A080376 Numbers where A080374 increases.

2, 4, 9, 24, 30, 34, 99, 189, 217, 282, 367, 738, 3302, 3427, 3644, 3793, 4612, 7970, 8688, 14357, 23283, 34202, 49414, 85633, 85787, 103520, 224659, 273413, 415069, 474029, 685903, 2386432, 2398788, 2959782, 4875380, 6169832, 9330121, 12768473, 13879771, 17681799

Offset: 1

Author

Labos Elemer, Feb 27 2003

Keywords

• nonn

Comments

Numbers where a consecutive prime-difference (prime(a(n)+1)-prime(a(n))) arises with a new prime-power factor.

Examples

			From _Michael De Vlieger_, May 12 2017: (Start)
Values of A080374 starting at a(n).
    n     a(n)   A080374(a(n))
    1       2    1
    2       4    2
    3       9    4
    4      24    12
    5      30    24
    6      34    168
    7      99    840
    8     189    2520
    9     217    27720
   10     282    471240
   11     367    942480
   12     738    12252240
   13    3302    24504480
   14    3427    465585120
   15    3644    2327925600
   16    3793    72165693600
   17    4612    216497080800
   18    7970    6278415343200
   19    8688    144403552893600
   20   14357    288807105787200
   21   23283    12418705548849600
   22   34202    509166927502833600
   23   49414    18839176317604843200
   24   85633    131874234223233902400
   25   85787    6989334413831396827200
...
(End)
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..64 (terms below 10^10)

Crossrefs

Cf. A001223, A080374.

Programs

Mathematica

s=1; Do[s1=s; s=LCM[s, Prime[n+1]-Prime[n]]; If[Greater[s, s1], Print[n]], {n, 1, 100000}]
(* Second program: *)
Most[Accumulate@ #2 + 1] & @@ Transpose@ Map[{First@ #, Length@ #} &, Split@ FoldList[LCM @@ {#1, #2} &, Differences@ Array[Prime, 10^4]]] (* Michael De Vlieger, May 12 2017 *)

PARI

lista(pmax) = {my(k = 1, p1 = 2, lcmmax = 1, lcm1 = 1, d); forprime(p2 = 3, pmax, d = p2 - p1;  lcm1 = lcm(lcm1, d); if(lcm1 > lcmmax, lcmmax = lcm1; print1(k, ", ")); p1 = p2; k++);} \\ Amiram Eldar, Jun 09 2024

Extensions

Edited by N. J. A. Sloane, May 13 2017 at the suggestion of Michael De Vlieger.

More terms from Amiram Eldar, Jun 09 2024