A072473 -id:A072473 - cp's OEIS frontend

A255171 First differences of A072473.

3, 4, 4, 6, 6, 2, 8, 4, 4, 6, 4, 8, 4, 2, 12, 2, 10, 6, 6, 6, 6, 2, 18, -2, 6, 10, 8, 6, 6, -2, 14, 0, 18, 2, 8, 8, 4, 10, 6, 6, 10, 0, 12, 2, 14, 0, 0, 14, 18, 12, 6, 6, 6, 2, 6, 0, 20, 0, 8

Offset: 1

Zak Seidov, Feb 15 2015

sign

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A072473, A072715.

a(n) = A072473(n+1) - A072473(n).

A255172 Integers k such that A072473(k) = A072473(k+1) = A072473(k+2) = A072473(k+3).

Original entry on oeis.org

126, 15190, 15924, 17180, 23047, 24410, 33553, 44049, 51019, 54358, 74240, 77460, 102494, 103168, 128522, 154518, 156386, 186056, 232346, 244086, 250216, 285095, 291306, 320942, 447634, 465803, 477517, 478415, 508078, 518164, 518861, 526764, 587712, 589208

Offset: 1

Zak Seidov, Feb 15 2015

nonn

Corresponding values of A072473(k): 900, 188790, 199218, 216630, 297504, 316572, 446890, 599998, 702990, 752700, 1052898, 1101480, 1488444, 1498860.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A072473, A072715, A255171.

{ my(p=0, q=0, k=0, r=0); for(n=1, 1e6, p=nextprime(1+p); q=nextprime(1+nextprime(1+q)); my(t=q-p); if(t<>r, r=t; k=0); k++; if(k>=4, print1(n+1-k, ", "))) } \\ Andrew Howroyd, Nov 11 2018

Terms a(19) and beyond from Andrew Howroyd, Nov 11 2018

A114042 Numbers n such that A072473(n)=A072473(n+1).

Original entry on oeis.org

32, 42, 46, 47, 56, 58, 61, 66, 71, 74, 76, 97, 103, 114, 118, 126, 127, 128, 161, 177, 180, 186, 195, 205, 232, 233, 267, 271, 290, 321, 326, 329, 331, 347, 359, 368, 370, 372, 383, 416, 423, 432, 437, 456, 508, 518, 520, 593, 594, 607, 633, 666, 675, 709

Offset: 1

Zak Seidov, Feb 01 2006

nonn

A072473: a(n) = p(2n) - p(n), where p(k) is the k-th prime.

Usually A072473(n) < A072473(n+1), only rarely this is not the case. Cf. A072473 a(n)=p(2n)-p(n), where p(k) is the k-th prime, A115867 numbers n such that A072473(n)=A072473(n+1).

Cf. A072473, A115867.

s={};Do[If[Prime[2n]-Prime[n]==Prime[2(n+1)]-Prime[n+1],AppendTo[s,n]],{n,1000}];s
SequencePosition[Table[Prime[2n]-Prime[n],{n,800}],{x_,x_}][[;;,1]] (* Harvey P. Dale, Jun 19 2024 *)

Entry revised by Robert G. Wilson v, Mar 16 2006

A066066 a(n) = prime(2n) - 2prime(n).

Original entry on oeis.org

-1, 1, 3, 5, 7, 11, 9, 15, 15, 13, 17, 15, 19, 21, 19, 25, 21, 29, 29, 31, 35, 35, 33, 45, 35, 37, 45, 49, 53, 55, 39, 49, 43, 59, 51, 57, 59, 57, 63, 63, 63, 71, 61, 71, 69, 81, 69, 57, 67, 83, 91, 91, 95, 91, 87, 87, 81, 99, 93, 97, 107, 97, 87, 97, 107, 109, 95, 95, 93, 111, 115, 109, 105, 111, 105, 115, 109, 117, 127, 123, 115

Offset: 1

Reinhard Zumkeller, Dec 01 2001

sign

a(n) = A022457(n) for n > 1.
a(n) = A031215(n)-A100484(n) = A072473(n)-A000040(n); see A179740 for primes. - Reinhard Zumkeller, Jul 25 2010
Asymptotically, a(n) ~ log(4) n, with log(4) = 2 log 2 = 1.38629436111989... = A016627. - M. F. Hasler, Oct 19 2013

Harry J. Smith, Table of n, a(n) for n = 1..1000

Table[Prime[2n]-2Prime[n],{n,100}] (* Harvey P. Dale, Aug 21 2016 *)

{ for (n = 1, 1000, a=prime(2*n) - prime(n)*2; write("b066066.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 09 2009

A066066(n)=prime(2*n)-2*prime(n) \\ M. F. Hasler, Oct 19 2013

A072715 Record differences between n-th prime and 2n-th prime.

