cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A140387 Binary encoding of the location of primes in integer sets r+30*n with remainder r=1,7,11,..,29.

Original entry on oeis.org

1, 32, 16, 129, 73, 36, 194, 6, 42, 176, 225, 12, 21, 89, 18, 97, 25, 243, 44, 44, 196, 34, 166, 90, 149, 152, 109, 66, 135, 225, 89, 169, 169, 28, 82, 210, 33, 213, 179, 170, 38, 92, 15, 96, 252, 171, 94, 7, 209, 2, 187, 22, 153, 9, 236, 197, 71, 179, 212, 197, 186
Offset: 1

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Author

Juri-Stepan Gerasimov, Jun 10 2008

Keywords

Comments

Classify all integers 30n+r with r= 1, 7, 11, 13, 17, 19, 23 or 29 as nonprime or prime and assign bit positions 0=LSB, 1, 2, 3, .., 7=MSB to the 8 remainders in the same order. Raise the bit if 30n+r is nonprime, erase it if 30n+r is prime.
The sequence interprets this as a number in base 2 and shows the decimal representation.

Examples

			For n=1, the 8 numbers 31 (r=1), 37 (r=7), 41 (r=11), 43 (r=17), 47 (r=17), 49 (r=19), 53 (r=23) and 59 (r=29) are prime, prime, prime, prime, prime, nonprime, prime, prime, prime, which is rendered into the binary 000100000 = 2^5=32=a(1).

Crossrefs

Cf. A132236, A132235, A132234, A039919, A132233, A132232, A136066, A140378.
Cf. A105052 (analog in base 10, prime = bit 1, remainder 1 = MSB), A140891 (analog in base 14, prime = bit 0, remainder 1 = LSB).

Extensions

Edited by R. J. Mathar, Jun 17 2008

A140889 Lengths of runs of consecutive primes and composites in A008364.

Original entry on oeis.org

1, 26, 1, 4, 1, 5, 1, 3, 1, 4, 1, 1, 1, 6, 1, 1, 1, 7, 1, 1, 1, 4, 2, 2, 1, 4, 1, 2, 1, 3, 1, 2, 2, 5, 1, 3, 1, 4, 1, 1, 1, 2, 1, 3, 1, 2, 3, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 3, 1, 4, 1, 3, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 2, 2, 1, 1, 1, 2, 2, 1, 5, 2, 4, 2, 4, 3, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jul 06 2008

Keywords

Comments

Primes can be classified according to their remainder modulo 210: remainder 1 (A073102), 11..113 (primes), 121 (composite), 127..139 (primes), 143 (composite), 149..167 (primes), 169 (composite), 173..181 (primes), 187 (composite), 191..199 (primes), or 209 (composite). In the sequence A008364 of all numbers (prime or composite) in any of these remainder classes, we look for runs of numbers that are successively prime or composite and place the lengths of these runs in this sequence.

Examples

			Groups of runs in A008364 are (1), (11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113), (121), (127, 131, 137, 139), (143), (149, 151, ... ), which is 1 composite followed by 26 primes followed by 1 composite followed by 4 primes etc.

Crossrefs

Cf. A140378.

Extensions

First number in the comment corrected and entries checked by R. J. Mathar, Apr 28 2010

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

Original entry on oeis.org

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

Views

Author

Harry E. Neel, Oct 16 2021

Keywords

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

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
Showing 1-3 of 3 results.

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