A229832
First term of smallest sequence of n consecutive weak primes.
Original entry on oeis.org
3, 19, 349, 2909, 15377, 128983, 1319411, 17797519, 94097539, 6927837559, 48486712787, 968068681519, 1472840004019, 129001208165719
Offset: 1
The primes 19 < (17+23)/2 and 23 < (19+29)/2 are the smallest pair of consecutive weak/early primes, so a(2) = 19.
A068843
Smallest prime in the first occurrence of a nondecreasing difference for a set of exactly n successive primes.
Original entry on oeis.org
37, 11, 17, 2, 1091, 2897, 13451, 448363, 7407287, 34400141, 255030533, 6564959561, 45605475961, 121054164221, 2552790756469
Offset: 1
The prime 2 starts the first run of exactly 4 nondecreasing gaps (1, 2, 2, 4) between the 5 primes (2, 3, 5, 7, 11). (The next gap would be of 2, smaller than 4.) Therefore a(4) = 2.
The prime 11 starts the first run of exactly 2 nondecreasing gaps (2, 4) between the 3 primes (11, 13, 17). (The preceding gap is 4 > 2 and the next gap would be 2 < 4.) Therefore a(2) = 11. The sequences of primes (2, 3, 5) as well as (5, 7, 11) are part of a longer run of nondecreasing gaps and are therefore not considered for the case n = 2.
The prime 17 starts the first run of exactly 3 nondecreasing gaps (2, 4, 4) between the 4 primes (17, 19, 23, 29). (The preceding gap is 4 > 2 and the next gap would be 2 < 4.) Therefore a(3) = 17. Again, primes which are part of a longer run cannot be considered.
The prime 37 starts the first run of exactly 1 nondecreasing gap (4) between the primes 37 and 41. (The preceding gap is 6 > 4 and the next gap would be 2 < 4.) Therefore a(1) = 37.
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(* used to find a(11) *) k = 0; While[p = Select[ Range[ k*10^6, (k + 1)*10^6 + 10^4], PrimeQ[ # ] & ]; l = Length[p]; d1 = Take[p, 1 - l] - Take[p, l - 1]; d2 = Take[d1, 2 - l] - Take[d1, l - 2]; s = Sign[ Sign[d2] + 1]; q = StringPosition[ ToString[s], StringDrop[ StringDrop[ ToString[ Table[1, {10}]], 1], -1]]; q == {}, k++ ]; p[[ (q[[1, 1]] + 1)/3 ]]
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A068843(n,c=n+1,g,o=2,P=2)={forprime(p=3,,c--;g>(g=-o+o=p)||next; c||break; c=n; P=p-g);P} \\ M. F. Hasler, May 16 2017
More terms from
Robert G. Wilson v, May 10 2002 and Dec 08 2002, who finds that a(12) > 2.7*10^9
A158940
First primes followed by sequences of exactly n monotonic decreasing prime gaps.
Original entry on oeis.org
2, 7, 31, 1637, 1831, 74653, 322171, 5051309, 11938793, 245333213, 245333159, 130272314561, 1273135176799, 23840790158827
Offset: 1
Alan Worley (aw(AT)xiboo.co.uk), Mar 31 2009
a(4)=1637 is the first prime to be followed by n=4 monotonic decreasing prime gaps: 20,6,4,2.
a(13)=1273135176799 is the first prime to be followed by n=13 monotonic decreasing prime gaps: 72,60,46,44,42,36,34,24,12,8,6,4,2.
Cf.
A158939 (monotonic increasing prime gaps)
A033625
Primes and gaps for the first 10 increasing consecutive gaps.
Original entry on oeis.org
94097537, 2, 94097539, 4, 94097543, 8, 94097551, 10, 94097561, 12, 94097573, 14, 94097587, 16, 94097603, 18, 94097621, 30, 94097651, 32, 94097683
Offset: 0
A133697
a(n) = smallest number k such that P(k)/P(k+1) > P(k+1)/P(k+2) > ... > P(k+n+1)/P(k+n+2), where P(k) = k-th prime = A000040(k).
