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%I #16 Sep 08 2022 08:45:08
%S 79,4259,322079,1391239,9416279,57229199,180621149,583332179,
%T 666475879,758206139,1090726199,1366313119,1780616609,3017394239,
%U 3297759379,3754381249,8119069289,9639336299,12901941509,16441316629,17940244339
%N Pinomial primes of order 4: primes of the form 3*x^4 + 1*x^3 + 4*x^2 + 1*x + 5, x a positive integer.
%C Note that only primes that end in 9 occur in the sequence of order 4. This is easy to prove by considering the ending digit combinations. actually, order 4 with odd x produces an even number so only even numbers need be considered.
%C For order 5, x<= 10, y = 3*x^5 + x^4 + 4*x^3 + x^2 + 5*x + 9. Sequence is 23 167 10559 54287 104561 314159 ...
%C Allowing any integral value for x, the sequence would read: 5, 59, 79, 769, 4259, 113279, 310379, 322079, 694039, 983789, 1391239, 1825129, 9416279,... (M. F. Hasler)
%C One could also reverse the order of polynomial coefficients, i.e. list primes p = 3 + 1*x + 4*x^2 + 1*x^3 + 5*x^4. For a base independent approach, one could take coefficients from the continued fraction expansion of Pi. - _M. F. Hasler_, Jun 17 2007
%H Vincenzo Librandi, <a href="/A078114/b078114.txt">Table of n, a(n) for n = 1..1000</a>
%F Only even x can give a prime.
%t Select[Table[3*n^4+n^3+4*n^2+n+5,{n,1,300}],PrimeQ] (* _Vincenzo Librandi_, Jul 15 2012 *)
%o (PARI) /* This program is hard-wired for order 4. Change the polynomial y for more terms. */ pinomial(n) = { ct=0; sr=0; for(x=1,n, /* odd x => even y */ y = 3*x^4 + x^3 + 4*x^2 + x + 5; /* y = 3*x^5 + x^4 + 4*x^3 + x^2 + 5*x + 9; /* order 5*/ if(isprime(y),ct+=1; print1(y" "); ); ) }
%o (PARI) polyprimes( N=25 /*# of terms*/, P=[3,1,4,1,5], X=1/*starting value*/, a=[] )={ local(t); P=Pol(P); while( #a<N, if( isprime( t=subst( P,x,X )) & !setsearch(Set(a),t), /*print([X,t]);*/ a=concat(a,t)); X++/*or: X=(X<=0)-X to get all integers*/); vecsort(a) } /* then polyprimes() gives the first 25 terms; polyprimes(50,[null],0) gives 50 terms including the term 5=P(x=0), etc. */ \\ _M. F. Hasler_, Jun 17 2007
%o (Magma) [a: n in [1..300] | IsPrime(a) where a is 3*n^4+n^3+ 4*n^2+ n+5]; // _Vincenzo Librandi_, Jul 15 2012
%K easy,nonn,base,less
%O 1,1
%A _Cino Hilliard_, Dec 04 2002
%E Edited by _M. F. Hasler_, Jun 17 2007
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