A116366 -id:A116366 - cp's OEIS frontend

A116367 Sums of rows of the triangle in A116366.

6, 18, 36, 70, 104, 158, 208, 282, 388, 468, 602, 728, 838, 984, 1174, 1382, 1536, 1772, 1986, 2170, 2448, 2698, 3008, 3386, 3684, 3940, 4258, 4530, 4868, 5528, 5910, 6370, 6712, 7340, 7710, 8234, 8776, 9258, 9832, 10424, 10866, 11658, 12128, 12694, 13180

Offset: 1

Reinhard Zumkeller, Feb 06 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

List([1..50], n-> Sum([1..n], k-> Primes[n+1] + Primes[k+1])); # G. C. Greubel, May 18 2019

[(&+[NthPrime(n+1) + NthPrime(k+1): k in [1..n]]): n in [1..50]]; // G. C. Greubel, May 18 2019

Table[Sum[Prime[n+1] + Prime[k+1], {k,1,n}], {n,1,50}] (* G. C. Greubel, May 18 2019 *)

vector(50, n, sum(k=1,n, prime(n+1) + prime(k+1))) \\ G. C. Greubel, May 18 2019

[sum(nth_prime(n+1) + nth_prime(k+1) for k in (1..n)) for n in (1..50)] # G. C. Greubel, May 18 2019

a(n) = Sum_{k=1..n} A116366(n,k).

a(n) = (n+1)*A000040(n+1) + A007504(n) - 2.

A116368 Central terms of the triangle in A116366.

Original entry on oeis.org

6, 12, 20, 30, 42, 54, 62, 76, 90, 102, 116, 130, 144, 154, 166, 190, 200, 218, 234, 246, 260, 276, 288, 320, 330, 342, 358, 372, 384, 408, 424, 448, 456, 486, 500, 516, 536, 550, 570, 588, 602, 624, 636, 654, 662, 690, 714, 730, 750, 774, 796, 810, 828, 850, 864, 882, 890, 918, 928

Offset: 1

Reinhard Zumkeller, Feb 06 2006

nonn

			For n=4, prime(2n) = prime(8) = 19, and prime(n+1) = prime(5) = 11, so a(4) = 19 + 11 = 30. - _Michael B. Porter_, Aug 15 2016

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A116366.

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

[NthPrime(2*n) + NthPrime(n+1): n in [1..70]]; // G. C. Greubel, May 18 2019

Table[Prime[2n] + Prime[n+1], {n, 1, 70}](* Terry D. Grant, Aug 15 2016 *)

vector(70, n, prime(2*n) + prime(n+1)) \\ G. C. Greubel, May 18 2019

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

a(n) = A000040(2n) + A000040(n+1).

a(n) = A116366(2*n-1, n).

A100484 The primes doubled; Even semiprimes.

Original entry on oeis.org

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.

A001043 Numbers that are the sum of 2 successive primes.

Original entry on oeis.org

5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, 138, 144, 152, 162, 172, 186, 198, 204, 210, 216, 222, 240, 258, 268, 276, 288, 300, 308, 320, 330, 340, 352, 360, 372, 384, 390, 396, 410, 434, 450, 456, 462, 472, 480, 492, 508, 520

Offset: 1

N. J. A. Sloane, R. K. Guy

Arithmetic derivative (see A003415) of prime(n)*prime(n+1). - Giorgio Balzarotti, May 26 2011
A008472(a(n)) = A191583(n). - Reinhard Zumkeller, Jun 28 2011
With the exception of the first term, all terms are even. a(n) is divisible by 4 if the difference between prime(n) and prime(n + 1) is not divisible by 4; e.g., prime(n) = 1 mod 4 and prime(n + 1) = 3 mod 4. In general, for a(n) to be divisible by some even number m > 2 requires that prime(n + 1) - prime(n) not be a multiple of m. - Alonso del Arte, Jan 30 2012

			2 + 3 = 5.
3 + 5 = 8.
5 + 7 = 12.
7 + 11 = 18.

