A051635 -id:A051635 - cp's OEIS frontend

A054832 Fifth term of weak prime sextet: p(m-3)-p(m-4) < p(m-2)-p(m-3) < p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m).

2939, 13499, 13921, 14983, 15401, 15413, 21433, 21577, 21893, 28297, 30911, 33247, 35617, 37747, 42257, 42611, 45841, 55681, 64693, 64951, 64969, 68227, 68239, 68917, 68927, 73973, 74231, 78623, 83137, 85549, 87359, 88037, 90947

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A051635, A054800-A054840.

Select[Partition[Prime[Range[8800]],6,1],Min[Differences[#,2]]>0&][[All,5]] (* Harvey P. Dale, Feb 22 2020 *)

A054836 Third term of weak prime septet: p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2) < p(m+4)-p(m+3).

Original entry on oeis.org

15383, 64927, 68213, 68903, 128987, 128993, 143519, 154087, 158009, 192383, 221723, 222403, 244471, 249737, 285301, 318683, 337283, 354377, 357839, 374189, 385397, 394733, 402587, 402593, 419603, 439171, 441923, 448387, 457403, 457679, 458197, 482513, 527987, 529819, 577537, 582767

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..1000, Oct 27 2018.

Cf. A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

a(n) = A151800(A054835(n)) = A151799(A054838(n)), A151800 = nextprime, A151799 = prevprime; A054836 = { m = A054829(n) | m = nextprime(A054829(n-1)) }. - M. F. Hasler, Oct 27 2018

More terms from M. F. Hasler, Oct 27 2018

A054839 Sixth term of weak prime septet: p(m-4)-p(m-5) < p(m-3)-p(m-4) < p(m-2)-p(m-3) < p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m).

Original entry on oeis.org

15413, 64969, 68239, 68927, 129011, 129023, 143551, 154127, 158047, 192431, 221747, 222461, 244507, 249779, 285377, 318713, 337313, 354401, 357913, 374239, 385433, 394759, 402613, 402631, 419651, 439217, 441971, 448451, 457433, 457711, 458239, 482539, 528013

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..233, Oct 27 2018.

Cf. A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

Transpose[Select[Partition[Prime[Range[50000]],7,1],Min[ Differences[ #,2]]> 0&]][[6]] (* Harvey P. Dale, Sep 27 2015 *)

a(n) = A151800(A054838(n)) = A151799(A054840(n)), A054839 = { m = A054832(n) | m = A151800(A054832(n-1)) } (A151800: nextprime, A151799: prevprime). - M. F. Hasler, Oct 27 2018

More terms from Harvey P. Dale, Sep 27 2015

A102552 a(n) = prime(n) - (prime(n+1) + prime(n-1))/2.

Original entry on oeis.org

0, -1, 1, -1, 1, -1, -1, 2, -2, 1, 1, -1, -1, 0, 2, -2, 1, 1, -2, 1, -1, -1, 2, 1, -1, 1, -1, -5, 5, -1, 2, -4, 4, -2, 0, 1, -1, 0, 2, -4, 4, -1, 1, -5, 0, 4, 1, -1, -1, 2, -4, 2, 0, 0, 2, -2, 1, 1, -4, -2, 5, 1, -1, -5, 4, -2, 4, -1, -1, -1, 1, 0, 1, -1, -1, 2, -2, -1, 4, -4, 4, -2, 1, -1, -1, 2, 1, -1, -4, 2, 2, -2, 2, -1, -3, 5, -8, 6, -2, 2, 0, 2, -2

Offset: 3

Yasutoshi Kohmoto, Feb 25 2005

sign

			a(6)=-1 because 13-(17+11)/2=-1.

Eric Weisstein, CRC Concise Encyclopedia of Mathematics, 1998, page 1321.

G. C. Greubel, Table of n, a(n) for n = 3..10000

Cf. A000040, A006562, A036263, A051634, A051635, A066875.

