A229832 -id:A229832 - cp's OEIS frontend

A051635 Weak primes: prime(n) < (prime(n-1) + prime(n+1))/2.

3, 7, 13, 19, 23, 31, 43, 47, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 167, 181, 193, 199, 229, 233, 241, 271, 283, 293, 313, 317, 337, 349, 353, 359, 383, 389, 401, 409, 421, 433, 443, 449, 463, 467, 491, 503, 509, 523, 547, 571, 577, 601, 619, 643, 647

Offset: 1

Felice Russo, Nov 15 1999

Primes prime(n) such that prime(n)-prime(n-1) < prime(n+1)-prime(n). - Juri-Stepan Gerasimov, Jan 01 2011
a(n) < A051634(n). a(n) ~ 2*prime(n). - Thomas Ordowski, Jul 25 2012
The inequality above is false. The least counterexample is a(19799) = 496291 > A051634(19799) = 496283. - Amiram Eldar, Nov 26 2023
Erdős called a weak prime an "early prime." He conjectured that there are infinitely many consecutive pairs of early primes, and offered $100 for a proof and $25000 for a disproof (Kuperberg 1992). See A229832 for a stronger conjecture. - Jonathan Sondow, Oct 13 2013

			7 belongs to the sequence because 7 < (5+11)/2.

A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Greg Kuperberg, The Erdos kitty: At least $9050 in prizes!, Newsgroups: rec.puzzles, sci.math, 1992. [Broken link]
Greg Kuperberg, The Erdos kitty: At least $9050 in prizes!, Newsgroups: rec.puzzles, sci.math, 1992. [Cached copy]
Wikipedia, Weak prime.

Cf. A006562, A051634, A229832.
Subsequence of A178943.
Cf. A225495 (multiplicative closure).

a051635 n = a051635_list !! (n-1)
a051635_list = g a000040_list where
   g (p:qs@(q:r:ps)) = if 2 * q < (p + r) then q : g qs else g qs
-- Reinhard Zumkeller, May 09 2013

Transpose[Select[Partition[Prime[Range[10^2]], 3, 1], #[[2]]<(#[[1]]+#[[3]])/2 &]][[2]] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
p=Prime[Range[200]]; p[[Flatten[1+Position[Sign[Differences[p, 2]], 1]]]]

p=2;q=3;forprime(r=5,1e3,if(2*qCharles R Greathouse IV, Jul 25 2011

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

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/2. - Alain Rocchelli, Mar 17 2024

More terms from James Sellers

A051634 Strong primes: prime(k) > (prime(k-1) + prime(k+1))/2.

Original entry on oeis.org

11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499, 521, 541, 557, 569, 587, 599, 613, 617, 631, 641, 659, 673, 701

Offset: 1

Felice Russo, Nov 15 1999

Prime(k) such that prime(k) - prime(k-1) > prime(k+1) - prime(k). - Juri-Stepan Gerasimov, Jan 01 2011
a(n) > A051635(n). - Thomas Ordowski, Jul 25 2012
The inequality above is false. The least counterexample is a(19799) = 496283 < A051635(19799) = 496291. - Amiram Eldar, Nov 26 2023
Conjecture: Limit_{N->oo} Sum_{n=1..N} (NextPrime(a(n))-a(n)) / a(N) = 1/4. [A heuristic proof is available at www.primepuzzles.net - Conjecture 91] - Alain Rocchelli, Nov 14 2022
A131499 is a subsequence. - Davide Rotondo, Oct 16 2023

			11 belongs to the sequence because 11 > (7 + 13)/2.

A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Carlos Rivera, Conjecture 91. A conjecture about strong primes, The Prime Puzzles & Problems Connection.

Cf. A006562, A051635, A229832.
Subsequence of A178943.
Cf. A225493 (multiplicative closure), A131499 (subsequence).

a051634 n = a051634_list !! (n-1)
a051634_list = f a000040_list where
   f (p:qs@(q:r:ps)) = if 2 * q > (p + r) then q : f qs else f qs
-- Reinhard Zumkeller, May 09 2013

q:= n-> isprime(n) and 2*n>prevprime(n)+nextprime(n):
select(q, [$3..1000])[];  # Alois P. Heinz, Jun 21 2023

Transpose[Select[Partition[Prime[Range[10^2]], 3, 1], #[[2]]>(#[[1]]+#[[3]])/2 &]][[2]] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
p=Prime[Range[200]]; p[[Flatten[1+Position[Sign[Differences[p,2]], -1]]]]

p=2;q=3;forprime(r=5,1e4,if(2*q>p+r,print1(q", "));p=q;q=r) \\ Charles R Greathouse IV, Jul 19 2011

from sympy import nextprime
def aupto(limit):
    alst, p, q, r = [], 2, 3, 5
    while q <= limit:
        if 2*q > p + r: alst.append(q)
        p, q, r = q, r, nextprime(r)
    return alst
print(aupto(701)) # Michael S. Branicky, Nov 17 2021

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/2. - Alain Rocchelli, Mar 17 2024

A158939 First primes followed by sequences of exactly n monotonic increasing prime gaps.

