A000101 -id:A000101 - cp's OEIS frontend

A066352 Pillai sequence: a(n) is the smallest term in A007924 requiring n primes.

0, 1, 4, 27, 1354, 401429925999155061

Offset: 0

Copied from www.primepuzzles.net by Frank Ellermann, Dec 19 2001

a(5) computed independently in 2007 by R. J. Mathar and Luca & Thangadurai, both using Thomas Nicely's tables.
On Cramer's conjecture, the number of primes required is O(log* n), where log* is the iterated logarithm, so the rate of growth of a(n) is tetrational in n. - Charles R Greathouse IV, Aug 28 2010
The next term likely has hundreds of millions of digits. - Charles R Greathouse IV, Jun 29 2015

			The greatest prime <= 27 is 23; the greatest prime <= 27-23 is 3; 27-23-3 = 1, so the Pillai representation of 27 is 23+3+1, which uses more terms than all preceding numbers.

S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp. 159-167.

Florian Luca and Ravindranathan Thangadurai, On an arithmetic function considered by Pillai, Journal de théorie des nombres de Bordeaux 21:3 (2009), pp. 695-701.
M. T. Marcos, Smarandache Prime Base representation, prime puzzle 141.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
S. S. Pillai, On an Arithmetic Function concerning Primes, Journal Of The Annamalai University, Vol-1 (1932), pp. 159-167 (pages 204-212 in pdf).

Cf. A007924.

A072491(n)=if(n<4,n>0,1+A072491(n-precprime(n)))
print1(r=0);for(n=1,1e7,if(A072491(n)>r,r=a(n);print1(", "n)))
\\ Charles R Greathouse IV, Feb 04 2013

a(n) = 2*p(m) - p(m-1) with minimal m = pi(a(n)) so that p(m) = a(n-1) + p(m-1), where p(n) is A008578(n).

Edited by Charles R Greathouse IV, Oct 28 2009

Entry rewritten by Charles R Greathouse IV, Aug 28 2010

A214935 Index of the primes of A205827, A000720(A205827(n)).

Original entry on oeis.org

1, 2, 4, 9, 30, 189, 217, 2225, 3385, 14357, 30802, 31545, 104071, 149689, 1094421, 1319945, 10655462, 23163298, 112228683, 182837804, 203615628, 486570087, 1094330259, 11992433550, 17883926781, 50070452577, 52302956123, 72178455400

Offset: 1

John W. Nicholson, Oct 28 2012

nonn

A000040(a(n)) = A205827(n).

With pi(x) being the prime counting function, A000720(x), for n from 1 to 3, a(n) = pi(A111870(n)) = A241542(n), for n from 5 to 28, a(n) = pi(A111870(n-1)) = A241542(n-1). - John W. Nicholson, May 10 2014

			a(4) = 9, A000040(9) = 23, and A205827(4) = 23.

John W. Nicholson, Table of n, a(n) for n = 1..38

Cf. A205827.

record=0;i=0;A214935=vectorsmall(38);for(n=1,75,current=(A000101[n]/A002386[n]*1.)^A005669[n];if(current>record,record=current;i++;A214935[i]=A005669[n]))

a(n) = pi(A205827(n)) = A000720(A205827(n)).

a(13)-a(28) from Donovan Johnson, Oct 28 2012

a(29)-a(38) from John W. Nicholson, Dec 01 2013

A053695 Differences between record prime gaps.

Original entry on oeis.org

1, 2, 2, 2, 6, 4, 2, 2, 12, 2, 8, 8, 20, 14, 10, 16, 2, 4, 14, 16, 6, 26, 30, 10, 2, 12, 14, 2, 32, 6, 4, 28, 16, 18, 28, 2, 10, 62, 8, 4, 6, 12, 4, 10, 14, 2, 16, 2, 6, 42, 6, 14, 50, 22, 42, 50, 12, 26, 2, 100, 10, 8, 208, 52, 14, 22, 4, 24, 24, 56, 28, 14, 72, 34, 12, 22

Offset: 1

Jeff Burch, Mar 23 2000

The largest known term of this sequence is a(63) = 1132 - 924 = 208. This seems rather strange for a(63) > 2*100+7 where 100 = max {a(k)| k < 63}. {1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,42,50,52,56,62,72,100,208} is the set of the distinct first 75 terms of the sequence. What is the smallest number m such that a(m) = 36? - Farideh Firoozbakht, May 30 2014

Conjecture: a(n) <= A005250(n). Based on the equivalent statement at A005250: A005250(n+1) / A005250(n) <= 2. - John W. Nicholson, Dec 30 2015

