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%I #39 Sep 08 2022 08:44:48
%S 1,1,2,2,3,3,4,5,5,6,7,7,8,9,10,10,11,12,12,13,14,15,16,17,17,18,18,
%T 19,21,22,23,23,25,25,26,27,28,29,30,30,32,32,33,33,35,37,38,38,39,40,
%U 40,42,43,44,45,45,46,47,47,49,51,52,52,53,55,56,58,58,59,60,61,62,63,64,65,66,67,68,70
%N a(n) = (prime(n+2)-1)/6 if this is an integer or (prime(n+2)+ 1)/6 otherwise.
%C Also number of partitions of n-th prime > 3 into a sum of 2's or 3's (inclusive or).
%C From _Wolfdieter Lang_, Mar 13 2012: (Start)
%C The primes of the form 6*k+1 are given in A002476.
%C For n >= 1 such that prime(n+2) is from A002476, one has 8*T(prime(n+2)-1) + 1 = r(n)^2, n >= 1, with the triangular numbers T(n) = A000217(n) and r(n) = A208296(n). Therefore, 24*prime(n+2)*a(n) + 1 = r(n)^2. E.g., n=2: prime(4)=7, a(2)=1, 8*21 + 1 = 13^2 = A208296(2)^2 = 24*7*1 + 1.
%C The primes of the form 6*k-1 are given in A007528.
%C For n >= 1 such that prime(n+2) is from A007528, one has 8*T(prime(n+2)) + 1 = r(n)^2. For T and r see the preceding comment. Therefore, 24*prime(n+2)*a(n) + 1 = r(n)^2. E.g., n=1, prime(3)=5, a(1)=1, 8*15 + 1 = 11^2 = A208296(1)^2 = 24*5*1 + 1.
%C (End)
%F a(n) = A103221(prime(n+2)). - _R. J. Mathar_, May 02 2010
%F a(n) = floor((prime(n+2)+3)/6), n >= 1, prime(n)=A000040(n). Consider the two cases prime(n+2) == 1 (mod 6) and == -1 (mod 6) separately. See the formula above. - _Wolfdieter Lang_, Mar 15 2012
%p From _R. J. Mathar_, May 02 2010: (Start)
%p A103221 := proc(n) a := 0 ; for t from 0 do if 2*t > n then return a; end if; if n-2*t mod 3 = 0 then a := a+1 ; end if; end do : end proc:
%p A024699 := proc(n) A103221(ithprime(n+2)) ; end proc: seq(A024699(n),n=1..120) ; (End)
%t pi6[n_]:=Module[{p=Prime[n+2],c},c=(p-1)/6;If[IntegerQ[c],c,(p+1)/6]]; Array[pi6,80] (* _Harvey P. Dale_, Aug 19 2013 *)
%t Table[Floor[(Prime[n + 2] + 3) / 6], {n, 100}] (* _Vincenzo Librandi_, Sep 06 2016 *)
%o (PARI) a(n) = (prime(n+2)+3)\6; \\ _Michel Marcus_, Sep 06 2016; after _Wolfdieter Lang_
%o (Magma) [(NthPrime(n+2)+3) div 6: n in [1..80]]; // _Vincenzo Librandi_, Sep 06 2016
%K nonn
%O 1,3
%A _Clark Kimberling_
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