A371201 a(n) = Sum_{k=prime(n)..prime(n+1)-1} k, with a(0) = 1.

1, 2, 7, 11, 34, 23, 58, 35, 82, 153, 59, 201, 154, 83, 178, 297, 333, 119, 381, 274, 143, 453, 322, 513, 740, 394, 203, 418, 215, 442, 1673, 514, 801, 275, 1435, 299, 921, 957, 658, 1017, 1053, 359, 1855, 383, 778, 395, 2454, 2598, 898, 455, 922, 1413, 479, 2455, 1521, 1557, 1593

Offset: 0

Author

Raul Prisacariu, Mar 15 2024

Keywords

• nonn
• easy

Comments

The sequence can be obtained graphically using the following grid walk rules. From an origin the first movement iteration consists of moving 1 unit in any direction. The n-th movement iteration consists of moving in the same direction n units. If n is a prime number, the movement iteration consists of first changing the movement direction by 90 degrees and then moving n units in the new direction. If n is a nonprime number, the movement iteration consists of moving n units in the same direction as the previous movement iteration. The sequence is obtained by measuring the length of each 90-degree turn.

a(0) is the length of the grid segment before doing any 90-degree turns and a(1) is the length of the first 90-degree turn.

Examples

			a(0) = 1.
a(1) = 2.
a(2) = 3 + 4 = 7.
a(3) = 5 + 6 = 11.
a(4) = 7 + 8 + 9 + 10 = 34.
a(5) = 11 + 12 = 23.
a(6) = 13 + 14 + 15 + 16 = 58.
a(7) = 17 + 18 = 35.
The natural numbers are summed in groups where each prime begins a new group,
  primes     v   v       v       v
         1   2   3   4   5   6   7   8   9  10  ...
        \-/ \-/ \-----/ \-----/ \-------------/
  a(n) = 1   2     7       11          34
    n  = 0   1     2       3           4
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000040, A054265, A138383.

Cf. A008837 (partial sums).

Programs

Maple

ithprime(0):=1:
a:= n-> ((j, k)-> (k-1+j)*(k-j)/2)(map(ithprime, [n, n+1])[]):
seq(a(n), n=0..56);  # Alois P. Heinz, Mar 16 2024

Mathematica

Join[{1},Table[Prime[n]+(Prime[n+1]+Prime[n])*(Prime[n+1]-Prime[n]-1)/2,{n,56}]] (* James C. McMahon, Apr 20 2024 *)

PARI

first(n) = {
	my(res = primes(n), t = 0);
	for(i = 1, n,
		res[i] = binomial(res[i],2) - t;
		t+=res[i];
	);
	res	
} \\ David A. Corneth, Mar 16 2024

Python

from sympy import nextprime, prime
def A371201(n):
    if n == 0: return 1
    q = nextprime(p:=prime(n))
    return (q-p)*(p+q-1)>>1 # Chai Wah Wu, Jun 01 2024

Formula

For n > 0, a(n) = A138383(n) - (prime(n+1) - prime(n)).

a(n) = binomial(prime(n+1), 2) - Sum_{k=0..n-1} a(k). - David A. Corneth, Mar 15 2024

a(n) = prime(n) + A054265(n), for n >= 1. - Michel Marcus, Mar 15 2024

a(n) = (prime(n+1)-prime(n))*(prime(n+1)+prime(n)-1)/2 for n>=1. - Chai Wah Wu, Jun 01 2024

Extensions

More terms from Michel Marcus, Mar 15 2024