A180138 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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), ..., .

LINKS

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

EXAMPLE

.\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

...

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}]

PROG

(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)]