A075402 -id:A075402 - cp's OEIS frontend

A075403 Smallest k such that the difference between the k-th triangular number and the following prime is equal to n.

2, 1, 5, 28, 10, 48, 13, 15, 65, 160, 58, 80, 106, 32, 162, 91, 94, 188, 70, 127, 62, 304, 201, 120, 394, 103, 141, 244, 382, 455, 301, 380, 521, 700, 625, 212, 721, 611, 681, 979, 277, 815, 1201, 1275, 569, 4588, 558, 2331, 5113, 1704, 1337, 2551, 2101

Offset: 0

Amarnath Murthy, Sep 23 2002

nonn

Index of the first occurrence of n in A075402.

Donovan Johnson, Table of n, a(n) for n = 0..500

Cf. A075402.

for(n=1,100,f=0:for(k=1,50000,if(nextprime(k*(k+1)/2)-k*(k+1)/2==n,f=k:break)):print1(f","))

Corrected and extended by Ralf Stephan, Mar 19 2003

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

A375754 a(n) is the difference between the next greater square of a prime and n*(n+1)/2.

Original entry on oeis.org

4, 3, 1, 3, 15, 10, 4, 21, 13, 4, 66, 55, 43, 30, 16, 1, 33, 16, 118, 99, 79, 58, 36, 13, 61, 36, 10, 151, 123, 94, 64, 33, 1, 280, 246, 211, 175, 138, 100, 61, 21, 100, 58, 15, 379, 334, 288, 241, 193, 144, 94, 43, 303, 250, 196, 141, 85, 28, 138, 79, 19, 318, 256

Offset: 0

Hugo Pfoertner, Aug 26 2024

			a(0) = A001248(1) - A000217(0) = 2^2 - 0 = 4,
a(1) = A001248(1) - A000217(1) = 2^2 - 1 = 3,
a(2) = A001248(1) - A000217(2) = 2^2 - 3 = 1,
a(3) = A001248(2) - A000217(3) = 3^2 - 6 = 3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000217, A001248, A075402, A375752, A375753.

NextPrime[Sqrt[#]]^2-#&/@Accumulate[Range[0,70]] (* Harvey P. Dale, Feb 16 2025 *)

a375754(n) = my(t=n*(n+1)/2, s=sqrtint(t), p=nextprime(s)); if(p^2-t < 0, p=nextprime(p+1)); p^2-t

cp's OEIS Frontend

A075403 Smallest k such that the difference between the k-th triangular number and the following prime is equal to n.

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

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A375754 a(n) is the difference between the next greater square of a prime and n*(n+1)/2.

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Mathematica

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