John W. Nicholson, Table of n, a(n) for n = 1..81
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression. International Mathematical Forum, Vol. 13, 2018, No. 2, 65-78, arXiv:1709.05508v3 [math.NT].
Eric Weisstein's World of Mathematics, Prime Gaps
Wikipedia, Prime gap

Cf. A000101, A001223, A002386, A005250, A053686.

m = 2; r = 0; Differences@ Reap[Monitor[Do[If[Set[d, Set[n, NextPrime[m]] - m] > r, Set[r, d]; Sow[d]]; m = n, {i, 10^7}], i]][[-1, -1]] (* Michael De Vlieger, Oct 30 2021 *)

a(n) = A005250(n+1) - A005250(n).

A005250(n+1) = 1 + Sum_{i=1..n} a(i). - John W. Nicholson, Dec 29 2015

Missing term 1 and more terms added by Farideh Firoozbakht, May 30 2014

a(75)-a(76) from John W. Nicholson, Feb 27 2018

A111943 Prime p with prime gap q - p of n-th record Cramer-Shanks-Granville ratio, where q is smallest prime larger than p and C-S-G ratio is (q-p)/(log p)^2.

Original entry on oeis.org

23, 113, 1327, 31397, 370261, 2010733, 20831323, 25056082087, 2614941710599, 19581334192423, 218209405436543, 1693182318746371

Offset: 1

N. J. A. Sloane, following emails from R. K. Guy and Ed Pegg Jr, Nov 27 2005

Primes less than 23 are anomalous and are excluded.
a(12) was discovered by Bertil Nyman in 1999.
Shanks conjectures that the ratio will never reach 1. Granville conjectures the opposite: that the ratio will exceed or come arbitrarily close to 2/e^gamma = 1.1229....
Firoozbakht's conjecture implies that the ratio is below 1-1/log(p) for all primes p>=11; see Th.1 of arXiv:1506.03042. In Cramér's probabilistic model of primes, the ratio is below 1-1/log(p) for almost all maximal gaps between primes; see A235402. - Alexei Kourbatov, Jan 28 2016

			-----------------------------
n   ratio                a(n)
-----------------------------
1   0.6103                23
2   0.6264               113
3   0.6575              1327
4   0.6715             31397
5   0.6812            370261
6   0.7025           2010733
7   0.7394          20831323
8   0.7953       25056082087
9   0.7975     2614941710599
10  0.8177    19581334192423
11  0.8311   218209405436543
12  0.9206  1693182318746371

R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, A8.

Andrew Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial J. 1 (1995), pp. 12-28.
Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, arXiv:1506.03042 [math.NT], 2015; J. Integer Sequences, 18 (2015), Article 15.11.2.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Daniel Shanks, On maximal gaps between successive primes, Math. Comp. 18 (88) (1964), 646-651.
Eric Weisstein's World of Mathematics, Prime Gaps.
Eric Weisstein's World of Mathematics, Cramer-Granville Conjecture.
Eric Weisstein's World of Mathematics, Shanks Conjecture (and Wolf Conjecture).

Subsequence of A002386.

Cf. A111870, A166363.

r=CSG=0;p=13;forprime(q=17,1e8,if(q-p>r,r=q-p; t=r/log(p)^2; if(t>CSG, CSG=t; print1(p", ")));p=q) \\ Charles R Greathouse IV, Apr 07 2013

Corrected and edited (p_n could be misinterpreted as the n-th prime) by Daniel Forgues, Nov 20 2009

Edited by Charles R Greathouse IV, May 14 2010

A008995 Increasing length runs of consecutive composite numbers (endpoints).

Original entry on oeis.org

4, 10, 28, 96, 126, 540, 906, 1150, 1360, 9586, 15726, 19660, 31468, 156006, 360748, 370372, 492226, 1349650, 1357332, 2010880, 4652506, 17051886, 20831532, 47326912, 122164968, 189695892, 191913030

Offset: 1

Mark Cramer (m.cramer(AT)qut.edu.au). Computed by Dennis Yelle (dennis(AT)netcom.com)

nonn

Netnews group rec.puzzles, circa Mar 01 1996 (I would like to get the exact reference).

MathForum rec.puzzles archive, Arithmetic Question 8 - consecutive.composites, June 2005.
User "Abigail", 1000 consectutive composites (sic), post in newsgroup rec.puzzles, Jun 19 1996.
Eric Weisstein's World of Mathematics, Prime Gaps.