Original entry on oeis.org

1, 4, 8, 12, 18, 24, 26, 34, 38, 42, 48, 52, 60, 64, 66, 78, 80, 90, 96, 102, 108, 114, 116, 134, 138, 148, 156, 162, 168, 180, 198, 200, 208, 216, 220, 230, 236, 242, 252, 264, 266, 280, 294, 312, 324, 330, 336, 342, 344, 350, 370, 378, 390, 394, 408, 420, 426

Offset: 1

Benoit Cloitre, Aug 07 2002

nonn

N. J. A. Sloane, Transforms

RECORDS transform of A072473.

A213926 prime(n^2) - prime(n).

Original entry on oeis.org

0, 4, 18, 46, 86, 138, 210, 292, 396, 512, 630, 790, 968, 1150, 1380, 1566, 1820, 2082, 2370, 2670, 3010, 3382, 3720, 4122, 4540, 4950, 5416, 5900, 6372, 6884, 7446, 8030, 8600, 9202, 9782, 10476, 11164, 11886, 12576, 13326, 14148, 14920, 15686, 16554, 17412

Offset: 1

Vincenzo Librandi, Mar 06 2013

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

Cf. A000040, A001223, A031131, A031165 - A031172, A072473, A092504.

[NthPrime(n^2)-NthPrime(n): n in [1..40]];

A213926 := proc(n) ithprime(n^2)-ithprime(n) ; end proc: seq(A213926(n), n=1..40) ;

Mathematica
```
Table[Prime[n^2] - Prime[n], {n, 40}]
```

a(n)=prime(n^2)-prime(n) \\ Charles R Greathouse IV, Mar 21 2014

a(n) = A000040(n^2) - A000040(n).

A258934 Half the difference between the 2n-th prime and the n-th prime, starting from n=2.

Original entry on oeis.org

2, 4, 6, 9, 12, 13, 17, 19, 21, 24, 26, 30, 32, 33, 39, 40, 45, 48, 51, 54, 57, 58, 67, 66, 69, 74, 78, 81, 84, 83, 90, 90, 99, 100, 104, 108, 110, 115, 118, 121, 126, 126, 132, 133, 140, 140, 140, 147, 156, 162, 165, 168, 171, 172, 175, 175, 185, 185, 189

Offset: 2

Federico Provvedi, Jun 15 2015

The differences between odd prime numbers are always even, so a(n) is well defined for n>=2.

Seiichi Manyama, Table of n, a(n) for n = 2..10000

Cf. A072473, A072715, A087461.

[(NthPrime(2*n)-NthPrime(n))/2: n in [2..60]]; // Bruno Berselli, Jun 15 2015

Table[(Prime[2 k] - Prime[k])/2, {k, 2, 60}]

[(nth_prime(2*n)-nth_prime(n))/2 for n in (2..60)] # Bruno Berselli, Jun 15 2015

a(n) = ( prime(2*n) - prime(n) ) / 2.

a(n) = A072473(n)/2.

A141657 a(n) = (prime(2m)-prime(m))/6 where m is the n-th positive integer such that (prime(2m)-prime(m))/6 is prime.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 19, 23, 71, 71, 83, 83, 89, 101, 103, 127, 163, 167, 191, 193, 199, 239, 239, 251, 263, 277, 307, 307, 307, 317, 331, 337, 389, 443, 463, 487, 509, 509, 557, 659, 691, 787, 797, 863, 881, 883, 887, 887, 953, 971, 977, 991, 1019, 1063, 1069

Offset: 1

Juri-Stepan Gerasimov, Sep 18 2008

nonn

			For m=4,  (p(2*4)  - p(4))/6  =  (19 -  7)/6 =  2 = a(1).
For m=5,  (p(2*5)  - p(5))/6  =  (29 - 11)/6 =  3 = a(2).
For m=10, (p(2*10) - p(10))/6 =  (71 - 29)/6 =  7 = a(3).
For m=15, (p(2*15) - p(15))/6 = (113 - 47)/6 = 11 = a(4).
For m=16, (p(2*16) - p(16))/6 = (131 - 53)/6 = 13 = a(5), etc.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000040, A072473.

