A051453

[Meeussen Wouter (bkarnd)]

%I A051453
%S A051453 1,2,3,4,5,7,9,19,25,31,47,89,127,139
%N A051453 Prime powers such that LCM[1,...,p^w] is prime.
%e A051453 89 is the 34th prime power;
LCM[1,..,89]=718766754945489455304472257065075294401 is also prime.
%Y A051453 A000961, A003418, A049536, A049537.
%K A051453 nonn
%O A051453 1,2
%A A051453 Labos E. (labos at ana1.sote.hu)

"Prime powers such that LCM[1, ..., p^w] is prime" 
should be "Prime powers such that 1+LCM[1,..,p^w] is prime".

Next terms are 1369, 2251, 3271.

I hit on a very similar sequence  : start of runs in A049537

A049537=Select[Range[1, 4000], PrimeQ[1 + LCM @@ Range[#]] &]

{1, 2, 3, 4, 5, 6, 7, 9, 10, 19, 20, 21, 22, 25, 26, 31, 47, 48, 89, 90, 91,
\
92, 93, 94, 95, 96, 127, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, \
1369, 1370, 1371, 1372, 2251, 2252, 2253, 2254, 2255, 2256, 2257, 2258,
2259, \
2260, 2261, 2262, 2263, 2264, 2265, 2266, 3271, 3272, 3273, 3274, 3275,
3276, \
3277, 3278, 3279, 3280, 3281, 3282, 3283, 3284, 3285, 3286, 3287, 3288,
3289, \
3290, 3291, 3292, 3293, 3294, 3295, 3296, 3297, 3298}

{First[#], Length[#]} & /@ Split[A049537, #1 === #2 - 1 &]

{{1, 7}, {9, 2}, {19, 4}, {25, 2}, {31, 1}, {47, 2}, {89, 8}, {127, 1}, 
{139, 10}, {1369, 4}, {2251, 16}, {3271, 28}}


the runs start at :
{1, 9, 19, 25, 31, 47, 89, 127, 139, 1369, 2251, 3271}
runs end at :
{7, 10, 22, 26, 31, 48, 96, 127, 148, 1372, 2266, 3298}

this doe not exactly coincide with the primes, but close:

Select[A049537, PrimeQ[#] &]
{2, 3, 5, 7, 19, 31, 47, 89, 127, 139, 2251, 3271}

the reason for the occurrence of 'runs' in A049537 is not fully clear to me.
I'll hold back on submitting until I see clear. Hints appreciated.

Wouter.





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