A102552:= func< n | (2*NthPrime(n)-NthPrime(n+1)-NthPrime(n-1))/2 >;
[A102552(n): n in [3..120]]; // G. C. Greubel, Feb 02 2025

a:=n->ithprime(n)-(ithprime(n+1)+ithprime(n-1))/2: seq(a(n),n=3..95); # Emeric Deutsch, Mar 02 2005

f[n_] := Prime[n] - (Prime[n - 1] + Prime[n + 1])/2; Table[f[n], {n, 3, 107}] (* Robert G. Wilson v, Sep 25 2006 *)
#[[2]]-(#[[1]]+#[[3]])/2&/@Partition[Prime[Range[2,110]],3,1] (* Harvey P. Dale, Sep 21 2013 *)

a(n) = prime(n)-(prime(n+1)+prime(n-1))/2;
vector(100,n,a(n+2)) \\ Joerg Arndt, Jan 20 2015

from sympy import sieve as p
def A102552(n): return p[n]-(p[n+1]+p[n-1])//2 # Karl-Heinz Hofmann, May 22 2024

a(n) = (1/2)*(A001223(n) - A001223(n+1)).

a(n) = -A036263(n-1)/2. - T. D. Noe, Oct 06 2006 [corrected by Georg Fischer, Oct 19 2023]

More terms from Emeric Deutsch, Mar 02 2005

A175102 1, followed by list of numbers n such that the number of strong primes and the number of weak primes are equal at the n-th prime.

Original entry on oeis.org

1, 60, 64, 41192, 41194, 41247, 41250, 41252, 41257, 41259, 41261, 41263, 41265, 41267, 41273, 41275, 41277, 41279, 41287, 41317, 41319, 41321, 41323, 41325, 41327, 41328, 41329, 41335, 41336, 41338, 41339, 41341, 41389, 41393, 41397, 41399, 41401, 41404, 41406, 41408, 41412, 41444, 41448, 42112

Offset: 1

G. L. Honaker, Jr., Dec 02 2010

nonn

Also, indices of zeros in A092243. - N. J. A. Sloane, Mar 13 2016

N. J. A. Sloane and Chai Wah Wu, Table of n, a(n) for n = 1..20000, terms for n = 1..446 from N. J. A. Sloane.
Chris Caldwell and G. L. Honaker, Jr., Prime Curio for 281

Cf. A051634, A051635, A092243.

my(c=1, q=3, r=2, s=0); print1(c, ", "); forprime(p=5, default(primelimit), c++;(s+=sign(r+0-2*(r=q)+q=p))||print1(c, ", ")) \\ M. F. Hasler, Dec 03 2010

More terms from Chris K. Caldwell

A054811 Fourth term of strong prime quintets: p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m).

Original entry on oeis.org

1667, 1787, 1867, 1871, 1997, 2381, 2473, 2531, 2539, 3457, 3461, 4217, 4517, 5279, 5417, 5441, 6043, 6659, 7243, 7307, 7757, 7877, 7933, 8167, 8521, 9613, 9619, 11057, 11393, 11593, 11831, 12409, 13877, 14827, 15137, 15551, 16061, 16333

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

First member of pairs of consecutive primes in A054807 (4th of strong prime quartets). - M. F. Hasler, Oct 27 2018

M. F. Hasler, Table of n, a(n) for n = 1..2000, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

a(n) = nextprime(A054810(n)) = prevprime(A054812(n)), nextprime = A151800, prevprime = A151799; A054811 = {m = A054807(n) | nextprime(m) = A054807(n+1)}. - M. F. Hasler, Oct 27 2018

A054812 Fifth term of strong prime quintets: p(m-3)-p(m-4) > p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1).

