A054835 -id:A054835 - cp's OEIS frontend

A229832 First term of smallest sequence of n consecutive weak primes.

3, 19, 349, 2909, 15377, 128983, 1319411, 17797519, 94097539, 6927837559, 48486712787, 968068681519, 1472840004019, 129001208165719

Offset: 1

Jonathan Sondow, Oct 13 2013

Erdős called a weak prime A051635 an "early prime," defined to be one which is less than the arithmetic mean of the prime before it and the prime after it. He conjectured that there are infinitely many consecutive pairs of early primes, and offered $100 for a proof and $25000 for a disproof. See Kuperberg 1992.
I make the stronger conjecture that the sequence a(n) is infinite.
a(1) = A051635(1), a(2) = A054820(1), a(3) = A054824(1), a(4) = A054829(1), a(5) = A054835(1).
a(n) is the prime following A158939(n+1). [Follows from the definitions] - Chris Boyd, Mar 28 2015

			The primes 19 < (17+23)/2 and 23 < (19+29)/2 are the smallest pair of consecutive weak/early primes, so a(2) = 19.

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. A051634, A051635, A054820, A054824, A054829, A054835, A158939.

a(n) = min{p(i): 2*p(i+j) < p(i+j-1) + p(i+j+1), j = 0,1,..,n-1}.

a(6) corrected by and a(7)-a(13) from Giovanni Resta, Jan 16 2014

a(14) from Giovanni Resta, Apr 19 2016

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

A054834 First term of weak prime septet: 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) < p(m+6)-p(m+5).

Original entry on oeis.org

15373, 64919, 68207, 68897, 128981, 128983, 143509, 154079, 157999, 192373, 221717, 222379, 244457, 249721, 285287, 318677, 337277, 354371, 357823, 374173, 385391, 394727, 402581, 402583, 419597, 439157, 441907, 448373, 457397, 457669, 458189, 482507, 527981, 529811, 577529, 582761, 655909

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..1000, Oct 27 2018
Shyam Sunder Gupta, Elegance of Squares, Cubes, and Higher Powers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 2, 29-81.

Cf. 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.

Select[Partition[Prime[Range[54000]],7,1],Min[Differences[#,2]]>0&][[All,1]] (* Harvey P. Dale, Mar 16 2020 *)

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

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

cp's OEIS Frontend

A229832 First term of smallest sequence of n consecutive weak primes.

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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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A054834 First term of weak prime septet: 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) < p(m+6)-p(m+5).

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