A000101 -id:A000101 - cp's OEIS frontend

A214756 a(n) = largest Ramanujan prime R_k in A104272 that is <= A002386(n).

2, 2, 2, 17, 71, 107, 503, 881, 1103, 1301, 9521, 15671, 19543, 31387, 155849, 360289, 370061, 492067, 1349147, 1356869, 2010553, 4652239, 17051297, 20831119, 47326519, 122164649, 189695483, 191912659

Offset: 1

John W. Nicholson, Jul 27 2012

nonn

While many values in A214757(n) are equal to A000101(n), here it seems the only value such that A002386(n) is equal to a(n) is a(1) = R_k = A002386(1) = 2.

See "Let rho(m) = A179196(m)" comment at A001223.

			A104272(94) = 1301 < 1327 = A002386(10), so a(10) = 1301.

Dana Jacobsen, Table of n, a(n) for n = 1..59

Cf. A002386, A104272, A001223.

Cf. A214925, A214926, A214757, A214924.

use ntheory ":all";  sub a_from_2386 { my $n = shift; $n = prev_prime($n) while !is_ramanujan_prime($n); $n } # Dana Jacobsen, Jul 13 2016

perl -Mntheory=:all -nE 'my $n=$1 if /(\d+)$/; $r=ramanujan_primes($n>1e6 ? $n-1e6 : 2, $n); say ++$x," ",$r->[-1];' b002386.txt # Dana Jacobsen, Jul 13 2016

Edited by N. J. A. Sloane, Aug 06 2012

a(16)-a(28) from Donovan Johnson, Nov 04 2012

A349995 Record gaps between odd squarefree semiprimes (A046388).

Original entry on oeis.org

6, 12, 16, 20, 22, 24, 26, 28, 32, 36, 38, 40, 44, 50, 52, 60, 64, 70, 74, 84, 90, 92, 100, 102, 116, 118, 120, 132, 136, 138, 140, 142, 146, 152, 154, 156, 164, 170, 184, 186, 210

Offset: 1

Hugo Pfoertner, Dec 25 2021

			  n  A350098(n)  A350099(n)  a(n)
  1      15          21        6
  2      21          33       12
  3      95         111       16
  4     267         287       20
  5    2369        2391       22

Lucas A. Brown, Python program.

Records in A341828.
Cf. A350098 lower ends of the record gaps, A350099 upper ends of the record gaps.
Cf. A000101, A002386, A005250, A085809, A114057.

a(35)-a(41) from Lucas A. Brown, Feb 29 2024

A053303 Length of maximal prime gap p_{k+1} - p_k with starting prime p_k < 10^n.

Original entry on oeis.org

4, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132

Offset: 1

Enoch Haga, Mar 05 2000

nonn

Prime gaps associated with A053302.

a(17) is probably 1220 and a(19) is probably 1296. - Robert G. Wilson v, Mar 16 2004

			a(1) = 4 from 7 to 11. a(2) = 8 from 89 to 97. a(3) = 20 from 887 to 907.
a(5)=72 because the 5-digit prime 31397 begins a gap of 72.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

p_k's are in A053302. Cf. A005250, A002386. Essentially the same as A038460.

a(16) from Eric W. Weisstein, Mar 05 2004

A073861 Smaller of pair of successive n-digit primes with maximal difference.

Original entry on oeis.org

3, 89, 887, 9551, 31397, 492113, 4652353, 47326693, 436273009, 4302407359, 42652618343, 738832927927, 7177162611713, 90874329411493, 218209405436543, 1693182318746371, 80873624627234849, 804212830686677669

Offset: 1

Amarnath Murthy, Aug 15 2002

A subsequence of A002386 (assuming that there is an n-digit term in A002386 for all values of n). - M. F. Hasler, Apr 28 2014

			a(3) = 887, the next prime is 907, 907-887=20 is the maximal possible difference of two 3-digit primes and no smaller pair exhibits this property.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]

