3*A008864(n); A000010(a(n)) = A006093(n); intersection of A001358 and A005843.
a(n) = A116366(n-1, n-1) for n>1. - Reinhard Zumkeller, Feb 06 2006
a(n) = A077017(n+1), n>1. - R. J. Mathar, Sep 02 2008
A078834(a(n)) = A000040(n). - Reinhard Zumkeller, Sep 19 2011
a(n) = A087112(n, 1). - Reinhard Zumkeller, Nov 25 2012
A000203(a(n)) = 3*n/2 + 3, n > 1. - Wesley Ivan Hurt, Sep 07 2013

Simpler definition.

A052147 a(n) = prime(n) + 2.

Original entry on oeis.org

4, 5, 7, 9, 13, 15, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk), Jan 24 2000

A048974, A052147, A067187 and A088685 are very similar after dropping terms less than 13. - Eric W. Weisstein, Oct 10 2003
A117530(n,2) = a(n) for n>1. - Reinhard Zumkeller, Mar 26 2006
a(n) = A000040(n) + 2 = A008864(n) + 1 = A113395(n) - 1 = A175221(n) - 2 = A175222(n) - 3 = A139049(n) - 4 = A175223(n) - 5 = A175224(n) - 6 = A140353(n) - 7 = A175225(n) - 8. - Jaroslav Krizek, Mar 06 2010
Left edge of the triangle in A065342. - Reinhard Zumkeller, Jan 30 2012
Union of A006512 and A107986. - David James Sycamore, Jul 08 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

A139690 is a subsequence.

Cf. A040976, A000040.

Filtered([1..300], k-> IsPrime(k) ) + 2; # G. C. Greubel, May 20 2019

a052147 = (+ 2) . a000040  -- Reinhard Zumkeller, Jul 03 2015

[p+2: p in PrimesUpTo(400)]; // Vincenzo Librandi, Nov 27 2013

seq(ithprime(n)+2,n=1..55); # Muniru A Asiru, Jul 08 2018

Prime[Range[70]]+2 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

a(n)=prime(n)+2 \\ Charles R Greathouse IV, Jan 19 2017

[nth_prime(n) +2 for n in (1..70)] # G. C. Greubel, May 20 2019

A087112 Triangle in which the n-th row contains n distinct semiprimes not listed previously with all prime factors from among the first n primes.

Original entry on oeis.org

4, 6, 9, 10, 15, 25, 14, 21, 35, 49, 22, 33, 55, 77, 121, 26, 39, 65, 91, 143, 169, 34, 51, 85, 119, 187, 221, 289, 38, 57, 95, 133, 209, 247, 323, 361, 46, 69, 115, 161, 253, 299, 391, 437, 529, 58, 87, 145, 203, 319, 377, 493, 551, 667, 841, 62, 93, 155, 217, 341, 403, 527, 589, 713, 899, 961

Offset: 1

Ray Chandler, Aug 21 2003

Terms through row n, sorted, will provide terms for A077553 through row n*(n+1)/2.

			Triangle begins:
   4;
   6,   9;
  10,  15,  25;
  14,  21,  35,  49;
  22,  33,  55,  77, 121;
  26,  39,  65,  91, 143, 169;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A100484 (left edge), A001248 (right edge), A143215 (row sums), A219603 (central terms of odd-indexed rows); A000040, A065342.

Cf. A276086, A370121.

a087112 n k = a087112_tabl !! (n-1) !! (k-1)
a087112_row n = map (* last ps) ps where ps = take n a000040_list
a087112_tabl = map a087112_row [1..]
-- Reinhard Zumkeller, Nov 25 2012

T := (n, k) -> ithprime(n) * ithprime(k):
seq(print(seq(T(n, k), k = 1..n)), n = 1..11);  # Peter Luschny, Jun 25 2024

Table[ Prime[j]*Prime[k], {j, 11}, {k, j}] // Flatten (* Robert G. Wilson v, Feb 06 2017 *)

A087112(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2); (prime(1+c) * prime(1+(n-binomial(1+c, 2)))); }; \\ Antti Karttunen, Feb 29 2024

The n-th row consists of n terms, prime(n)*prime(i), i=1..n.
T(n, k) = A000040(n) * A000040(k).
For n >= 2, a(n) = A276086(A370121(n-1)). - Antti Karttunen, Feb 29 2024

A065305 Triangular array giving means of two odd primes: T(n,k) = (n-th prime + k-th prime)/2, n >= k >= 2.

