A180138 - OEIS

%I #22 Jul 09 2024 19:42:22

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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, ..., .

%C 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, ...,

%C 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, ..., .

%C 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..., .

%C 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), ..., .

%H Robert G. Wilson v, <a href="/A180138/b180138.txt">Table of n, a(n) for n = 1..10153</a>

%e .\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

%e .b\

%e .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

%e .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

%e .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

%e .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

%e .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

%e .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

%e .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

%e .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

%e 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

%e 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

%e 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

%e 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

%e ...

%t 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

%t (* 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}]

%o (Python)

%o from sympy import isprime, nextprime, prevprime

%o def sum2succ(n):

%o   if n <= 5: return n == 5

%o   return not isprime(n//2) and n == prevprime(n//2) + nextprime(n//2)

%o def T(b, e):

%o   k, powb = 1, b**e

%o   while not sum2succ(k*powb): k += 1

%o   return k

%o def atodiag(maxd): # maxd antidiagonals

%o   return [T(b-e, e) for b in range(2, maxd+2) for e in range(b-1)]

%o print(atodiag(13)) # _Michael S. Branicky_, May 05 2021

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

%K base,nonn,tabl

%O 1,1

%A _Zak Seidov_ & _Robert G. Wilson v_, Aug 15 2010