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Numbers k such that Sum_{j=0..13} (k + j)^j is a prime.
2

%I #16 Nov 22 2024 16:26:19

%S 25,43,55,109,131,379,389,587,617,649,683,697,719,971,1013,1279,1291,

%T 1727,1735,1823,1853,2005,2059,2087,2167,2395,2399,2561,2647,2783,

%U 2957,2983,3011,3127,3233,3245,3323,3335,3433,3631,3763,3827,3931,3995,4211,4345,4367

%N Numbers k such that Sum_{j=0..13} (k + j)^j is a prime.

%H Chai Wah Wu, <a href="/A138091/b138091.txt">Table of n, a(n) for n = 1..1618</a>

%p q:= n-> isprime(add((n+j)^j, j=0..13)):

%p select(q, [$1..5000])[]; # _Alois P. Heinz_, Nov 22 2024

%t a={};Do[If[PrimeQ[n^70+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7+(n+8)^8+(n+9)^9+(n+10)^10+(n+11)^11+(n+12)^12+(n+13)^13],AppendTo[a,n]],{n,10^3*3}];a

%t Select[Range[3000],PrimeQ[Total[Range[#,#+13]^Range[0,13]]]&] (* _Harvey P. Dale_, Apr 25 2018 *)

%o (Python)

%o from sympy import isprime

%o A138091_list, m = [], [6227020800, 44068147200, 181142438400, 564307430400, 1475073815040, 3408641107200, 7182564530400, 14081919023520, 26048741640120, 45924510262992, 77755456075656, 127171611204708, 201851662963039, 312086923782438]

%o for n in range(1,2*10**5+1):

%o for i in range(13):

%o m[i+1]+= m[i]

%o if isprime(m[-1]):

%o A138091_list.append(n) # _Chai Wah Wu_, Nov 06 2014

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, May 06 2008