A052187 -id:A052187 - cp's OEIS frontend

A116497 Indices n such that the difference between the n-th prime and the next larger prime is 200.

20226285, 23169912, 25441017, 26172843, 27841352, 29657240, 30714253, 31998495, 33419255, 36637876, 37995065, 38182448, 38758272, 39019864, 40022755, 40296600, 40339473, 40985817, 41293073, 42116899, 45474429

Offset: 1

Zak Seidov, Feb 18 2006

nonn

			p(20226286) - p(20226285) = 378044179 - 378043979 = 200.

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A116493, A116495, A116496.
The actual primes are given in A204807. - M. F. Hasler, Jan 19 2012
See also A052187 and references therein.

lista(nn) = {my(i=0, q=2); forprime(p=3, nn, i++; if(p-q==200, print1(i, ", ")); q=p); } \\ Jinyuan Wang, Jan 29 2020

A204807(n) = prime(a(n)). - M. F. Hasler, Jan 19 2012

A054342 First occurrence of distances of equidistant lonely primes. Each equidistant prime is at the same distance (or has the same gap) from the preceding prime and the next prime.

Original entry on oeis.org

5, 53, 211, 20201, 16787, 69623, 255803, 247141, 3565979, 6314447, 4911311, 12012743, 23346809, 43607429, 34346287, 36598607, 51042053, 460475569, 652576429, 742585297, 530324449, 807620777, 2988119339, 12447231899, 383204683, 4470608101, 5007182863, 36589015601

Offset: 1

Harvey P. Dale, May 06 2000

nonn

Or, least balanced primes: starting with 2nd term, 53, the smallest prime such that the distances to the next smallest and next largest primes are both equal to 6n.
The distances corresponding to the above terms are 2, 6, 12, 18, 24, ..., 192, 198, 204, 210, 218, 224.
a(1) is the smallest prime p such that {p-2, p, p+2} are three consecutive primes. For n>1, a(n) is the smallest prime p such that {p-6*(n-1), p, p+6*(n-1)} are three consecutive primes. - Jeppe Stig Nielsen, Apr 16 2022

			211 is an equidistant lonely prime with distance 12. This is the first occurrence of the distance 12, thus 211 is in the sequence.
20201 is a least balanced prime because it is the third term in the sequence and is separated from both the next lower and next higher primes by 3 * 6 = 18.
Here is the beginning of the table of equidistant lonely primes.
Equivalent to 3 consecutive primes in arithmetic progression.
* indicates a maximal gap. This table gives rise to A058867, A058868 and the present sequence.
  Gap  First occurrence
  ---  ----------------
    2*         5
    6*        53
   12*       211
   18      20201
   24*     16787
   30*     69623
   36     255803
   42*    247141
   48*   3565979
   54    6314447
   60*   4911311
   66*  12012743
   72*  23346809
   78   43607429
   84*  34346287
   90*  36598607
   96*  51042053
  102  460475569
  108  652576429

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..53 (based on A052187 b-file)

Cf. A006562, A052187, A058867, A058868, A103709.

a(1) = A052187(1) + 2. For n>1, a(n) = A052187(n) + 6*(n-1). - Jeppe Stig Nielsen, Apr 16 2022

More terms from Jud McCranie, Jun 13 2000
Further terms from Harvey Dubner (harvey(AT)dubner.com), Sep 11 2004
Entry revised by N. J. A. Sloane, Jul 23 2006
4 further terms from Walter Neumann (neumann(AT)math.columbia.edu), Aug 14 2006
a(28) corrected, and terms after a(28) moved from Data section to b-file by Jeppe Stig Nielsen, Apr 16 2022

A124596 Primes p such that q-p = 30, where q is the next prime after p.

Original entry on oeis.org

4297, 4831, 5351, 5749, 6491, 6917, 7253, 7759, 7963, 8389, 8893, 13063, 13187, 13933, 13967, 14251, 14983, 16381, 16573, 17627, 18553, 18869, 20563, 21283, 21347, 21617, 23633, 23689, 24251, 25189, 26053, 26597, 27299, 27367, 27551, 28319, 28979, 29537

Offset: 1

N. J. A. Sloane, Dec 19 2006

nonn

These are the primes p = prime(k) where A001223(k) = 30. - N. J. A. Sloane, Dec 25 2019

Primes p such that r-q=q-p=30, where p, q, r are three successive primes, are A052195 = (69593, 110651, 134609, 228647, 237791, 250889, 303157, 24907,...). Primes p such that s-r=r-q=q-p=30, where p, q, r, s are four successive primes, are A052243 = (642427, 1058861, 3431903, 4176587, 4560121, 4721047, 5072269, 5145403, ...). - Zak Seidov, Dec 20 2006, edited by M. F. Hasler, Apr 04 2013

Remi Eismann, Table of n, a(n) for n = 1..10000
K. Soundararajan, Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18. DOI: 10.1090/S0273-0979-06-01142-6
K. Soundararajan, Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim, arXiv:math/0605696 [math.NT], 2006.
Index entries for primes, gaps between

Cf. A001223, A052195, A052243.

