# A051453

Meeussen Wouter (bkarnd) wouter.meeussen at vandemoortele.com
Fri Apr 18 12:55:48 CEST 2003

```%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.

===============================
This email is confidential and intended solely for the use of the individual to whom it is addressed.
If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited.
You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

```