A083242 For n >= 3, a(n-3) + a(n-2) + a(n-1) + a(n) = prime(n); a(0) = 0, a(1) = 1, a(2) = 1.
0, 1, 1, 3, 2, 5, 3, 7, 4, 9, 9, 9, 10, 13, 11, 13, 16, 19, 13, 19, 20, 21, 19, 23, 26, 29, 23, 25, 30, 31, 27, 39, 34, 37, 29, 49, 36, 43, 35, 53, 42, 49, 37, 63, 44, 53, 39, 75, 56, 57, 41, 79, 62, 59, 51, 85, 68, 65, 53, 91, 72, 67, 63, 105, 76, 69, 67, 119
Offset: 0
Examples
a(43) + a(44) + a(45) + a(46) = 63 + 44 + 53 + 39 = 199 = p[46]
Links
- Robert Israel, Table of n, a(n) for n = 0..10000
- Robert Israel, Plot of a(n) - prime(n)/4 for n = 1 .. 200000. n == 0, 1, 2, 3 (mod 4) in red, orange, green, blue respectively.
Programs
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Maple
f:= proc(n) option remember; ithprime(n) - procname(n-1) - procname(n-2)-procname(n-3) end proc: f(0):= 0: f(1):= 1: f(2):= 1: map(f, [$0..100]); # Robert Israel, Aug 20 2024
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Mathematica
f[x_] := Prime[x]-f[x-1]-f[x-2]-f[x-3] {f[0]=0, f[1]=1, f[2]=1}; Table[f[w], {w, 0, 20}]
Formula
From Robert Israel, Aug 20 2024: (Start)
a(4*k) = Sum_{j=1..k} A001223(4*j-1).
a(4*k + 1) = 1 + Sum_{j=1..k} A001223(4*j).
a(4*k + 2) = Sum_{j=0..k} A001223(4*j+1).
a(4*k + 3) = 1 + Sum_{j=0..k} A001223(4*j+2). (End)