Cf. A000101, A008950, A008996.

maxGap = 1; Reap[ Do[ gap = Prime[n + 1] - (p = Prime[n]); If[gap > maxGap, Print[p + gap - 1]; Sow[p + gap - 1]; maxGap = gap], {n, 2, 10^8}]][[2, 1]] (* Jean-François Alcover, Jun 12 2013 *)

a(n) = A000101(n+1)-1.

A058193 Smallest prime p such that there is a gap of 6n between p and the next prime.

Original entry on oeis.org

23, 199, 523, 1669, 4297, 9551, 16141, 28229, 35617, 43331, 162143, 31397, 188029, 461717, 404851, 360653, 1444309, 2238823, 492113, 1895359, 1671781, 1357201, 3826019, 11981443, 13626257, 17983717, 39175217, 37305713, 52721113

Offset: 1

Labos Elemer, Nov 28 2000

nonn

			d = 72 appears after 31397, while smaller d = 54, 60, 66 come later, following primes 35617, 43331, 162143, respectively.

Amiram Eldar, Table of n, a(n) for n = 1..240
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Index entries for primes, gaps between.

Cf. A101232, A062529, A000230.

Module[{nn=32*10^5,prs,gps},prs=Prime[Range[nn]];gps=Differences[prs];Table[SelectFirst[Thread[{Most[prs],gps}],#[[2]]==6n&],{n,30}]][[;;,1]] (* Harvey P. Dale, Mar 03 2025 *)

a(n) = {p=3; q = nextprime(p+1); while((q-p) != 6*n, p = q; q = nextprime(q+1)); p;} \\ Michel Marcus, Mar 12 2016

a(n) = A000230(3n).

Offset corrected by M. F. Hasler, Apr 09 2013

A062529 Smallest prime p such that there is a gap of 2^n between p and the next prime.

Original entry on oeis.org

2, 3, 7, 89, 1831, 5591, 89689, 3851459, 1872851947, 1999066711391, 22790428875364879

Offset: 0

Labos Elemer, Jun 25 2001

nonn

a(11) <= 79419801290172271035479303914142441 and  a(12) <= 55128448018333565337014555712123010955456071077000028555991469751. - Abhiram R Devesh, Aug 09 2014
From Zhining Yang, Dec 02 2022: (Start)
a(11) = 5333419265419188034369535864125349, 34 digits, discovered by Helmut Spielauer in 2013
a(12) = 55128448018333565337014555712123010955456071077000028555991469751, 65 digits, discovered by Helmut Spielauer in 2013
a(13) = 192180552346991956641101827551986346298837407139466361414211497406670710665021150917759713696699494356609164354068319457039591759, 129 digits, discovered by Dana Jacobsen in 2016
a(14) = 267552521*631#/210 - 9606, 268 digits, discovered by Dana Jacobsen in 2016
a(15) = 2717*1303#/268590 - 16670, 552 digits, discovered by Dana Jacobsen in 2014
a(16) = 7079*3559#/9870 - 36310, 1517 digits, discovered by Michiel Jansen, Pierre Cami, and Jens Kruse Andersen in 2013
a(17) = 1111111111111111111*9059#/(11#*5237) - 86522, 3899 digits, discovered by Hans Rosenthal in 2017
a(11) to a(17) were searched from Thomas R. Nicely's homepage. (End)
Importantly, the values in the previous comment are only upper bounds on a(11)-a(17), and are (almost certainly) not the correct values. As of this comment, the largest prime gap length whose first occurrence is known is 1676 < 2^11. - Brian Kehrig, May 01 2025

			a(2)=7 because 7 and 11 are consecutive primes with difference 2^2=4.
a(3)=89 because 89 and 97 are consecutive primes with difference 2^3=8.

Cino Hilliard, TwinPrimes Java code.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101].
Thomas R. Nicely, Other tables of prime gaps

Cf. A000230, A062530, A101232, A002386.

f[n_] := Block[{k = 1}, While[Prime[k + 1] != n + Prime[k], k++ ]; Prime[k]]; Do[ Print[ f[2^n]], {n, 0, 10}] (* Robert G. Wilson v, Jan 13 2005 *)

import sympy
n=0
while n>=0:
    p=2
    while sympy.nextprime(p)-p!=(2**n):
        p=sympy.nextprime(p)
    print(p)
    n=n+1
    p=sympy.nextprime(p)
## Abhiram R Devesh, Aug 09 2014

a(n) = A000230(2^(n-1)). - R. J. Mathar, Jan 12 2007

a(n) = A000230(2^(n-1)) = Min{p|nextprime(p)-p = 2^n} [may need adjusting since offset has been changed].

a(10) sent by Robert G. Wilson v, Jan 13 2005

a(11)-a(12) removed by Brian Kehrig, May 01 2025

A127596 Numbers k such that 1 + Sum_{i=1..k-1} A001223(i)*(-1)^i = 0.

