Ridouane Oudra - cp's OEIS frontend

User: Ridouane Oudra

Ridouane Oudra has authored 2 sequences.

A384210 Number of numbers <= n of the form p * m^2, where p is a prime and m is an integer >= 1.

0, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 8, 8, 9, 10, 11, 12, 12, 12, 13, 13, 13, 13, 14, 15, 16, 16, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 22, 23, 23, 24, 25, 25, 26, 26, 27, 28, 28, 28, 28, 28, 28, 29, 29, 30, 30, 31, 31, 31, 31, 32, 33, 33, 33, 34, 35, 36, 36, 37

Offset: 1

Ridouane Oudra, May 22 2025

nonn

Partial sums of A358769.
First differs from A055038 at a(30).
a(A229125(n)) = n.

Ridouane Oudra, Table of n, a(n) for n = 1..10000

Cf. A229125, A358769, A000720, A008836, A001221.

with(numtheory): A358769:=n-> add(nops(factorset(d)), d in divisors(n)) mod 2:
seq(add(A358769(i), i=1..n), n=1..100);

a(n) = sum(k=1, n, isprime(core(k))); \\ Michel Marcus, May 29 2025

from math import isqrt
from sympy import primepi
def A384210(n): return sum(primepi(n//y**2) for y in range(1,isqrt(n)+1)) # Chai Wah Wu, Jun 06 2025

a(n) = Sum_{i=1..n} A358769(i).
a(n) = Sum_{i=1..floor(sqrt(n))} pi(floor(n/i^2)), where pi = A000720.
a(n) = - Sum_{i=1..n} lambda(i)*omega(i)*floor(n/i), where lambda = A008836 and omega = A001221.

A377484 a(n) = Product_{d|n, d>1} (d - 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 10, 6, 21, 16, 36, 10, 330, 12, 78, 112, 315, 16, 1360, 18, 2052, 240, 210, 22, 53130, 96, 300, 416, 6318, 28, 146160, 30, 9765, 640, 528, 816, 1570800, 36, 666, 912, 560196, 40, 639600, 42, 27090, 39424, 990, 46, 37456650, 288, 42336, 1600, 45900, 52, 1874080, 2160

Offset: 1

Ridouane Oudra, Oct 29 2024

			a(12) = (2-1)*(3-1)*(4-1)*(6-1)*(12-1) = 1*2*3*5*11 = 330.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007955, A020696.

Cf. A000079, A005329, A027871, A027872, A027875, A027879, A175787.

with(numtheory): seq(mul(d-1, d in divisors(n) minus {1}), n=1..80);

a[n_] := Times @@ (Rest@ Divisors[n] - 1); Array[a, 60] (* Amiram Eldar, Nov 01 2024 *)

a(n) = my(d=divisors(n)); prod(k=2, #d, d[k]-1); \\ Michel Marcus, Oct 30 2024

a(n) = Product_{k=2..A000005(n)} (A027750(n,k) - 1).
a(p^n) = Product_{k=1..n} (p^k - 1), where p is prime, and n an integer.
a(2^n) = A005329(n).
a(3^n) = A027871(n).
a(5^n) = A027872(n).
a(7^n) = A027875(n).
a(11^n) = A027879(n).
From Amiram Eldar, Nov 02 2024: (Start)
a(n) = n-1 if and only if n is in A175787 (i.e., n = 4 or n is prime).
a(n) == 1 (mod 2) if and only if n is a power of 2 (A000079). (End)

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User: Ridouane Oudra

A384210 Number of numbers <= n of the form p * m^2, where p is a prime and m is an integer >= 1.

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