A180138 Table, t, read by antidiagonals: t(b,e) is the smallest k such that k*b^e is a sum of two successive primes.

5, 5, 4, 5, 4, 2, 5, 2, 2, 1, 5, 1, 7, 6, 7, 5, 2, 4, 2, 2, 4, 5, 6, 1, 10, 9, 10, 2, 5, 1, 18, 1, 2, 8, 20, 1, 5, 2, 2, 10, 4, 8, 2, 26, 9, 5, 3, 2, 15, 30, 12, 12, 25, 22, 15, 5, 18, 1, 20, 2, 18, 2, 12, 11, 10, 8, 5, 1, 6, 6, 22, 19, 4, 1, 36, 6, 16, 4, 5, 4, 1, 24, 6, 16, 6, 28, 4, 12, 10, 8, 2

Offset: 1

Author

Zak Seidov & Robert G. Wilson v, Aug 15 2010

Keywords

• base
• nonn
• tabl

Comments

1st row: A180130, 2nd row: A180131, 3rd row: bisection of A180130, 4th row: A180132, 5th row: A180133, 6th row: A180134, 7th row: trisection of A180130, 8th row: bisection of A180131, 9th row: A179975, 10th row: A180135, 11th row: A180136 and 12th row: A180137; 1st column: A010716.

The k-th term == 1 10, 12, 24, 30, 32, 36, 58, 68, 74, 81, 105, 155, 278, 303, 315, 331, 419, 437, 439, 632, 638, 752, 857, 863, 906, 924, 950, ..., .

Increasing terms: {5, 6, 10, 20, 26, 72, 104, 118, 306, 320, 348, 572, 824, 828, 972, 1054, 1110, 1540, 5, 7, 10, 18, 20, 26, 30, 36, 52, 66, 72, 120, 132, 168, 266, 574, 640, 776, 1600, 1938, 2616, 3124, 3306, 4440, ...,

which occurs at the k-th term: 5, 6, 10, 20, 26, 72, 104, 118, 306, 320, 348, 572, 824, 828, 972, 1054, 1110, 1540, 5, 7, 10, 18, 20, 26, 30, 36, 52, 66, 72, 120, 132, 168, 266, 574, 640, 776, 1600, 1938, 2616, 3124, 3306, 4440, 1, 13, 25, 31, 35, 44, 50, 75, 114, 117, 119, 166, 187, 267, 289, 615, 1416, 1575, 2069, 3463, 4840, 5968, 7709, 9695, ..., .

Increasing terms by antidiagonals: t(2,0)=5, t(4,2)=t(2,4)=7, t(5,3)=t(3,5)=10, t(3,6)=20, t(3,7)=26, t(7,4)=30, t(5,8)=36, t(3,13)=72, t(7,12)=120, t(5,15)=132, t(11,13)=168, t(13,12)=266, t(17,19)=574, t(17,37)=640, t(23,34)=776, t(13,52)=1600, t(25,59)=1938, t(13,86)=2616. t(29,81)=3124, t(43,82)=3306, t(37,103)=4440..., .

Corresponding primes are twin primes for t(18,2), t(24,2), t(54,6), t(60,5), t(72,6), t(102,8), t(114,1), t=(126,1), ..., .

Examples

			.\e..0...1...2...3...4...5...6...7...8...9..10..11..12..13..14..15..16..17..18..19..20..21..22..23..24..25
.b\
.2...5...4...2...1...7...4...2...1...9..15...8...4...2...1..25..19..11..12...6...3..10...5..35..33..52..26
.3...5...4...2...6...2..10..20..26..22..10..16...8...8..72..24...8..18...6...2...6...2..10..20..20..22..20
.4...5...2...7...2...9...8...2..25..11...6..10..35..52..13..14..15..19..47..13..84..21..35...9..23..49..52
.5...5...1...4..10...2...8..12..12..36..12..28..66..30...6..18.132..36.108..34..14..48..60..12..22.150..30
.6...5...2...1...1...4..12...2...1...4...3...5...8...7..34...8..11..33..26..13...9..13..90..15..40..30...5
.7...5...6..18..10..30..18...4..28...4..30..30..60.120..38..12...6..52.120..70..10.102..60..70..10.186.174
.8...5...1...2..15...2..19...6...5..52..28..15..45..13..42..35..46..49..26..24...3..18..15..21..62..32...4
.9...5...2...2..20..22..16...8..24..18...2...2..20..22..52.104..84..38.102.100..30.192..46..22..84.176..30
10...5...3...1...6...6...6..14...6...9..19..21..21..42..93..21...6..11...2..12.111..37..39..63..38..42..24
11...5..18...6..24...6..32..40..26..20..94..50..26..10.168..30..18.196.126..70.166..30..54.130..26..50..10
12...5...1...1...2..18...8..13...6...2..11..11..39..20..12...1...8...9..31.182..24...2.126.128..66...9..86
13...5...4..24...4...8..22..40...4..14..16..28..10.266..40..20..46.112.156..12..20.228..26...2.220..60.140
...
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10153

Crossrefs

Cf. A180130, A180131, A180132, A180133, A180134, A179975, A180135, A180136, A180137.

Programs

Mathematica

t[b_, e_] := Block[{k = 1, hnp = b^e/2}, While[ h = k*hnp; PrimeQ@h || NextPrime[h, -1] + NextPrime@h != 2 h, k++ ]; k]; Table[ t[b - e, e], {b, 2, 14}, {e, 0, b - 2}] // Flatten
(* to find twins other than 2&3 *) gQ[b_, e_, k_] := Block[{h = k*b^e/2}, NextPrime@h - NextPrime[h, -1] < 3 ]; Do[ If[ gQ[b - e, e, k], Print[{b - e, e}]], {b, 2, 143}, {e, 0, b - 2}]

Python

from sympy import isprime, nextprime, prevprime
def sum2succ(n):
  if n <= 5: return n == 5
  return not isprime(n//2) and n == prevprime(n//2) + nextprime(n//2)
def T(b, e):
  k, powb = 1, b**e
  while not sum2succ(k*powb): k += 1
  return k
def atodiag(maxd): # maxd antidiagonals
  return [T(b-e, e) for b in range(2, maxd+2) for e in range(b-1)]
print(atodiag(13)) # Michael S. Branicky, May 05 2021