A217916 a(n) = prime(n) - Sum_{k=1..A002024(n)-1} a(n-k), where A002024(m) = [sqrt(2*m)+1/2] is "n appears n times".

2, 1, 4, 2, 5, 6, 4, 4, 9, 12, 2, 10, 8, 11, 16, 6, 8, 12, 14, 15, 18, 6, 10, 14, 20, 18, 17, 22, 2, 10, 24, 18, 26, 20, 27, 24, 6, 8, 14, 30, 24, 28, 30, 29, 28, 2, 18, 20, 18, 32, 28, 34, 32, 39, 34, 6, 8, 20, 26, 22, 34, 38, 48, 36, 41, 38, 14, 12, 18, 22, 30

Offset

1,1

Comments

Form this sequence into a number triangle, then it seems that:

(1) the odd numbers only appear adjacent to the main diagonal;

(2) the number 2 only appears in column 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..10011

Formula

Row sums equal A011756(n) = prime(n(n+1)/2).

Example

Start with a(1) = 2, from then on, the sum of A002024(n) consecutive terms prior to and including a(n) generates the n-th prime number
(A002024 begins: [1, 2,2, 3,3,3, 4,4,4,4, 5,5,5,5,5, ...]).
n=1: 2 = 2; (start)
n=2: 3 = 2 + 1; (sum of 2 terms = prime)
n=3: 5 = 1 + 4;     "
n=4: 7 = 1 + 4 + 2; (sum of 3 terms = prime)
n=5: 11 = 4 + 2 + 5;    "
n=6: 13 = 2 + 5 + 6;    "
n=7: 17 = 2 + 5 + 6 + 4; (sum of 4 terms = prime)
n=8: 19 = 5 + 6 + 4 + 4;    "
n=9: 23 = 6 + 4 + 4 + 9;    "
n=10: 29 = 4 + 4 + 9 + 12;    "
n=11: 31 = 4 + 4 + 9 + 12 + 2; (sum of 5 terms = prime)
...
As a triangle, the sequence begins:
2;
1, 4;
2, 5, 6;
4, 4, 9, 12;
2, 10, 8, 11, 16;
6, 8, 12, 14, 15, 18;
6, 10, 14, 20, 18, 17, 22;
2, 10, 24, 18, 26, 20, 27, 24;
6, 8, 14, 30, 24, 28, 30, 29, 28;
2, 18, 20, 18, 32, 28, 34, 32, 39, 34;
6, 8, 20, 26, 22, 34, 38, 48, 36, 41, 38;
14, 12, 18, 22, 30, 28, 42, 44, 54, 40, 47, 46;
4, 22, 22, 20, 32, 32, 34, 46, 50, 62, 44, 49, 50;
12, 12, 26, 30, 24, 38, 44, 36, 64, 56, 72, 50, 55, 52;
6, 22, 18, 32, 32, 30, 44, 48, 38, 76, 66, 74, 54, 61, 58;
2, 18, 26, 24, 40, 42, 38, 54, 56, 44, 82, 70, 82, 60, 65, 66;
4, 16, 28, 38, 26, 50, 44, 42, 56, 66, 58, 86, 72, 86, 74, 69, 68;
4, 24, 20, 36, 48, 34, 54, 50, 48, 70, 70, 64, 92, 80, 92, 86, 73, 74;
2, 14, 26, 26, 46, 50, 44, 56, 56, 66, 74, 72, 68, 98, 86, 100, 92, 79, 96;
2, 12, 22, 36, 32, 52, 58, 56, 60, 62, 72, 76, 78, 80, 108, 104, 102, 96, 85, 98; ...
Notice the positions of the odd numbers and of the number 2;
the only odd numbers appear adjacent to the main diagonal and
the number 2 only appears in the first column.

Prog

(PARI) /* Using the recurrence (slow) */
{a(n)=if(n<1, 0, if(n==1, 2, prime(n)-sum(k=1, floor(1/2+sqrt(2*n))-1, a(n-k))))}
for(n=1, 120, print1(a(n), ", "))
(PARI) /* Print as a Triangle of M Rows (much faster) */
{M=20; A=vector(M*(M+1)/2); }
{a(n)=A[n]=if(n<1, 0, if(n==1, 2, prime(n)-sum(k=1, floor(1/2+sqrt(2*n))-1, A[n-k])))}
for(n=1, M, for(k=1, n, print1(a(n*(n-1)/2+k), ", ")); print(""))

Crossrefs

Cf. A011756, A217917 (odd numbers that appear), A217918 (positions of 2), A217919 (rows in which 2 appears), A217920 (column 1).

Sequence in context: A339220 A161293 A328163 * A057923 A147763 A342415

Adjacent sequences: A217913 A217914 A217915 * A217917 A217918 A217919

Keyword

nonn

Author

Paul D. Hanna, Mar 04 2013

Status

approved