Original entry on oeis.org

2, 4, 14, 22, 28, 233, 249, 261, 488, 497, 511, 515, 519, 526, 531, 534, 548, 562, 620, 633, 635, 2985, 3119, 3123, 3128, 3157, 4350, 4358, 4392, 4438, 4474, 4484, 4606, 4610, 4759, 5191, 12493, 1761067, 2785124, 2785152, 2785718, 2785729, 2867471

Offset: 1

Manuel Valdivia, Apr 03 2007

nonn

Or, with prime(0) = 1, numbers k such that Sum_{i=0..k-1} (prime(i+1)-prime(i))*(-1)^i = Sum_{i=0..k-1} (A008578(i+1)-A008578(i))*(-1)^i = 0.

There are 313 terms < 10^7, 846 terms < 10^8, 1161 terms < 10^9.

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

Jinyuan Wang, Table of n, a(n) for n = 1..1161 (first 846 terms from Klaus Brockhaus)
Eric Weisstein's World of Mathematics, Andrica's Conjecture
Eric Weisstein's World of Mathematics, Prime Difference Function

Cf. A001223 (differences between consecutive primes), A008578 (prime numbers at the beginning of the 20th century), A000101 (increasing gaps between primes, upper end), A002386 (increasing gaps between primes, lower end).

Cf. A282178 (prime(a(n))), A330545, A330547.

S=0; Do[j=Prime[n+1]; i=Prime[n]; d[n]=j-i; S=S+(d[n]*(-1)^n); If[S+1==0, Print[Table[j|PrimePi[j]|S+1]]], {n,1,10^7,1}]

{m=10^8; n=1; p=1; e=1; s=0; while(nKlaus Brockhaus, Apr 29 2007 */

Edited by Klaus Brockhaus, Apr 29 2007

A182514 Primes prime(n) such that (prime(n+1)/prime(n))^n > n.

Original entry on oeis.org

2, 3, 7, 113, 1327, 1693182318746371

Offset: 1

Thomas Ordowski, May 04 2012

nonn

The Firoozbakht conjecture: (prime(n+1))^(1/(n+1)) < prime(n)^(1/n), or prime(n+1) < prime(n)^(1+1/n), prime(n+1)/prime(n) < prime(n)^(1/n), (prime(n+1)/prime(n))^n < prime(n).
Using the Mathematica program shown below, I have found no further terms below 2^27. I conjecture that this sequence is finite and that the terms stated are the only members. - Robert G. Wilson v, May 06 2012 [Warning: this conjecture may be false! - N. J. A. Sloane, Apr 25 2014]
I conjecture the contrary: the sequence is infinite. Note that 10^13 < a(6) <= 1693182318746371. - Charles R Greathouse IV, May 14 2012
[Stronger than Firoozbakht] conjecture: All (prime(n+1)/prime(n))^n values, with n >= 5, are less than n*log(n). - John W. Nicholson, Dec 02 2013, Oct 19 2016
The Firoozbakht conjecture can be rewritten as (log(prime(n+1)) / log(prime(n)))^n < (1+1/n)^n. This suggests the [weaker than Firoozbakht] conjecture: (log(prime(n+1))/log(prime(n)))^n < e. - Daniel Forgues, Apr 26 2014
All a(n) <= a(6) are in A002386, A205827, and A111870.
The inequality in the definition is equivalent to the inequality prime(n+1)-prime(n) > log(n)*log(prime(n)) for sufficiently large n. - Thomas Ordowski, Mar 16 2015
Prime indices, A000720(a(n)) = 1, 2, 4, 30, 217, 49749629143526. - John W. Nicholson, Oct 25 2016

			7 is in the list because, being the 4th prime, and 11 the fifth prime, we verify that (11/7)^4 = 6.09787588507... which is greater than 4.
11 is not on the list because (13/11)^5 = 2.30543740804... and that is less than 5.

Farhadian, R. (2017). On a New Inequality Related to Consecutive Primes. OECONOMICA, vol 13, pp. 236-242.

