Displaying 41-50 of 123 results found.
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Smallest start for a run of at least n composite numbers.
+10
5
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
COMMENTS
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
REFERENCES
Amarnath Murthy, Some more conjectures on primes and divisors, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001.
EXAMPLE
a(5) = 24 as 24 is the first of the five consecutive composite numbers 24, 25, 26, 27, 28.
MATHEMATICA
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 *)
Increasing gaps among twin primes: the smallest prime of the second twin pair.
+10
5
5, 11, 29, 59, 101, 347, 419, 809, 2549, 6089, 13679, 18911, 24917, 62927, 188831, 688451, 689459, 851801, 2870471, 4871441, 9925709, 14658419, 17384669, 30754487, 32825201, 96896909, 136286441, 234970031, 248644217, 255953429
EXTENSIONS
Terms a(3)-a(41) are given by Rathbun (1998).
Terms up to a(72) are listed in Kourbatov (2013), terms up to a(75) on Oliveira e Silva's website, added by Max Alekseyev, Nov 06 2015
Increasing gaps among twin primes: size.
+10
5
0, 4, 10, 16, 28, 34, 70, 148, 166, 208, 280, 370, 496, 628, 922, 928, 1006, 1450, 1510, 1528, 1720, 1900, 2188, 2254, 2830, 2866, 3010, 3100, 3178, 3478, 3802, 4768, 5290, 6028, 6280, 6472, 6550, 6646, 7048, 7978, 8038, 8992, 9310, 9316, 10198, 10336, 10666, 10708
EXTENSIONS
Terms 0, 4 prepended, missing term 1006 inserted, and more terms added from A113274 by Max Alekseyev, Nov 05 2015
Nondecreasing gaps between primes.
+10
5
1, 2, 2, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 8, 14, 14, 14, 18, 20, 22, 34, 34, 36, 36, 36, 44, 52, 52, 72, 86, 86, 96, 112, 114, 118, 132, 132, 148, 154, 154, 154, 180, 210, 220, 222, 234, 248, 250, 250, 282, 288, 292, 320, 336, 336, 354, 382, 384, 394, 456, 464, 468, 474, 486, 490, 500, 514, 516, 532, 534, 540, 582, 588, 602, 652, 674, 716, 766, 778
COMMENTS
All terms of A005250 are in the sequence, but some terms of A005250 appear in this sequence more than once. a(n) is the gap between the n-th and (n+1)-th sublists of prime numbers defined in A348178. - Ya-Ping Lu, Oct 19 2021
REFERENCES
R. K. Guy, Unsolved problems in number theory.
EXAMPLE
a(21) = a(22) = 34 because prime(218) - prime(217) = prime(1060) - prime(1059) = 34 and prime(n+1) - prime(n) is less than 34, for n < 1059 and n not equal to 217.
MATHEMATICA
f[n_] := Prime[n+1]-Prime[n]; v={}; Do[ If[f[n]>=If[n==1, 1, v[[ -1]]], v1=n; v=Append[v, f[v1]]; Print[v]], {n, 105000000}]
DeleteDuplicates[Differences[Prime[Range[10^7]]], Greater] (* Harvey P. Dale, Jan 17 2024 *)
PROG
(Python)
from sympy import nextprime; p, r = 2, 0
while r < 778:
q = nextprime(p); g = q - p
if g >= r: print(g, end = ', '); r = g
11, 11, 11, 29, 97, 127, 569, 937, 1151, 1367, 9613, 15727, 19681, 31481, 156007, 360769, 370387, 492251, 1349669, 1357333, 2010881, 4652507, 17051981, 20831639, 47326913, 122165059, 189695893, 191913047
Record gaps between odd squarefree semiprimes ( A046388).
+10
5
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
EXAMPLE
1 15 21 6
2 21 33 12
3 95 111 16
4 267 287 20
5 2369 2391 22
CROSSREFS
Cf. A350098 lower ends of the record gaps, A350099 upper ends of the record gaps.
Length of maximal prime gap p_{k+1} - p_k with starting prime p_k < 10^n.
+10
4
4, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132
COMMENTS
Prime gaps associated with A053302.
EXAMPLE
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.
Record gaps between consecutive primes that repeat at least once before a new record occurs.
+10
4
2, 4, 6, 14, 34, 36, 52, 86, 132, 154, 250, 336
COMMENTS
Scan the sequence of prime differences ( A001223) looking for new records, but append the record difference to the present sequence only if the difference appears at least twice in A001223 before it is beaten by a new record. - N. J. A. Sloane, Dec 30 2007 The sequence of primes where these gaps first appear is A133788.
EXTENSIONS
More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 13 2006
Smaller of pair of successive n-digit primes with maximal difference.
+10
4
3, 89, 887, 9551, 31397, 492113, 4652353, 47326693, 436273009, 4302407359, 42652618343, 738832927927, 7177162611713, 90874329411493, 218209405436543, 1693182318746371, 80873624627234849, 804212830686677669
FORMULA
a(n) = max { p in A002386 | nextprime(p) < 10^n } (under the assumption given in the comment). - M. F. Hasler, Apr 28 2014
EXAMPLE
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.
MATHEMATICA
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 *)
Middle q of three consecutive primes p,q,r, such that one adjacent prime is near, the other is far and the ratio of the differences (whichever of (r-q)/(q-p) or (q-p)/(r-q) is greater than 1) sets a record.
+10
4
3, 29, 113, 139, 199, 523, 1151, 1669, 2971, 6947, 10007, 16141, 25471, 40639, 79699, 102761, 173359, 265621, 404851, 838249, 1349533, 1562051, 6371537, 7230479, 27980987, 42082303, 53231051, 70396589, 192983851, 253878617, 390932389, 465828731, 516540163, 1692327137
COMMENTS
Are there entries other than a(3) for which the smaller difference exceeds 2?
EXAMPLE
a(3) = 113 because the ratio (113-109)/(127-113) = 2/7 = 0.28571.. is smaller than the previous minimum produced by (31-29)/(29-23) = 1/3 = 0.33333...
PROG
(PARI) a084105(limit)={my(p1=2, p2=3, r=0); forprime(p3=5, limit, my(q=max((p2-p1)/(p3-p2), (p3-p2)/(p2-p1))); if(q>r, r=q; print1(p2, ", ")); p1=p2; p2=p3)};
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