Select[Table[(Prime[2n]-Prime[n])/6,{n,1000}],PrimeQ[#]&] (* Harvey P. Dale, Jun 18 2023 *)

19, 23 inserted, one instance of 331 removed, 577 removed and extended by R. J. Mathar, Oct 04 2008
Definition clarified by Harvey P. Dale, Jun 18 2023
Definition clarified by David A. Corneth, Jun 18 2023

A142338 Nonprimes of the form (p(2*n)-p(n))/4, where p(n)=n-th prime.

Original entry on oeis.org

1, 6, 12, 15, 16, 20, 24, 27, 33, 39, 42, 45, 45, 50, 52, 54, 55, 63, 63, 66, 70, 70, 70, 78, 81, 84, 86, 102, 105, 108, 110, 115, 117, 117, 118, 121, 121, 132, 133, 138, 141, 146, 148, 150, 156, 158, 165, 168, 168, 171, 180, 180, 182, 198, 203, 205, 205, 210, 210

Offset: 1

Juri-Stepan Gerasimov, Sep 18 2008

nonn

Terms are in order of n. The sequence has repetitions and is not monotonic: e.g. a(71) = 249 and a(72) = 248. - Robert Israel, Nov 09 2020

			If n=2, then (p(2*2)-p(2))/4=(7-3)/4=1=a(1).
If n=6, then (p(2*6)-p(6))/4=(37-13)/4=6=a(2).
If n=11, then (p(2*11)-p(11))/4=(79-31)/4=12=a(3).
If n=13, then (p(2*13)-p(13))/4=(101-41)/4=15=a(4).
If n=14, then (p(2*14)-p(14))/4=(107-43)/4=16=a(5), etc.

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

Cf. A000040, A072473, A141468.

q:= 1: p:= 1: count:= 0: R:= NULL:
while count < 100 do
  q:= nextprime(q); p:= nextprime(nextprime(p));
  v:= (p-q)/4;
  if v::integer and not isprime(v) then count:= count+1; R:= R, v fi
od:
R; # Robert Israel, Nov 09 2020

59 and 87 removed by R. J. Mathar, Oct 04 2008

A255174 a(n) = prime(3n) - prime(2n).

Original entry on oeis.org

2, 6, 10, 18, 18, 24, 30, 36, 42, 42, 58, 62, 66, 74, 84, 92, 94, 100, 106, 108, 126, 124, 148, 136, 150, 158, 168, 170, 178, 182, 194, 192, 206, 220, 222, 234, 234, 236, 246, 250, 256, 268, 284, 286, 298, 308, 320, 324, 332, 322, 326, 342, 360, 360, 376, 384

Offset: 1

Zak Seidov, Feb 15 2015

nonn

The sequence is not monotonic since, for instance, a(22) = 124 < 126 = a(21).

Cf. A072473.

[NthPrime(3*n) - NthPrime(2*n): n in [1..80]]; // Vincenzo Librandi, Feb 16 2015

Mathematica
```
Table[Prime[3n]-Prime[2n], {n,100}]
```

More terms from Vincenzo Librandi, Feb 16 2015

cp's OEIS Frontend

A255171 First differences of A072473.

Views

Author

Keywords

Links

Crossrefs

Formula

A255172 Integers k such that A072473(k) = A072473(k+1) = A072473(k+2) = A072473(k+3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A114042 Numbers n such that A072473(n)=A072473(n+1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A066066 a(n) = prime(2*n) - 2*prime(n).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

PARI

A072715 Record differences between n-th prime and 2n-th prime.

Views

Author

Keywords

Links

Crossrefs

A213926 prime(n^2) - prime(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A258934 Half the difference between the 2n-th prime and the n-th prime, starting from n=2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A141657 a(n) = (prime(2*m)-prime(m))/6 where m is the n-th positive integer such that (prime(2*m)-prime(m))/6 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A142338 Nonprimes of the form (p(2*n)-p(n))/4, where p(n)=n-th prime.

Views

Author

A066066 a(n) = prime(2n) - 2prime(n).

A141657 a(n) = (prime(2m)-prime(m))/6 where m is the n-th positive integer such that (prime(2m)-prime(m))/6 is prime.

A255174 a(n) = prime(3n) - prime(2n).