Archimedeans Problems Drive, Eureka, 26 (1963), 12.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Albert Frank & Philippe Jacqueroux, International Contest, 2001. Item 22
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968
N. J. A. Sloane and Brady Haran, Eureka Sequences, Numberphile video (2021)

Subsequence of A050936.

Cf. A000040 (primes), A031131 (first differences), A092163 (bisection), A100479 (bisection).

a001043 n = a001043_list !! (n-1)
a001043_list = zipWith (+) a000040_list $ tail a000040_list
-- Reinhard Zumkeller, Oct 19 2011

[(NthPrime(n+1) + NthPrime(n)): n in [1..100]]; // Vincenzo Librandi, Apr 02 2011

Primes:= select(isprime,[2,seq(2*i+1,i=1..1000)]):
n:= nops(Primes):
Primes[1..n-1] + Primes[2..n]; # Robert Israel, Aug 29 2014

Table[Prime[n] + Prime[n + 1], {n, 55}] (* Ray Chandler, Feb 12 2005 *)
Total/@Partition[Prime[Range[60]], 2, 1] (* Harvey P. Dale, Aug 23 2011 *)
Abs[Differences[Table[(-1)^n Prime[n], {n, 60}]]] (* Alonso del Arte, Feb 03 2016 *)

p=2;forprime(q=3,1e3,print1(p+q", ");p=q) \\ Charles R Greathouse IV, Jun 10 2011

is(n)=precprime((n-1)/2)+nextprime(n/2)==n&&n>2 \\ Charles R Greathouse IV, Jun 21 2012

BB = primes_first_n(56)
L = []
for i in range(55): L.append(BB[1 + i] + BB[i])
L # Zerinvary Lajos, May 14 2007

a(n) = prime(n) + prime(n + 1) = A000040(n) + A000040(n+1).
a(n) = A116366(n, n - 1) for n > 1. - Reinhard Zumkeller, Feb 06 2006
a(n) = 2*A024675(n-1), n>1. - R. J. Mathar, Jan 12 2024

More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000

A001747 2 together with primes multiplied by 2.

Original entry on oeis.org

2, 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

Offset: 1

N. J. A. Sloane

When supplemented with 8, may be considered the "even primes", since these are the even numbers n = 2k which are divisible just by 1, 2, k and 2k. - Louis Zuckerman (louis(AT)trapezoid.com), Sep 12 2000
Sequence gives solutions of sigma(n) - phi(n) = n + tau(n) where tau(n) is the number of divisors of n.
Numbers n such that sigma(n) = 3*(n - phi(n)).
Except for 2, orders of non-cyclic groups k (in A060679(n)) such that x^k==1 (mod k) has only 1 solution 2<=x<=k. - Benoit Cloitre, May 10 2002
Numbers n such that A092673(n) = 2. - Jon Perry, Mar 02 2004
Except for initial terms, this sequence = A073582 = A074845 = A077017. Starting with the term 10, they are identical. - Robert G. Wilson v, Jun 15 2004
Together with 8 and 16, even numbers n such that n^2 does not divide (n/2)!. - Arkadiusz Wesolowski, Jul 16 2011
Twice noncomposite numbers. - Omar E. Pol, Jan 30 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A060679, A009530, A098764.

Equals {2} UNION {A100484}.

Concatenation([2], List([1..60], n-> 2*Primes[n])); # G. C. Greubel, May 18 2019

[2] cat [2*NthPrime(n): n in [1..60]]; // G. C. Greubel, May 18 2019

Join[{2},2*Prime[Range[60]]] (* Harvey P. Dale, Jul 23 2013 *)

print1(2);forprime(p=2,97,print1(", "2*p)) \\ Charles R Greathouse IV, Jan 31 2012

[2]+[2*nth_prime(n) for n in (1..60)] # G. C. Greubel, May 18 2019

a(n) = A001043(n) - A001223(n+1), except for initial term.
a(n) = A116366(n-2,n-2) for n>2. - Reinhard Zumkeller, Feb 06 2006
A006093(n) = A143201(a(n+1)) for n>1. - Reinhard Zumkeller, Aug 12 2008
a(n) = 2*A008578(n). - Omar E. Pol, Jan 30 2012, and Reinhard Zumkeller, Feb 16 2012

A048448 a(n) = prime(n-1) + prime(n+1) (assuming prime(i) = 0 for i < 1).