Original entry on oeis.org

3, 4, 5, 5, 6, 7, 7, 8, 9, 11, 8, 9, 10, 12, 13, 10, 11, 12, 14, 15, 17, 11, 12, 13, 15, 16, 18, 19, 13, 14, 15, 17, 18, 20, 21, 23, 16, 17, 18, 20, 21, 23, 24, 26, 29, 17, 18, 19, 21, 22, 24, 25, 27, 30, 31, 20, 21, 22, 24, 25, 27, 28, 30, 33, 34, 37, 22, 23, 24, 26, 27, 29, 30

Offset: 2

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Oct 29 2001

			3; 4,5; 5,6,7; 7,8,9,11; ...

Reinhard Zumkeller, First 125 rows of triangle, flattened
Index entries for sequences related to Goldbach conjecture

Cf. A065306.
a(n, k) = A065342(n, k)/2 [but note different offsets]
Cf. A098090 (left edge), A065091 (right edge), A000040.

import Data.List (inits)
a065305 n k = a065305_tabl !! (n-2) !! (k - 1)
a065305_row n = a065305_tabl !! (n-2)
a065305_tabl = zipWith (map . (flip div 2 .) . (+))
                       a065091_list $ tail $ inits a065091_list
-- Reinhard Zumkeller, Aug 02 2015, Jan 30 2012

seq(seq((ithprime(i)+ithprime(j))/2,j=2..i),i=2..20)

A116366 Triangle read by rows: even numbers as sums of two odd primes.

Original entry on oeis.org

6, 8, 10, 10, 12, 14, 14, 16, 18, 22, 16, 18, 20, 24, 26, 20, 22, 24, 28, 30, 34, 22, 24, 26, 30, 32, 36, 38, 26, 28, 30, 34, 36, 40, 42, 46, 32, 34, 36, 40, 42, 46, 48, 52, 58, 34, 36, 38, 42, 44, 48, 50, 54, 60, 62, 40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74, 44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82

Offset: 1

Reinhard Zumkeller, Feb 06 2006

T(n,k) = 2*A065305(n,k) = A065342(n+1,k+1);
Row sums give A116367; central terms give A116368;
T(n,1) = A113935(n+1);
T(n,n-2) = A048448(n) for n>2;
T(n,n-1) = A001043(n) for n>1;
T(n,n) = A001747(n+2) = A100484(n+1).

			Triangle begins:
  6;
  8,  10;
  10, 12, 14;
  14, 16, 18, 22;
  16, 18, 20, 24, 26;
  20, 22, 24, 28, 30, 34;
  22, 24, 26, 30, 32, 36, 38;
  26, 28, 30, 34, 36, 40, 42, 46;
  32, 34, 36, 40, 42, 46, 48, 52, 58;
  34, 36, 38, 42, 44, 48, 50, 54, 60, 62;
  40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74;
  44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82; etc. - _Bruno Berselli_, Aug 16 2013

G. C. Greubel, Rows n = 1..100 of triangle, flattened
Index entries for sequences related to Goldbach conjecture

Cf. A001043, A001747, A048448, A065305, A065342, A100484, A113935, A116367, A116368.

[NthPrime(n+1)+NthPrime(k+1): k in [1..n], n in [1..15]]; // Bruno Berselli, Aug 16 2013

Table[Prime[n+1] + Prime[k+1], {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, May 12 2019 *)

{T(n,k) = prime(n+1) + prime(k+1)}; \\ G. C. Greubel, May 12 2019

[[nth_prime(n+1) + nth_prime(k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 12 2019

T(n,k) = prime(n+1) + prime(k+1), 1 <= k <= n.

cp's OEIS Frontend

A100484 The primes doubled; Even semiprimes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

GAP

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A052147 a(n) = prime(n) + 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

Sage

A087112 Triangle in which the n-th row contains n distinct semiprimes not listed previously with all prime factors from among the first n primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A065305 Triangular array giving means of two odd primes: T(n,k) = (n-th prime + k-th prime)/2, n >= k >= 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Maple

A116366 Triangle read by rows: even numbers as sums of two odd primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula