A138383 -id:A138383 - cp's OEIS frontend

A137174 First differences of A138383.

0, 6, 4, 25, -13, 37, -25, 49, 73, -98, 146, -49, -73, 97, 121, 36, -218, 266, -109, -133, 314, -133, 193, 229, -350, -193, 217, -205, 229, 1241, -1169, 289, -530, 1168, -1144, 626, 36, -301, 361, 36, -698, 1504, -1480, 397, -385, 2069, 144, -1708, -445, 469

Offset: 1

Odimar Fabeny, May 11 2008

Cf. A138383.

Differences[Join[{3},(#[[2]]-#[[1]]) (Total[#]+1)/2&/@Partition[Prime[ Range[ 60]],2,1]]] (* Harvey P. Dale, Oct 27 2020 *)

More terms from Harvey P. Dale, Oct 27 2020

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

Original entry on oeis.org

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

Raul Prisacariu, Mar 15 2024

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.

			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

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

Cf. A000040, A054265, A138383.

Cf. A008837 (partial sums).

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

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 *)

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

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

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

More terms from Michel Marcus, Mar 15 2024

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A137174 First differences of A138383.

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

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