Original entry on oeis.org
1, 7, 69, 420, 1796, 12073, 101397, 1139211, 5440508, 320620306, 2058187481, 36451609409, 54594153615, 4100904808215
Offset: 0
Philippe LALLOUET (philip.lallouet(AT)orange.fr), Jan 04 2008
P(1)=2, P(2)=3, P(3)=5; 2/3 > 3/5, hence a(0)=1.
17/19 > 19/23 > 23/29 is the first double inequality satisfied by consecutive primes, hence a(1)=7 as 17=P(7).
347/349 > 349/353 > 353/359 > 359/367 is the first triple inequality satisfied by consecutive primes, hence a(2)=69 as 347=P(69).
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(* for the 6th term *) n = 12000; While[ Prime[n]/Prime[n + 1] < Prime[n + 1]/Prime[n + 2] || Prime[n + 1]/Prime[n + 2] < Prime[n + 2]/Prime[n + 3] || Prime[n + 2]/Prime[n + 3] < Prime[n + 3]/Prime[n + 4] || Prime[n + 3]/Prime[n + 4] < Prime[n + 4]/Prime[n + 5] || Prime[n + 4]/Prime[n + 5] < Prime[n + 5]/Prime[n + 6] || Prime[n + 5]/Prime[n + 6] < Prime[n + 6]/Prime[n + 7] || Prime[n + 6]/Prime[n + 7] < Prime[n + 7]/Prime[n + 8], n++ ]; Print[n] (* Robert G. Wilson v, Mar 01 2008 *)
A348927
a(n) is the smallest prime followed by n prime gaps in arithmetic progression with a common difference of 2.
Original entry on oeis.org
2, 5, 17, 347, 2903, 15373, 128981, 95285633, 2426256797, 41193770837, 706748065831, 68439250465123, 68439250465123
Offset: 1
a(2)=5, the 2 prime gaps after 5 are 2,4.
a(3)=17, the 3 prime gaps after 17 are 2,4,6.
a(4)=347, the 4 prime gaps after 347 are 2,4,6,8.
a(5)=2903, the 5 prime gaps after 2903 are 6,8,10,12,14.
Cf.
A349121 (same, but starting prime gap = 2),
A001223 (prime gaps),
A036263 (2nd differences),
A158939 (monotonic increasing),
A006560 (consecutive primes in arithmetic progression).
First column of
A094749 (except first few terms).
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Join[{p=2},Table[While[Union@Differences[NextPrime[p,Range[0,n]],2]!={2},p=NextPrime@p];p,{n,2,7}]] (* Giorgos Kalogeropoulos, Nov 05 2021 *)
A349121
a(n) is the smallest prime p, such that p + k + k^2 are consecutive primes for 0 <= k <= n, but not for k>n.
Original entry on oeis.org
3, 5, 17, 347, 13901, 665111, 128981, 113575727, 2426256797, 137168442221, 4656625081181, 101951758179851, 484511389338941, 221860944705726407
Offset: 1
a(4)=347, because it is the smallest prime such that 347 + 2 = 349, 349 + 4 = 353, 353 + 6 = 359, 359 + 8 = 367 are 5 consecutive primes.
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Table[k=2;While[Flatten[k+(s={Range[0,n]})+s^2]!=NextPrime[k,Range[0,n]]||NextPrime[k,n+1]==k+n+1+(n+1)^2,k=NextPrime@k];k,{n,7}] (* Giorgos Kalogeropoulos, Nov 10 2021 *)
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isok(p, n) = my(q=p); for (k=1, n, my(r = p+k+k^2); if (nextprime(q+1) != r, return (0)); q=r); return(1);
a(n) = my(p=2); while (!isok(p, n), p=nextprime(p+1)); p; \\ Michel Marcus, Nov 09 2021
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