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A080376 Numbers where A080374 increases.

%I A080376 #27 Jun 10 2024 08:52:14
%S A080376 2,4,9,24,30,34,99,189,217,282,367,738,3302,3427,3644,3793,4612,7970,
%T A080376 8688,14357,23283,34202,49414,85633,85787,103520,224659,273413,415069,
%U A080376 474029,685903,2386432,2398788,2959782,4875380,6169832,9330121,12768473,13879771,17681799
%N A080376 Numbers where A080374 increases.
%C A080376 Numbers where a consecutive prime-difference (prime(a(n)+1)-prime(a(n))) arises with a new prime-power factor.
%H A080376 Amiram Eldar, <a href="/A080376/b080376.txt">Table of n, a(n) for n = 1..64</a> (terms below 10^10)
%e A080376 From _Michael De Vlieger_, May 12 2017: (Start)
%e A080376 Values of A080374 starting at a(n).
%e A080376     n     a(n)   A080374(a(n))
%e A080376     1       2    1
%e A080376     2       4    2
%e A080376     3       9    4
%e A080376     4      24    12
%e A080376     5      30    24
%e A080376     6      34    168
%e A080376     7      99    840
%e A080376     8     189    2520
%e A080376     9     217    27720
%e A080376    10     282    471240
%e A080376    11     367    942480
%e A080376    12     738    12252240
%e A080376    13    3302    24504480
%e A080376    14    3427    465585120
%e A080376    15    3644    2327925600
%e A080376    16    3793    72165693600
%e A080376    17    4612    216497080800
%e A080376    18    7970    6278415343200
%e A080376    19    8688    144403552893600
%e A080376    20   14357    288807105787200
%e A080376    21   23283    12418705548849600
%e A080376    22   34202    509166927502833600
%e A080376    23   49414    18839176317604843200
%e A080376    24   85633    131874234223233902400
%e A080376    25   85787    6989334413831396827200
%e A080376 ...
%e A080376 (End)
%t A080376 s=1; Do[s1=s; s=LCM[s, Prime[n+1]-Prime[n]]; If[Greater[s, s1], Print[n]], {n, 1, 100000}]
%t A080376 (* Second program: *)
%t A080376 Most[Accumulate@ #2 + 1] & @@ Transpose@ Map[{First@ #, Length@ #} &, Split@ FoldList[LCM @@ {#1, #2} &, Differences@ Array[Prime, 10^4]]] (* _Michael De Vlieger_, May 12 2017 *)
%o A080376 (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
%Y A080376 Cf. A001223, A080374.
%K A080376 nonn
%O A080376 1,1
%A A080376 _Labos Elemer_, Feb 27 2003
%E A080376 Edited by _N. J. A. Sloane_, May 13 2017 at the suggestion of _Michael De Vlieger_.
%E A080376 More terms from _Amiram Eldar_, Jun 09 2024