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A100478 Pentanacci pi function: a(1)=a(2)=a(3)=a(4)=a(5)=1; for n>5, a(n) = pi(Sum_{j=1..5} a(n-j)) where pi = A000720.

%I A100478 #20 Feb 16 2025 08:32:55
%S A100478 1,1,1,1,1,3,4,4,6,7,9,10,11,14,15,17,19,21,23,24,27,30,30,32,34,36,
%T A100478 37,39,40,42,44,46,47,47,48,50,51,53,53,54,55,56,58,58,60,61,62,62,62,
%U A100478 63,63,64,65,66,66,66,66,66,66,66,66,66,66,66,66,66,66,66,66,66,66,66,66
%N A100478 Pentanacci pi function: a(1)=a(2)=a(3)=a(4)=a(5)=1; for n>5, a(n) = pi(Sum_{j=1..5} a(n-j)) where pi = A000720.
%C A100478 Starting with other values of a(1), a(2), a(3), a(4), a(5) what behaviors are possible? Does the sequence always stick at a single integer after some point, or can it go into a loop, or is there a third pattern?
%C A100478 a(n) is equal to 66 for 54 <= n <= 10^7. - _G. C. Greubel_, Apr 06 2023
%H A100478 G. C. Greubel, <a href="/A100478/b100478.txt">Table of n, a(n) for n = 1..10000</a>
%H A100478 Andrew Booker, <a href="https://t5k.org/nthprime/index.php#piofx">The Nth Prime Page</a>.
%H A100478 I. Flores, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/5-3/flores.pdf">k-Generalized Fibonacci numbers</a>, Fib. Quart., 5 (1967), 258-266.
%H A100478 V. E. Hoggatt, Jr. and M. Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-4/hoggatt-a.pdf">Diagonal sums of generalized Pascal triangles</a>, Fib. Quart., 7 (1969), 341-358, 393.
%H A100478 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>
%F A100478 a(n) = pi(a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5)) with a(1) = a(2) = a(3) = a(4) = a(5) = 1.
%e A100478 a(6) = pi(a(1)+a(2)+a(3)+a(4)+a(5)) = pi(1+1+1+1+1) = pi(5) = 3.
%e A100478 a(7) = pi(a(2)+a(3)+a(4)+a(5)+a(6)) = pi(1+1+1+1+3) = pi(7) = 4.
%e A100478 a(8) = pi(a(3)+a(4)+a(5)+a(6)+a(7)) = pi(1+1+1+3+4) = pi(10) = 4.
%e A100478 a(9) = pi(a(4)+a(5)+a(6)+a(7)+a(8)) = pi(1+1+3+4+4) = pi(13) = 6.
%e A100478 a(10) = pi(a(5)+a(6)+a(7)+a(8)+a(9)) = pi(1+3+4+4+6) = pi(18) = 7.
%t A100478 a[n_]:= a[n]= If[n<6,1,PrimePi[Sum[a[n-j], {j,5}]]];
%t A100478 Table[a[n], {n,80}] (* _Robert G. Wilson v_, Dec 03 2004 *)
%o A100478 (SageMath)
%o A100478 @CachedFunction
%o A100478 def a(n): # a = A100478
%o A100478     if (n<6): return 1
%o A100478     else: return prime_pi(sum(a(n-j) for j in range(1,6)))
%o A100478 [a(n) for n in range(1, 81)] # _G. C. Greubel_, Apr 06 2023
%Y A100478 Cf. A001591, A038607.
%K A100478 easy,nonn
%O A100478 1,6
%A A100478 _Jonathan Vos Post_, Nov 22 2004
%E A100478 Edited and extended by _Robert G. Wilson v_, Dec 03 2004