A270310 -id:A270310 - cp's OEIS frontend

A269364 Difference between the number of occurrences of prime gaps not divisible by 3, versus number of prime gaps that are multiples of 3, up to n-th prime gap: a(n) = A269849(n) - A269850(n).

1, 2, 3, 4, 5, 6, 7, 8, 7, 8, 7, 8, 9, 10, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10, 11, 12, 13, 14, 15, 16, 17, 16, 17, 18, 19, 18, 17, 18, 17, 16, 17, 18, 19, 20, 21, 20, 19, 20, 21, 22, 21, 22, 23, 22, 21, 20, 21, 20, 21, 22, 23, 24, 25, 26, 27, 28, 27, 28, 29, 30, 29, 30, 29, 28, 29, 28, 29, 30, 31, 32, 33, 34, 35, 34

Offset: 1

Antti Karttunen, Mar 17 2016

nonn

This is related to "Lemke Oliver-Soundararajan bias", term first used by Terence Tao March 14, 2016 in his blog.

Antti Karttunen, Table of n, a(n) for n = 1..50000
Robert J. Lemke Oliver and Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
Terence Tao, Biases between consecutive primes, blog entry March 14, 2016

Cf. A001223, A137264, A269849, A269850, A270189, A270190.

Cf. also A270310, A038698.

a(n) = sum(k=1, n, ((prime(k+1) - prime(k)) % 3) != 0) - sum(k=1, n, ((prime(k+1) - prime(k)) % 3) == 0); \\ Michel Marcus, Mar 18 2016

(define (A269364 n) (- (A269849 n) (A269850 n)))

a(n) = A269849(n) - A269850(n).

A270311 Indices of primes ending with the same decimal digit as the previous or next prime.

Original entry on oeis.org

34, 35, 42, 43, 53, 54, 61, 62, 68, 69, 80, 81, 82, 83, 101, 102, 106, 107, 115, 116, 125, 126, 127, 128, 138, 139, 141, 142, 145, 146, 154, 155, 157, 158, 172, 173, 175, 176, 177, 178, 191, 192, 193, 194, 204, 205, 222, 223, 233, 234, 258, 259, 260, 266, 267, 269, 270, 279, 280, 289, 290, 306, 307

Offset: 1

Francois Alcover, Mar 15 2016

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

Cf. A270310.

Select[Range@ 307, Function[k, Or[k == Mod[NextPrime@ Prime@ #, 10], k == Mod[NextPrime[Prime@ #, -1], 10]]]@ Mod[Prime@ #, 10] &] (* Michael De Vlieger, Mar 15 2016 *)
PrimePi/@Select[Partition[Prime[Range[350]],3,1],Mod[#[[2]],10]==Mod[#[[1]],10]||Mod[#[[2]],10]==Mod[#[[3]],10]&][[;;,2]] (* Harvey P. Dale, Mar 07 2024 *)

More terms from Michael De Vlieger, Mar 15 2016

A272043 a(n) is the shyest prime in base n.

Original entry on oeis.org

2, 3, 31, 13, 523, 31, 3833, 491, 5483, 523, 18149, 661, 44657, 3833, 18869, 7333, 165479, 5483, 153953, 20411, 129127, 18149, 538651, 7079, 932257, 44657, 417037, 52639, 2223773, 18869, 3124217, 175229, 1993763, 165479, 2794811, 50461, 8678963, 153953

Offset: 1

Andy Martin, Apr 18 2016

Terminology: consider pairs of final digits of consecutive primes (a,b). Then of all pairs (3,1) is found last in the prime sequence, corresponding to (523, 541). This is termed the shyest pair, with 523 the shyest prime.
Consider final digit pairs (a,b) of consecutive primes.
There are three unique pairs: (2, 3) (3, 5) (5, 7)
For the remaining 16 pairs, record the first observed primes corresponding to the pair:
Initial prime           -- Second prime (mod 10) ---
      (mod 10)          1        3        7        9
            1     181,191   11, 13   31, 37  401,409
            3     523,541  283,293   13, 17   23, 29
            7       7, 11   47, 53  337,347   17, 19
            9      29, 31   19, 23   89, 97  139,149
523,541 is the largest pair, thus the last to occur in the sequence of primes. The first member of this pair is the shyest prime, base 10. (Note that if we consider two digit pairs (ab, cd) then 40191937, 40192037 is the shyest pair for base 10.)
For base 3 the table is:
Initial prime      Second prime (mod 3)
      (mod  3)     0         1       2
            0      -         -     3,5
            1      -     31,37    7,11
            2      2,3     5,7   23,29
and 31 is the shyest prime base 3.

Giovanni Resta, Table of n, a(n) for n = 1..200
Erica Klarreich, Mathematicians Discover Prime Conspiracy, Quanta Magazine, March 13, 2016
Robert J. Lemke Oliver and Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
Andy Martin, Ruby code output of first 35 terms and additional data
Terence Tao, Biases between consecutive primes, blog entry March 14, 2016.

Cf. A269364, A270310.

a[n_] := Block[{g,p,m,q,k, e= First /@ Select[ Tally[ Mod[ Prime@ Range[n* 100], n]], #[[2]] > 50 &], A}, A = Association@ Table[{i,j} -> 0, {i,e}, {j,e}]; g = Length[e]^2; m=p=2; While[g > 0, q = NextPrime@p; k = Mod[{p, q}, n]; If[ Lookup[A, Key@k, 1] == 0, A[k] = 1; g--]; m=p; p=q]; m]; Array[a, 25] (* Giovanni Resta, Apr 19 2016 *)

require 'Prime'
# Ruby Code
# Generates Hash with first occurrences of all possible pairs (a,b)
# of final digits for consecutive primes in specified base.
def gen_hash(h, base)
  last_prime = 2
  iteration = last_found = 0
  Prime.each() do |prime|
    # This check could be improved & may be invalid for bases above 35.
    return if (iteration+=1) > 10000 && iteration > 2 * last_found
    next if prime == 2
    l =  last_prime.to_s(base)[-1]
    p =  prime.to_s(base)[-1]
    if h[[l,p]].nil?
      h[[l,p]] = [last_prime,prime]
      last_found = iteration
    end
    last_prime = prime
  end
end
puts "First Prime  Second Prime  Base  Difference  Different  Final Digits In"
puts "                                                 Pairs    Base Notation"
puts "          2             3     1           1          1              1 1"
# For bases above 35 additional programming needed.
2.upto(35){|base|
  gen_hash(h = Hash.new, base)
  p0 = h.values.sort.last[0]
  p1 = h.values.sort.last[1]
  printf("%11d  %12d  %4d  %10d %10d              %s %s\n",
  p0, p1, base, p1 - p0, h.length, p0.to_s(base)[-1], p1.to_s(base)[-1])
}

a(22)-a(38) from Giovanni Resta, Apr 19 2016

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A269364 Difference between the number of occurrences of prime gaps not divisible by 3, versus number of prime gaps that are multiples of 3, up to n-th prime gap: a(n) = A269849(n) - A269850(n).

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A270311 Indices of primes ending with the same decimal digit as the previous or next prime.

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A272043 a(n) is the shyest prime in base n.

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