A073737 -id:A073737 - cp's OEIS frontend

A073736 Sum of primes whose index is congruent to n (mod 3); equals the partial sums of A073737 (in which sums of three successive terms form the primes).

1, 2, 3, 6, 9, 14, 19, 26, 33, 42, 55, 64, 79, 96, 107, 126, 149, 166, 187, 216, 237, 260, 295, 320, 349, 392, 421, 452, 499, 530, 565, 626, 661, 702, 765, 810, 853, 922, 973, 1020, 1095, 1152, 1201, 1286, 1345, 1398, 1485, 1556, 1621, 1712, 1785, 1854, 1951

Offset: 0

Paul D. Hanna, Aug 07 2002

For purposes of this sequence, 1 is treated as a prime. - Harvey P. Dale, Jul 24 2013

			a(10) = p_10 + p_7 + p_4 + p_1 = 29 + 17 + 7 + 2 = 55.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A073737.

a073736 n = a073736_list !! n
a073736_list = scanl1 (+) a073737_list
-- Reinhard Zumkeller, Apr 28 2013

a[0] = 1; a[-1] = 0; a[-2] = 0; p[0] = 1; p[n_?Positive] := Prime[n]; a[n_] := a[n] = p[n] - a[n-1] - a[n-2]; Table[a[n], {n, 0, 60}] // Accumulate (* Jean-François Alcover, Jun 25 2013 *)
Sort[Flatten[Accumulate/@Transpose[Partition[Join[{1},Prime[Range[61]]], 3]]]] (* Harvey P. Dale, Jul 24 2013 *)

a(n) = Sum_{m<=n, m=n (mod 3)} p_m, where p_m is the m-th prime; that is, a(3n+k) = p_(3n) + p_(3(n-1)) + p_(3(n-2)) + ... + p_k, for 0<=k<3, where a(0)=1 and the 0th prime is taken to be 1.

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.

Original entry on oeis.org

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

Labos Elemer, Apr 24 2003

			a(43) + a(44) + a(45) + a(46) = 63 + 44 + 53 + 39 = 199 = p[46]

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.

Cf. A001223, A034467, A073737, A014687, A083240.

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

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}]

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)

A103781 Sum of any four successive terms is prime, a(1)=a(2)=0,a(3)=1.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 5, 3, 6, 5, 9, 9, 8, 11, 13, 11, 12, 17, 19, 13, 18, 21, 21, 19, 22, 27, 29, 23, 24, 31, 31, 27, 38, 35, 37, 29, 48, 37, 43, 35, 52, 43, 49, 37, 62, 45, 53, 39, 74, 57, 57, 41, 78, 63, 59, 51, 84, 69, 65, 53, 90, 73, 67, 63, 104, 77, 69, 67, 118, 83, 79, 69

Offset: 1

Zak Seidov, Feb 15 2005

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A073737, A083242.

(*seed*)b4 = {0, 0, 1}; Do[x = Prime[n] - (b4[[ -1]] + b4[[ -2]] + b4[[ -3]]); b4 = Append[b4, x], {n, 1, 200}]; b4
nxt[{a_, b_, c_}] := {b, c, NextPrime[a + b + c] - (a+b + c)}; NestList[nxt, {0, 0, 1}, 100][[All, 1]] (* Harvey P. Dale, Sep 20 2022 *)

a(n) = A000040(n-3) - a(n-1) - a(n-2) - a(n-3). - Jason Yuen, Sep 01 2024

cp's OEIS Frontend

A073736 Sum of primes whose index is congruent to n (mod 3); equals the partial sums of A073737 (in which sums of three successive terms form the primes).

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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.

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A103781 Sum of any four successive terms is prime, a(1)=a(2)=0,a(3)=1.

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