OFFSET
0,2
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.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
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
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Raul Prisacariu, Mar 15 2024
EXTENSIONS
More terms from Michel Marcus, Mar 15 2024
STATUS
approved