Original entry on oeis.org

1669, 1789, 1871, 1873, 1999, 2383, 2477, 2539, 2543, 3461, 3463, 4219, 4519, 5281, 5419, 5443, 6047, 6661, 7247, 7309, 7759, 7879, 7937, 8171, 8527, 9619, 9623, 11059, 11399, 11597, 11833, 12413, 13879, 14831, 15139, 15559, 16063, 16339

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

Second member of pairs of consecutive primes in A054807 (4th term of strong prime quartets). - M. F. Hasler, Oct 27 2018

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

spqQ[c_]:=Module[{d=Differences[c]},d[[1]]>d[[2]]>d[[3]]>d[[4]]]; Transpose[ Select[Partition[Prime[Range[2000]],5,1],spqQ]][[5]] (* Harvey P. Dale, Jan 01 2013 *)

a(n) = nextprime(A054811(n)); A054811 = {m = A054807(n) | prevprime(m) = A054807(n-1)}; nextprime = A151800, prevprime = A151799. - M. F. Hasler, Oct 27 2018

A054813 First term of strong prime sextets: p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3) > p(m+5)-p(m+4).

Original entry on oeis.org

1831, 2477, 3413, 9551, 21433, 22973, 25189, 26053, 32143, 33359, 33893, 39047, 40771, 41203, 44221, 47251, 48787, 55849, 57751, 66977, 70079, 74231, 74653, 74687, 75083, 75109, 82913, 84263, 87811, 88339, 88609, 103723, 103843, 106219, 106921, 108139, 110881, 112979, 118093

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..2500, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

a(n) = A151799(A054814(n)), A054813 = { m = A054808(n) | m = A151799(A054808(n+1)) }, where A151799 = next smaller prime. - M. F. Hasler, Oct 27 2018

More terms and offset corrected to 1 by M. F. Hasler, Oct 27 2018

A054814 Second term p(m) of strong prime sextets: p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3).

Original entry on oeis.org

1847, 2503, 3433, 9587, 21467, 22993, 25219, 26083, 32159, 33377, 33911, 39079, 40787, 41213, 44249, 47269, 48799, 55871, 57773, 67003, 70099, 74257, 74687, 74699, 75109, 75133, 82939, 84299, 87833, 88379, 88643, 103769, 103867, 106243, 106937, 108161, 110899, 112997, 118127, 120371

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..2500, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

Subsequence of A054808.

Select[Partition[Prime[Range[12000]],6,1],Max[Differences[#,2]]<0&][[;;,2]] (* Harvey P. Dale, Jun 17 2023 *)

a(n) = A151800(A054813(n)) = A151799(A054815(n)), A151800 = nextprime, A151799 = prevprime; A054814 = { m = A054809(n) | m = nextprime(A054809(n-1)) }. - M. F. Hasler, Oct 27 2018

Edited and offset changed to 1 by M. F. Hasler, Oct 26 2018

A054824 Second term of weak prime quintets: p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2).

Original entry on oeis.org

349, 677, 1429, 1489, 1621, 2207, 2239, 2689, 2909, 2917, 4093, 4129, 4933, 5573, 5927, 6271, 6473, 6703, 6829, 8089, 8171, 8233, 8933, 10333, 10733, 11779, 12109, 12281, 13469, 13477, 13903, 13907, 14083, 14629, 14657, 14951, 14957, 15077

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

a(1) = A229832(3). - Jonathan Sondow, Oct 13 2013

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A051635, A054800-A054840, A229832.

wpqQ[{a_,b_,c_,d_,e_}]:=b-aHarvey P. Dale, Jul 29 2019 *)

cp's OEIS Frontend

A054832 Fifth term of weak prime sextet: p(m-3)-p(m-4) < p(m-2)-p(m-3) < p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m).

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A054836 Third term of weak prime septet: p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2) < p(m+4)-p(m+3).

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A054839 Sixth term of weak prime septet: p(m-4)-p(m-5) < p(m-3)-p(m-4) < p(m-2)-p(m-3) < p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m).

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A102552 a(n) = prime(n) - (prime(n+1) + prime(n-1))/2.

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A175102 1, followed by list of numbers n such that the number of strong primes and the number of weak primes are equal at the n-th prime.

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A054811 Fourth term of strong prime quintets: p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m).

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A054812 Fifth term of strong prime quintets: p(m-3)-p(m-4) > p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1).

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A054813 First term of strong prime sextets: p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3) > p(m+5)-p(m+4).

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A054814 Second term p(m) of strong prime sextets: p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3).