A048760 -id:A048760 - cp's OEIS frontend

A345202 Decimal expansion of gamma + zeta(2), where gamma is Euler's constant (A001620).

2, 2, 2, 2, 1, 4, 9, 7, 3, 1, 7, 4, 9, 7, 5, 9, 2, 9, 7, 0, 7, 8, 9, 2, 7, 2, 5, 6, 7, 2, 8, 4, 2, 7, 6, 2, 0, 2, 6, 1, 1, 0, 9, 2, 3, 7, 1, 4, 6, 7, 2, 2, 0, 3, 6, 5, 4, 1, 3, 2, 5, 4, 6, 4, 2, 5, 4, 8, 7, 5, 1, 9, 7, 1, 8, 0, 8, 6, 5, 5, 4, 4, 7, 7, 0, 5, 7

Offset: 1

Amiram Eldar, Jun 10 2021

The value of the sum (see the Formula section) discovered in 1926 by the Italian mathematician and historian of science Giovanni Enrico Eugenio Vacca (1872-1953).

			2.22214973174975929707892725672842762026110923714672...

G. Vacca, Nuova serie per la costante di Eulero, C=0,577..., Rendiconti, Accademia Nazionale dei Lincei, Roma, Classe di Scienze Fisiche, Matematiche e Naturali, Serie 6, Vol. 3 (1926), pp. 19-20.

Ettore Carruccio, Giovanni Vacca, matematico, storico e filosofo della scienza, Bollettino dell'Unione Matematica Italiana, Serie 3, Vol. 8 (1953), pp. 448-456.
Tanguy Rivoal, Is Euler's constant a value of an arithmetic special function?, 2017.

Cf. A001620, A013661, A048760.

RealDigits[EulerGamma + Pi^2/6, 10, 100][[1]]

Equals Sum_{k>=1} (1/floor(sqrt(k))^2 - 1/k) (Vacca, 1926).
Equals Sum_{k>=1} f(k)/k^2, where f(k) = Sum_{j=1..2*k} j/(j + k^2).
Equals A001620 + A013661.

A386472 The maximum exponent in the prime factorization of the largest divisor of n whose exponents in its prime factorization are squares.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jul 23 2025

All the terms are by definition squares.

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

Cf. A048760, A051903, A386469.

a[n_] := Floor[Sqrt[Max[FactorInteger[n][[;; , 2]]]]]^2; a[1] = 0; Array[a, 100]

a(n) = if(n == 1, 0, sqrtint(vecmax(factor(n)[,2]))^2);

a(n) = A051903(A386469(n)).
a(n) = A048760(A051903(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} (2*k+1)*(1-1/zeta((k+1)^2)) = 1.2383138701540899647042996... .

A263727 Largest square number less than or equal to the n-th Fibonacci number.

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 4, 9, 16, 25, 49, 81, 144, 225, 361, 576, 961, 1521, 2500, 4096, 6724, 10816, 17689, 28561, 46225, 74529, 121104, 196249, 316969, 514089, 831744, 1345600, 2175625, 3523129, 5702544, 9223369, 14922769, 24157225, 39087504, 63234304, 102333456

Offset: 0

Eli Jaffe, Oct 24 2015

			For a(8), Fibonacci(8) = 21, the largest square under 21 is 16, so a(8) = 16.

Cf. A000045, A048760, A061287.

Floor[Sqrt[Fibonacci[Range[40]]]]^2 (* Alonso del Arte, Oct 24 2015 *)

a(n) = sqrtint(fibonacci(n))^2; \\ Michel Marcus, Oct 25 2015

a(n) = floor(sqrt(Fibonacci(n)))^2.
a(n) = A061287(n)^2. - Michel Marcus, Oct 25 2015
a(n) = A048760(A000045(n)). - Michel Marcus, Nov 11 2015

A350493 a(n) = floor(sqrt(prime(n)))^2 mod n.

Original entry on oeis.org

0, 1, 1, 0, 4, 3, 2, 0, 7, 5, 3, 0, 10, 8, 6, 1, 15, 13, 7, 4, 1, 20, 12, 9, 6, 22, 19, 16, 13, 10, 28, 25, 22, 19, 4, 0, 33, 30, 27, 9, 5, 1, 40, 37, 16, 12, 8, 4, 29, 25, 21, 17, 13, 9, 36, 32, 28, 24, 20, 16, 12, 41, 37, 33, 29, 25, 56, 52, 48, 44, 40, 36

Offset: 1

Simon Strandgaard, Jan 01 2022

nonn

			a(4) = A065730(4) mod 4 =  4 mod 4 = 0;
a(5) = A065730(5) mod 5 =  9 mod 5 = 4;
a(6) = A065730(6) mod 6 =  9 mod 6 = 3;
a(7) = A065730(7) mod 7 = 16 mod 7 = 2.

Cf. A000040, A048760, A065730.

Table[PowerMod[Floor[Sqrt[Prime[n]]],2,n],{n,72}] (* Stefano Spezia, Jan 02 2022 *)

a(n) = (sqrtint(prime(n))^2) % n;
vector(20,n,a(n))

from sympy import prime, integer_nthroot
def a(n): return (integer_nthroot(prime(n), 2)[0]**2)%n
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Jan 02 2022

require 'prime'
values = []
Prime.first(20).each_with_index do |prime, i|
    values << ((Integer.sqrt(prime) ** 2) % (i + 1))
end
p values

a(n) = A065730(n) mod n.

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A345202 Decimal expansion of gamma + zeta(2), where gamma is Euler's constant (A001620).

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A386472 The maximum exponent in the prime factorization of the largest divisor of n whose exponents in its prime factorization are squares.

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A263727 Largest square number less than or equal to the n-th Fibonacci number.

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A350493 a(n) = floor(sqrt(prime(n)))^2 mod n.

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