Table[Last[Sort[{#[[2]]-#[[1]],#[[1]],#[[2]]}&/@Partition[Prime[Range[PrimePi[10^i]+1,PrimePi[10^(i+1)]]],2,1]]][[2]],{i,7}] (* Harvey P. Dale, Jan 23 2010 *)

A073861 = A000040 o A241623. - M. F. Hasler, Apr 28 2014

a(n) = max { p in A002386 | nextprime(p) < 10^n } (under the assumption given in the comment). - M. F. Hasler, Apr 28 2014

Corrected error and added terms Harvey P. Dale, Jan 23 2010

a(9)-a(18) from Donovan Johnson, Nov 29 2010

A086977 Increasing peaks in the prime gap sequence A000230.

Original entry on oeis.org

199, 1831, 5591, 30593, 81463, 82073, 162143, 173359, 404597, 542603, 544279, 1100977, 1444309, 2238823, 5845193, 6752623, 6958667, 11981443, 13626257, 49269581, 83751121, 147684137, 166726367, 378043979, 895858039, 1872851947

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is the smaller of the two consecutive primes having a late occurring prime gap g = p_k+1 - p_k. All even gaps smaller than g occur at a smaller prime. Also, the next even gap g+2 also occurs earlier.

			1831 is in this list because the next prime is 1847, giving a prime gap of 16. All even gaps less than 16 occur before this (for smaller primes) and the next even gap, 18, also occurs earlier.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 144.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

Cf. A000230, A001223, A001632, A038664, A086978, A086979, A086980, A002386.

lst={};b=max=2;Do[a=2;While[NextPrime@a-a!=2n,a=NextPrime@a];If[a=max,AppendTo[lst,b]];b=a;If[b>max,max=b],{n,40}];lst (* Giorgos Kalogeropoulos, Aug 18 2021 *)

A086979 Increasing peaks in the prime gap sequence A038664.

Original entry on oeis.org

46, 282, 738, 3302, 7970, 8028, 14862, 15783, 34202, 44773, 44903, 85787, 110224, 165326, 402884, 460883, 474029, 786922, 887313, 2959782, 4875380, 8321465, 9330121, 20226285, 45808557, 92276646, 114867712, 201745031, 265878477

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is Pi(p_k), the number of primes up to and including p_k, where p_k is the initial prime of a prime gap g = p_k+1 - p_k. All even gaps smaller than g occur at a smaller prime and the next even gap g+2 also occurs earlier.

			282 is in this list because the 282nd prime is 1831, the next prime is 1847, giving a prime gap of 16. All even gaps less than 16 occur before this (for smaller primes) and the next even gap, 18, also occurs earlier.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 144.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps.

Cf. A000230, A001223, A001632, A002386, A038664, A086977, A086978, A086980.

A086980 Late occurring prime gaps in the prime gap sequence A001223.

Original entry on oeis.org

12, 16, 32, 38, 46, 56, 66, 70, 74, 80, 88, 94, 102, 108, 116, 124, 134, 144, 150, 158, 166, 186, 194, 200, 228, 256, 264, 278, 294, 298, 316, 328, 334, 362, 370, 388, 422, 436, 442, 452, 466, 472, 482, 488, 510, 520, 536, 568, 576, 580, 590, 608, 628, 632

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is the gap g = p_k+1 - p_k between consecutive primes with all even gaps smaller than g occurring at a smaller prime and the next even gap g+2 also occurring earlier.

			16 is in this list because the first time a prime gap of 16 occurs is between consecutive primes 1831 and 1847. All even prime gaps less than 16 occur for a smaller prime. The next even prime gap of 18 also occurs earlier.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 144.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

Cf. A000230, A001223, A001632, A038664, A086977, A086978, A086979, A002386.

A130642 Numbers n such that 1 + Sum{k=1..n/2} A001223(2k-1)*(-1)^k = 0.

Original entry on oeis.org

2, 6, 14, 190, 194, 200, 306, 462, 468, 474, 478, 490, 560, 1208, 1890, 1938, 23716, 23850, 25226, 25834, 25968, 26642, 26650, 26998, 48316, 311888, 311922, 313946, 331540, 331762, 331782, 377078, 377518, 377666, 377674, 377748, 378422, 378428

Offset: 1

Manuel Valdivia, Jun 20 2007

Sequence has 170 terms < 10^8.

