A375754 -id:A375754 - cp's OEIS frontend

A375753 a(n) is the difference between n*(n+1)/2 and the next smaller square of a prime.

2, 1, 6, 12, 3, 11, 20, 6, 17, 29, 42, 56, 71, 15, 32, 2, 21, 41, 62, 84, 107, 11, 36, 62, 17, 45, 74, 104, 135, 167, 32, 66, 101, 137, 174, 212, 251, 291, 20, 62, 105, 29, 74, 120, 167, 215, 264, 314, 365, 9, 62, 116, 171, 227, 284, 30, 89, 149, 42, 104, 167, 231, 296, 2

Offset: 3

Hugo Pfoertner, Aug 26 2024

			a(3) = A000217(3) - A001248(1) = 6 - 2^2 = 2,
a(4) = A000217(4) - A001248(2) = 10 - 3^2 = 1.

Karl-Heinz Hofmann, Table of n, a(n) for n = 3..10000

Cf. A000217, A001248, A375752, A375754.

a375753(n) = my(t=n*(n+1)/2, s=sqrtint(t)); t-precprime(s)^2

from sympy import prevprime, integer_nthroot
def A375753(n): return (t:=n*(n+1)//2) - prevprime(integer_nthroot(t,2)[0]+1)**2
# Karl-Heinz Hofmann, Aug 27 2024

A375752 a(n) is the difference between T=n*(n+1)/2 and the largest prime not exceeding T.

Original entry on oeis.org

0, 1, 3, 2, 2, 5, 5, 2, 2, 5, 5, 2, 2, 7, 5, 2, 4, 9, 11, 2, 2, 5, 7, 8, 2, 5, 5, 2, 2, 5, 5, 4, 2, 11, 5, 2, 2, 7, 9, 2, 16, 5, 7, 2, 12, 5, 5, 2, 16, 5, 5, 2, 2, 9, 13, 16, 2, 11, 7, 2, 2, 5, 11, 2, 4, 5, 5, 4, 8, 5, 7, 2, 8, 7, 9, 2, 2, 23, 11, 2, 12, 17, 11, 12, 2

Offset: 2

Hugo Pfoertner, Aug 26 2024

Cf. A000217, A007917, A065384, A075402, A375753, A375754.

Table[T=n(n+1)/2;T-If[PrimeQ[T],T,NextPrime[T,-1]],{n,2,86}] (* James C. McMahon, Sep 27 2024 *)

a(n) = my(t=n*(n+1)/2); t - precprime(t)

from sympy import prevprime
def A375752(n): return (t:=n*(n+1)//2) - prevprime(t) # Karl-Heinz Hofmann, Aug 27 2024

a(n) = A000217(n) - A065384(n). - Michel Marcus, Aug 27 2024

cp's OEIS Frontend

A375753 a(n) is the difference between n*(n+1)/2 and the next smaller square of a prime.

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