A348394 Primes preceding record runs of composites coprime to 30 (A007775).

7, 47, 113, 317, 523, 1327, 9551, 15683, 19609, 25471, 31397, 155921, 360653, 370261, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783, 387096133, 1294268491, 1453168141, 2300942549, 3842610773, 4302407359, 10726904659, 20678048297, 22367084959, 25056082087, 42652618343, 127976334671, 182226896239

Offset: 1

Author

Harry E. Neel, Oct 16 2021

Keywords

• nonn

Comments

"There are 8 potential primes modulo 30...." Using only the potential prime locations within this domain there are no consecutive integers within the domain until an integer is determined to have a prime factor, here the first such integer is 49. When an integer is determined to be composite then there is a "gap" within the succession of primes.

While the location of the first few record consecutive integers differ from established maximal gaps, they quickly become the same. It is not known if they continue to remain the same or if some variation may occur. Here the record number of composites will always be lower because the count of composites are only those that are within this domain.

Hugo van der Sanden greatly expanded the data contained in this sequence.

Examples

			The next number coprime to 30 after 7 is 11, giving a run of 0 composites.
47 is followed by 49 = 7^2 and 53 (prime), a run of 1 composite.
113 is followed by 119 = 7*17, 121 = 11^2, and 127 (prime), a run of 2 composites.
The first few entries correspond to the following table. The table contains the order in which record composites occur (n), the number of composites between successive primes (gap size), the prime preceding the record composites (1st prime), the prime following the record composites (2nd prime) and the merit of the gap (merit) rounded to 4 decimals. The merit is the gap size divided by the natural log of the 1st prime (gap size / log(1st prime)).
   n  gap size 1st prime  2nd prime   gap merit
   1,    0,        7,          11,      0.0000
   2,    1,       47,          53,      0.2597
   3,    2,      113,         127,      0.4231
   4,    3,      317,         331,      0.5209
   5,    4,      523,         541,      0.6390
   6,    8,     1327,        1361,      1.1126
   7,    9,     9551,        9587,      0.9821
   8,   10,    15683,       15727,      1.0352
   9,   12,    19609,       19661,      1.2141
  10,   13,    25471,       25523,      1.2814
  11,   18,    31397,       31469,      1.7384
  12,   22,   155921,      156007,      1.8399
  13,   24,   360653,      360749,      1.8756
  ...
  38,  125, 182226896239, 182226896713,  4.8209
		

Links

• C. K. Caldwell, Table of Known Maximal Gaps
• P. H. Fry, J. Nesheivat and B. K. Szymanski, Computing Twin Primes and Brun's Constant: A Distributed Approach, IEEE Computer Society Press, 1998, pages 42-49.

Crossrefs

Cf. A002386, A007775, A140378, A000040, A001223, A038510.

Programs

Mathematica

Block[{m = Select[Range[29], CoprimeQ[#, 30] &], s, t}, s = Reap[Array[Map[If[! PrimeQ[#], Sow[#]] &, 30 # + m] &, 2^20]][[-1, -1]]; Set[{s, t}, Transpose@ #] &@ Tally@ Array[NextPrime[s[[#]], -1] &, Length@ s]; Map[s[[FirstPosition[t, #][[1]]]] &, Union@ FoldList[Max, t]] ] (* Michael De Vlieger, Oct 25 2021 *)

PARI

isok(x) = vecsearch([1, 7, 11, 13, 17, 19, 23, 29], x%30);
nbc(n, v) = {my(i=n+1, c= v[i], nb=0); while(!isprime(c), nb++; i++; if (i>#v, return(-1)); c = v[i]); nb;}
lista(nn) = {my(v = [2..nn], m=-1, nb); v = select(x->isok(x), v); v = apply(isprime, v); for (n=1, #v-1, if (isprime(v[n]), nb = nbc(n, v); if (nb==-1, break); if (nb > m, print1(v[n], ", "); m = nb);););} \\ Michel Marcus, Oct 21 2021