Being prime(n) = 1 + Sum{k=1..n-1}A000040(k)*(-1)^Floor(k/2), for n/2 odd and, prime(n) = (1 + Sum{k=1..n- 1}A000040(k)*(-1)^Floor(k/2))*(-1), for n/2 even.

			1 + ( -A001223(1)) = 1+(-1) = 0, hence 2 is a term.
1 + ( -A001223(1) + A001223(3) - A001223(5)) = 1+(-1+2-2) = 0, hence 6 is a term.

Cf. A127596, A128039, A001223, A000101, A002386.

S=0; a=0; Do[S=S+(Prime[2*k]-Prime[2*k-1])*(-1)^k; If[1+S==0, a++; Print[a, " ", 2*k]], {k, 1, 10^8, 1}]

A130643 Numbers n such that 1 - Sum{k=1..n/2} A001223(2k-1)*(-1)^k = 0.

Original entry on oeis.org

4, 8, 12, 22, 38, 302, 308, 464, 472, 476, 1186, 1884, 2006, 2026, 2106, 23636, 23656, 23698, 25984, 25990, 26706, 26924, 27000, 311914, 311938, 313866, 313880, 331676, 332002, 332676, 377102, 377634, 377670, 379026, 379090, 379108, 387618, 389076

Offset: 1

Manuel Valdivia, Jun 20 2007

Sequence has 177 terms < 10^8.

Being prime(n) = 1 - Sum{k=1..n-1}A000040(k)*(-1)^Floor(k/2), for n/2 even and, prime(n) = (1 - Sum{k=1..n- 1}A000040(k)*(-1)^Floor(k/2))*(-1), for n/2 odd.

			1 - ( -A001223(1) + A001223(3)) = 1-(-1+2) = 0, hence 4 is a term.
1 - ( -A001223(1) + A001223(3) - A001223(5) + A001223(7)) = 1-(-1+2-2+2) = 0, hence 8 is a term.

Cf. A127596, A128039, A001223, A000101, A002386.

S=0; a=0; Do[S=S+(Prime[2*k]-Prime[2*k-1])*(-1)^k; If[1-S==0, a++; Print[a, " ", 2*k]], {k, 1, 10^8, 1}]

A214924 Number of primes <= A214756(n).

Original entry on oeis.org

1, 1, 1, 7, 20, 28, 96, 152, 185, 212, 1179, 1829, 2217, 3382, 14350, 30780, 31528, 40929, 103498, 104047, 149674, 325845, 1094396, 1319933, 2850163, 6957867, 10539421, 10655453

Offset: 1

John W. Nicholson, Jul 29 2012

nonn

a(n) = pi(A214756(n)).

			A214756(5) = 71, so a(5) = primepi(A214756(5)) = primepi(71) = 20.

Cf. A000101, A104272, A214756, A001223.

Cf. A002386, A214925, A214926, A214757.

a(n) = A000217(A214756(n))

Extension to a(28) added by John W. Nicholson, Nov 11 2013

cp's OEIS Frontend

A214756 a(n) = largest Ramanujan prime R_k in A104272 that is <= A002386(n).

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A349995 Record gaps between odd squarefree semiprimes (A046388).

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A053303 Length of maximal prime gap p_{k+1} - p_k with starting prime p_k < 10^n.

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A073861 Smaller of pair of successive n-digit primes with maximal difference.

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Mathematica

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A086977 Increasing peaks in the prime gap sequence A000230.

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Mathematica

A086979 Increasing peaks in the prime gap sequence A038664.

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A086980 Late occurring prime gaps in the prime gap sequence A001223.

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A130642 Numbers n such that 1 + Sum{k=1..n/2} A001223(2k-1)*(-1)^k = 0.

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Mathematica

A130643 Numbers n such that 1 - Sum{k=1..n/2} A001223(2k-1)*(-1)^k = 0.