See also A052187 and references therein.

max = 30000;
Reap[For[p = 2; q = 3, p < max, p = q, q = NextPrime[p]; If[q - p == 30, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Sep 02 2018 *)
Select[Partition[Prime[Range[5000]],2,1],#[[2]]-#[[1]]==30&][[All,1]] (* Harvey P. Dale, Dec 25 2019 *)

A227888 Smallest odd k such that k2^n-1, k2^n-1+2j, k2^n-1+4j or k2^n-1-2j, k2^n-1, k2^n-1+2j are consecutive primes in arithmetic progression for some j.

Original entry on oeis.org

3, 1, 19, 3, 19, 273, 93, 113, 87, 35, 31, 143, 31, 15, 315, 779, 207, 347, 91, 327, 291, 351, 195, 39, 1911, 971, 1083, 435, 1345, 593, 183, 1295, 291, 2553, 735, 1113, 31, 131, 61, 209, 379, 567, 2331, 1907, 4429, 23, 453, 1517, 2281, 2187, 1441, 4847, 1975

Offset: 1

Pierre CAMI, Oct 26 2013

nonn

			3*2^1-1-2=3 3*2^1-1=5 3*2^1-1+2=7 so a(1)=3.
3*2^4-1=47 3*2^4-1+6=53 3*2^4-1+12=59 so a(4)=3.

Pierre CAMI, Table of n, a(n) for n = 1..325

Cf. A052187.

A224325 First of three consecutive primes in arithmetic progression with gap of 6n, and such that a(n) > a(n-1).

Original entry on oeis.org

47, 199, 20183, 40039, 69593, 255767, 689467, 3565931, 6314393, 9113263, 12012677, 23346737, 43607351, 69266033, 75138781, 324237847, 460475467, 652576321, 742585183, 747570079, 807620651, 2988119207, 12447231761

Offset: 1

M. F. Hasler, Apr 03 2013

nonn

Without the condition on monotonicity, this would be essentially the same as A052187, but there 255767 is followed by 247099, while monotonicity here gives 689467. Similarly, following a(9) = A052187(10) = 6314393 we have a(10) = 9113263, while A052187(11) = 4911251. The next term which is not matching is a(14) = 69266033 vs A052187(15) = 34346203. One may notice that the two terms differ approximately by a factor of 2.

			a(1) = A047948(1) = 47 is the least prime p(k) such that p(k+1) - p(k) = p(k+2) - p(k+1) = 6.
a(2) = A052188(1) = 199 is the least prime p(k) > 47 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 12.
a(3) = A052189(1) = 20183 is the least prime p(k) > 199 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 18.
a(4) = A052190(1) = 40039 is the least prime p(k) > 20183 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 24.
a(5) = A052195(1) = 69593 is the least prime p(k) > 40039 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 30.

Pierre CAMI, Table of n, a(n) for n = 1..23

Cf. A224324 (gaps of 30n).

g=6;o=2;forprime(p=2,,o+g==(o=p)||next;nextprime(p+1)==p+g||next;print1(p-g",");g+=6)

A224324 First of three consecutive primes with a gap of 30n.

Original entry on oeis.org

69593, 4911251, 36598517, 530324329, 4470607951, 65997364441, 264860525297, 1521870803867, 4934972897089, 6537587646371, 65110291176851, 130520768860387, 26923643849563, 187891466722493, 4096358893763537

Offset: 1

M. F. Hasler, Apr 03 2013

Terms a(5) through a(7) obtained from A052187, computed by Donovan Johnson.

Cf. A052187 and the monotonic variant A224325.

g=30;o=2;forprime(p=2,,o+g==(o=p)||next;nextprime(p+1)==p+g||next;print1(p-g",");g+=30)

a(n) = A052187(5n+1).

a(8)-a(10) added by Paul Zimmermann, Mar 15 2018, using the values from A052187

a(11)-a(15) from Jerry M Lagrou, Jan 01 2024

cp's OEIS Frontend

A116497 Indices n such that the difference between the n-th prime and the next larger prime is 200.

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A054342 First occurrence of distances of equidistant lonely primes. Each equidistant prime is at the same distance (or has the same gap) from the preceding prime and the next prime.

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A124596 Primes p such that q-p = 30, where q is the next prime after p.

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A227888 Smallest odd k such that k*2^n-1, k*2^n-1+2*j, k*2^n-1+4*j or k*2^n-1-2*j, k*2^n-1, k*2^n-1+2*j are consecutive primes in arithmetic progression for some j.

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A224325 First of three consecutive primes in arithmetic progression with gap of 6n, and such that a(n) > a(n-1).

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A224324 First of three consecutive primes with a gap of 30n.

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PARI

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Extensions

A227888 Smallest odd k such that k2^n-1, k2^n-1+2j, k2^n-1+4j or k2^n-1-2j, k2^n-1, k2^n-1+2j are consecutive primes in arithmetic progression for some j.