Original entry on oeis.org

3, 2, 17, 347, 2903, 15373, 128981, 1319407, 17797517, 94097537, 6927837557, 48486712783, 968068681511, 1472840004017, 129001208165717

Offset: 1

Alan Worley (aw(AT)xiboo.co.uk), Mar 31 2009

For n > 1, a(n) is the prime preceding A229832(n-1). [Follows from the definitions] - Chris Boyd, Mar 28 2015

Banks, Freiberg, & Turnage-Butterbaugh show that a(n) exists for each n. - Charles R Greathouse IV, Jun 30 2022

			a(8)=1319407 is the first prime to be followed by n=8 monotonic increasing prime gaps: 4,8,10,14,16,18,32,34.
a(14)=1472840004017 is the first prime to be followed by n=14 monotonic increasing prime gaps: 2,4,6,8,10,12,14,28,30,38,48,64,66,74.

William D. Banks, Tristan Freiberg, and Caroline L. Turnage-Butterbaugh, Consecutive primes in tuples (2013), arXiv:1311.7003 [math.NT].
Carlos Rivera, Puzzle 11. Distinct, Increasing & Decreasing Gaps, The Prime Puzzles and Problems Connection.

Cf. A158940 (monotonic decreasing prime gaps), A229832.

Cf. A133697.

is(p,k,g=0)=my(q=nextprime(p+1));if(g>=q-p,0,if(k>1,is(q,k-1,q-p),q-p>=nextprime(q+1)-q))
a(n)=forprime(p=2,,if(is(p,n),return(p))) \\ Charles R Greathouse IV, Nov 02 2012

a(15) from Giovanni Resta, Apr 19 2016

A054835 Second term of weak prime septet: 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) < p(m+5)-p(m+4).

Original entry on oeis.org

15377, 64921, 68209, 68899, 128983, 128987, 143513, 154081, 158003, 192377, 221719, 222389, 244463, 249727, 285289, 318679, 337279, 354373, 357829, 374177, 385393, 394729, 402583, 402587, 419599, 439163, 441913, 448379, 457399, 457673, 458191, 482509, 527983, 529813, 577531, 582763, 655913

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, A229832.

Cf. 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(1) = A229832(5). - Jonathan Sondow, Oct 13 2013

a(n) = A151800(A054834(n)) = A151799(A054836(n)), A151800 = nextprime, A151799 = prevprime; A054835 = { m = A054828(n) | m = nextprime(A054828(n-1)) }. - M. F. Hasler, Oct 27 2018

More terms from M. F. Hasler, Oct 27 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 *)

A054829 Second term of weak prime sextet: 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

2909, 13469, 13903, 14951, 15377, 15383, 21401, 21559, 21863, 28279, 30871, 33203, 35593, 37693, 42223, 42571, 45823, 55663, 64663, 64921, 64927, 68209, 68213, 68899, 68903, 73943, 74203, 78583, 83093, 85517, 87317, 88003, 90911

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

a(1) = A229832(4). - Jonathan Sondow, Oct 13 2013

Cf. A051635, A054800-A054840, A229832.

wps[{a_,b_,c_,d_,e_,f_}]:=b-aHarvey P. Dale, Jun 25 2013 *)

A054820 Second term of weak prime quartet: p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1).

Original entry on oeis.org

19, 43, 83, 109, 229, 283, 313, 349, 353, 383, 401, 443, 463, 503, 571, 643, 677, 683, 829, 859, 883, 911, 1033, 1063, 1093, 1193, 1231, 1279, 1303, 1321, 1373, 1429, 1433, 1453, 1489, 1493, 1553, 1609, 1621, 1627, 1699, 1879, 1999, 2029, 2089, 2161

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

Lesser of a pair p(m), p(m+1) of consecutive weak primes A051635. See A229832 for more comments, references and links. - Jonathan Sondow, Oct 13 2013

a(1) = A229832(2). - Jonathan Sondow, Oct 13 2013

Cf. A051635, A054800-A054840, A229832.

A235874 First term of the earliest sequence of n consecutive strong primes.

Original entry on oeis.org

11, 37, 1657, 1847, 74687, 322193, 5051341, 11938853, 245333213, 397597169, 130272314657, 1273135176871

Offset: 1

Giovanni Resta, Jan 16 2014

nonn

A strong prime is a prime p(n) such that p(n) > (p(n-1) + p(n+1))/2.

			a(2) = 37 because the two consecutive primes 37 and 41 are both strong and are the first such pair.

Cf. A051634, A229832.

cp's OEIS Frontend

A051635 Weak primes: prime(n) < (prime(n-1) + prime(n+1))/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A051634 Strong primes: prime(k) > (prime(k-1) + prime(k+1))/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A158939 First primes followed by sequences of exactly n monotonic increasing prime gaps.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A054835 Second term of weak prime septet: 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) < p(m+5)-p(m+4).

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A054829 Second term of weak prime sextet: 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).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A054820 Second term of weak prime quartet: p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1).

Views

Author

Keywords

Comments

Crossrefs

A235874 First term of the earliest sequence of n consecutive strong primes.

Views

Author

Keywords

Comments

Examples