A359403 - OEIS

%I #8 Dec 30 2022 06:31:08

%S 149,11681,164729,1102459049,1262144387420489,

%T 1472236648286964521369622528399544939174411840147874772641,

%U 1755578637259143234191361824800363140073127359051977856583921

%N Primes that are the concatenation of the first m consecutive k-th powers.

%C The following comments were moved from A284377 when this sequence was created.

%C From _XU Pingya_, Mar 25 2017: (Start)

%C There are 2 primes in the first 1000 terms of A284377, a(3) = 11681 and a(637) = 11681256...164648481361 (which has 6368 digits).

%C Let cp(k, n) denote the concatenation of the first n k-th powers. Note that there are no primes in cp(1, n) = A007908(n) up to n = 64000 or cp(3, n) = A019522(n) up to n = 31152. But there is a prime in cp(2, n) = A019521(n): cp(2, 3) = A019521(3) = 149. Conjecture: if k is odd, no "smaller primes" (< 10000 digits, or non-gigantic primes) exist in cp(k, n); if k is even, then such primes may occur. So far, the expanded search supports this conjecture. For k from 5 to 500, when k is an odd number, we have found no "smaller primes" in cp(k, n); for even k, the following primes have been found:

%C k = 6, cp(6, 3) = 164729.

%C k = 10, cp(10, 3) = 1102459049.

%C k = 18, cp(18, 3) = 1262144387420489.

%C k = 72, cp(72, 3) = 1472236648286964521369622528399544939174411840147874772641.

%C k = 76, cp(76, 3) = 1755578637259143234191361824800363140073127359051977856583921.

%C k = 108, cp(108, 3) = 13245185536...198527451848561 (a 86-digit prime).

%C k = 124, cp(124, 7) = 12126764793...315965558474401 (a 463-digit prime).

%C k = 432, cp(432, 7) = 11109067877...198797825539201 (a 1605-digit prime).

%C However, the next search contradicted the above conjecture. For k from 501 to 1100, we get two counterexamples:

%C k = 543, cp(543,13) = 12879304828...134856329859397 (a 5325-digit prime).

%C k = 815, cp(815,7) = 12184974969...385767566149943 (a 3021-digit prime).

%C And finally, by testing all cp(k, n) less than 10^10000 (using Mathematica's "PrimeQ[*]" function), it is confirmed that there are 15 "smaller primes" in all concatenations of k-th powers (k >= 1), of which 13 are listed above, and the other two are

%C k = 2140, cp(2140, 3) = 11600260630...045844060490801 (a 1608-digit prime) and

%C k = 2176, cp(2176, 3) = 11099690731...621473725585921 (a 1696-digit prime). (End)

%Y Cf. A007908, A019521, A019522, A284377.

%K nonn,base

%O 1,1

%A _Michel Marcus_, Dec 30 2022