Original entry on oeis.org

2, 3, 7, 10, 16, 20, 28, 32, 40, 48, 54, 66, 72, 80, 88, 96, 106, 114, 126, 132, 140, 150, 156, 168, 180, 190, 200, 208, 212, 220, 236, 244, 264, 270, 286, 290, 306, 314, 324, 336, 346, 354, 370, 374, 388, 392, 408, 422, 438, 452, 460, 468, 474, 490, 498, 514

Offset: 0

Patrick De Geest, May 15 1999

nonn

Starting from prime sequence add previous and next term yielding generation 2.
a(n) = A116366(n,n-2) for n>2. - Reinhard Zumkeller, Feb 06 2006
Arithmetic derivative (see A003415) of prime(n-1)*prime(n+1) for n > 1. - Giorgio Balzarotti, May 26 2011

G. C. Greubel, Table of n, a(n) for n = 0..10000

Generation 1 is the 'prime sequence A000040'. See A048449-A048466. See also A047844.

Concatenation([2,3], List([2..60], n-> Primes[n-1] + Primes[n+1])); # G. C. Greubel, May 18 2019

[2,3] cat [NthPrime(n-1) + NthPrime(n+1): n in [2..60]];  // G. C. Greubel, May 18 2019

Table[If[n < 2, Prime[n+1], Prime[n+1] + Prime[n-1]], {n, 0, 60}]
Join[{2,3},First[#]+Last[#]&/@Partition[Prime[Range[60]],3,1]] (* Harvey P. Dale, Jan 25 2016 *)

ithprime(i)+ithprime(i+2) $ i = 1..54 // Zerinvary Lajos, Feb 26 2007

je=[2,3]; for(n=1,60,je=concat(je, prime(n)+prime(n+2))); je \\ modified by G. C. Greubel, May 18 2019

[2,3] + [nth_prime(n-1) + nth_prime(n+1) for n in (2..60)] # G. C. Greubel, May 18 2019

A113935 a(n) = prime(n) + 3.

Original entry on oeis.org

5, 6, 8, 10, 14, 16, 20, 22, 26, 32, 34, 40, 44, 46, 50, 56, 62, 64, 70, 74, 76, 82, 86, 92, 100, 104, 106, 110, 112, 116, 130, 134, 140, 142, 152, 154, 160, 166, 170, 176, 182, 184, 194, 196, 200, 202, 214, 226, 230, 232, 236, 242, 244, 254, 260, 266, 272, 274

Offset: 1

Jorge Coveiro, Jan 30 2006

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

Cf. A098090, A116366.

List([1..70], n-> Primes[n]+3); # G. C. Greubel, May 18 2019

[p+3: p in PrimesUpTo(500)]; // Vincenzo Librandi, Nov 27 2013

Prime[Range[70]]+3 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

PARI
```
for(x=1,100,print1(prime(x)+3,","))
```

[nth_prime(n)+3 for n in (1..70)] # G. C. Greubel, May 18 2019

a(n) = A116366(n-1,1) for n>1. - Reinhard Zumkeller, Feb 06 2006
a(n) = 2*A098090(n-1) for n > 1. - Reinhard Zumkeller, Sep 14 2006
a(n) = A000040(n) + 3 = A008864(n) + 2 = A052147(n) + 1 = A175221(n) - 1 = A175222(n) - 2 = A139049(n) - 3 = A175223(n) - 4 = A175224(n) - 5 = A140353(n) - 6 = A175225(n) - 7. - Jaroslav Krizek, Mar 06 2010

cp's OEIS Frontend

A116367 Sums of rows of the triangle in A116366.

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A116368 Central terms of the triangle in A116366.

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A100484 The primes doubled; Even semiprimes.

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A001043 Numbers that are the sum of 2 successive primes.

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A001747 2 together with primes multiplied by 2.

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A048448 a(n) = prime(n-1) + prime(n+1) (assuming prime(i) = 0 for i < 1).

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