Reza Farhadian, A New Conjecture On the primes, Preprint, 2016.
R. Farhadian, and R. Jakimczuk, On a New Conjecture of Prime Numbers Int. Math. Forum, vol. 12, 2017, pp. 559-564.
Luan Alberto Ferreira, Some consequences of the Firoozbakht's conjecture, arXiv:1604.03496v2 [math.NT], 2016.
Luan Alberto Ferreira, Hugo Luiz Mariano, Prime gaps and the Firoozbakht Conjecture, São Paulo J. Math. Sci. (2018), 1-11.
A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015.
A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2.
Carlos Rivera, Conjecture 78. P_n^((P_n+1/P_n)^n) <= n^P_n, 2016.
Nilotpal Kanti Sinha, On a new property of primes that leads to a generalization of Cramer's conjecture, arXiv:1010.1399 [math.NT], 2010.
Matt Visser, Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap, arXiv:1904.00499 [math.NT], 2019.
Wikipedia, Firoozbakht’s conjecture

Cf. A111870.

Prime[Select[Range[1000], (Prime[# + 1]/Prime[#])^# > # &]] (* Alonso del Arte, May 04 2012 *)
firoozQ[n_, p_, q_] := n * Log[q] > Log[n] + n * Log[p]; k = 1; p = 2; q = 3; While[ k < 2^27, If[ firoozQ[k, p, q], Print[{k, p}]]; k++; p = q; q = NextPrime@ q] (* Robert G. Wilson v, May 06 2012 *)

n=1;p=2;forprime(q=3,1e6,if((q/p*1.)^n++>n, print1(p", "));p=q) \\ Charles R Greathouse IV, May 14 2012

for(n=1,75,if((A000101[n]/A002386[n]*1.)^A005669[n]>=A005669[n], print1(A002386[n],", "))) \\ Each sequence is read in as a vector as to overcome PARI's primelimit \\ John W. Nicholson, Dec 01 2013

q=3;n=2; forprime(p=5, 10^9,result=(p/q)^n/(n*log(n));if(result>1, print(q," ",p, " ", n, " ", result));n++;q=p) \\ for stronger than Firoozbakht conjecture \\ John W. Nicholson, Mar 16 2015, Oct 19 2016

a(6) from John W. Nicholson, Dec 01 2013

A030296 Smallest start for a run of at least n composite numbers.

Original entry on oeis.org

4, 8, 8, 24, 24, 90, 90, 114, 114, 114, 114, 114, 114, 524, 524, 524, 524, 888, 888, 1130, 1130, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 9552, 9552, 15684, 15684, 15684, 15684, 15684, 15684, 15684, 15684, 19610, 19610, 19610

Offset: 1

Eric W. Weisstein

a(n) is even, since a(n)-1 is a prime > 2, by the minimality of a(n). - Jonathan Sondow, May 31 2014

Except for a(1), records occur at even values of n, and each term appears an even number of times consecutively. (Proof. A maximal run of composites must begin and end at even numbers.) - Jonathan Sondow, May 31 2014

			a(5) = 24 as 24 is the first of the five consecutive composite numbers 24, 25, 26, 27, 28.

Amarnath Murthy, Some more conjectures on primes and divisors, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001.

Donovan Johnson, Table of n, a(n) for n = 1..1475 (terms < 4*10^18)
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

Cf. A008950, A008995, A008996, A000101, A002386, A104138.

a[n_] := a[n] = For[p1 = a[n-1]-1; p2 = NextPrime[p1], True, p1 = p2; p2 = NextPrime[p1], If[ p2-p1-1 >= n, Return[p1+1]]]; a[1] = 4; Table[a[n], {n, 1, 43}] (* Jean-François Alcover, May 24 2012 *)
Module[{nn=20000,cmps},cmps=Table[If[CompositeQ[n],1,0],{n,nn}];Table[ SequencePosition[ cmps,PadRight[{},k,1],1][[1,1]],{k,50}]] (* Harvey P. Dale, Jan 01 2022 *)

a(n) = A104138(n) + 1. - Jonathan Sondow, May 31 2014

cp's OEIS Frontend

A066352 Pillai sequence: a(n) is the smallest term in A007924 requiring n primes.

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A214935 Index of the primes of A205827, A000720(A205827(n)).

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A053695 Differences between record prime gaps.

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A111943 Prime p with prime gap q - p of n-th record Cramer-Shanks-Granville ratio, where q is smallest prime larger than p and C-S-G ratio is (q-p)/(log p)^2.

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A008995 Increasing length runs of consecutive composite numbers (endpoints).

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A058193 Smallest prime p such that there is a gap of 6n between p and the next prime.

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A062529 Smallest prime p such that there is a gap of 2^n between p and the next prime.

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