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A002386
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Increasing gaps between primes (lower end): primes p(k) where p(k+1)-p(k) exceeds p(j+1)-p(j) for all j<k.
(Formerly M0858 N0327)
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43
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2, 3, 7, 23, 89, 113, 523, 887, 1129, 1327, 9551, 15683, 19609, 31397, 155921, 360653, 370261, 492113, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783, 387096133, 436273009, 1294268491
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| See the links by Jens Kruse Andersen et al. for very large gaps.
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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 6.1, Table 1.
M. Kraitchik, Recherches sur la Th\'{e}orie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 14.
T. R. Nicely: New maximal prime gaps and first occurrences, Math. Comput. 68,227 (1999) 1311-1315.
D. Shanks, On maximal gaps between successive primes, Math. Comp., 18 (1964), 646-651.
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).
J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.
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LINKS
| M. F. Hasler and N. J. A. Sloane, Table of n, a(n) for n=1..75
Jens Kruse Andersen, The Top-20 Prime Gaps
Jens Kruse Andersen, New record prime gap
Jens Kruse Andersen, Maximal gaps
T. R. Nicely, List of prime gaps
Tomas Oliveira e Silva, Gaps between consecutive primes
Hans Rosenthal and Jens Kruse Andersen, A prime megagap
Eric Weisstein's World of Mathematics, Prime Gaps
Index entries for primes, gaps between
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FORMULA
| a(n)=A000101(n)-A005250(n)=A008950(n-1)-1 - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Dec 13 2007
A000720(a(n)) = A005669(n).
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MATHEMATICA
| s = {2}; gm = 1; Do[p = Prime[n]; g = Prime[n + 1] - p; If[g > gm, Print[p]; AppendTo[s, p]; gm = g], {n, 2, 1000000}]; s (* From Jean-François Alcover , Mar 31 2011 *)
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PROG
| (PARI) a(n)=local(p, g); if(n<2, 2*(n>0), p=a(n-1); g=nextprime(p+1)-p; while(p=nextprime(p+1), if(nextprime(p+1)-p>g, break)); p) - Michael Somos Feb 07 2004
(PARI) p=q=2; g=0; until( g<(q=nextprime(1+p=q))-p & print1(q-g=q-p, ", "), ) \\ - M. F. Hasler, Dec 13 2007
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CROSSREFS
| Cf. A001223, A000101 (upper ends), A005250 (record gaps), A000230.
Cf. A005669, A134266, A070866.
Sequence in context: A088173 A129739 A163834 * A000230 A133429 A087770
Adjacent sequences: A002383 A002384 A002385 * A002387 A002388 A002389
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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