cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A355264 a(n) = n * largest-nth-power(n, 2) = n * A000188(n), where largest-nth-power(n, e) is the largest positive integer b such that b^e divides n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 16, 27, 10, 11, 24, 13, 14, 15, 64, 17, 54, 19, 40, 21, 22, 23, 48, 125, 26, 81, 56, 29, 30, 31, 128, 33, 34, 35, 216, 37, 38, 39, 80, 41, 42, 43, 88, 135, 46, 47, 192, 343, 250, 51, 104, 53, 162, 55, 112, 57, 58, 59, 120, 61, 62, 189, 512
Offset: 1

Views

Author

Peter Luschny, Jul 12 2022

Keywords

Links

Crossrefs

Cf. A000027, A000188, A007913.
Cf. A001620, A306016.

Programs

  • Maple
    with(NumberTheory): seq(n*LargestNthPower(n, 2), n = 1..64);
  • Mathematica
    Table[n*Times @@ (#1^Floor[#2/2] & @@@ FactorInteger[n]), {n, 64}] (* Michael De Vlieger, Jul 12 2022 *)
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,2] + f[i,2]\2));} \\ Amiram Eldar, Sep 21 2023

Formula

Multiplicative with a(p^e) = p^(e+floor(e/2)). - Amiram Eldar, Jul 13 2022
From Amiram Eldar, Sep 21 2023: (Start)
Dirichlet g.f.: zeta(s-1) * zeta(2*s-3)/ zeta(2*s-2).
Sum_{k=1..n} a(k) ~ (3*n^2/(4*Pi^2)) * (2*log(n) + 6*gamma - 4*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620). (End)

A360165 a(n) is the sum of the square roots of the unitary divisors of n that are odd squares minus the sum of the square roots of the unitary divisors of n that are even squares.

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, 1, 4, 1, 1, -1, 1, 1, 1, -3, 1, 4, 1, -1, 1, 1, 1, 1, 6, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -4, 1, 1, 1, 1, 1, 1, 1, -1, 4, 1, 1, -3, 8, 6, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 4, -7, 1, 1, 1, -1, 1, 1, 1, 4, 1, 1, 6, -1, 1, 1, 1, -3, 10, 1
Offset: 1

Views

Author

Amiram Eldar, Jan 29 2023

Keywords

Comments

The unitary analog of A347176.

Links

Crossrefs

Cf. A001620, A306016, A347176, A360162, A360164.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], 1, p^(e/2) + 1]; f[2, e_] := If[OddQ[e], 1, 1 - 2^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, if(f[i, 2]%2, 1, 1 - f[i, 1]^(f[i, 2]/2)), if(f[i, 2]%2, 1, f[i, 1]^(f[i, 2]/2) + 1))); }

Formula

a(n) = Sum_{d|n, gcd(d, n/d)=1, d odd square} (-1)^(d+1)*sqrt(d).
a(n) = A360164(n) - 2 * A360162(n).
Multiplicative with a(2^e) = 1 - 2^(e/2) if e is even and 1 otherwise, and for p > 2, a(p^e) = p^(e/2) + 1 if e is even and 1 if e is odd.
Dirichlet g.f.: (zeta(s)*zeta(2*s-1)/zeta(3*s-1))*(2^(3*s)-2^(s+2)+2)/(2^(3*s)-2).
Sum_{k=1..n} a(k) ~ (n/Pi^2)*(log(n) + 3*gamma - 1 + 4*log(2) - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

A358336 Multiplicative sequence with a(p^e) = ((p-1) * (1 + e*(e+1)/2) + e) * p^(e-1) for prime p and e > 0.

Original entry on oeis.org

1, 3, 5, 12, 9, 15, 13, 40, 30, 27, 21, 60, 25, 39, 45, 120, 33, 90, 37, 108, 65, 63, 45, 200, 90, 75, 153, 156, 57, 135, 61, 336, 105, 99, 117, 360, 73, 111, 125, 360, 81, 195, 85, 252, 270, 135, 93, 600, 182, 270, 165, 300, 105, 459, 189, 520, 185, 171, 117, 540, 121, 183, 390, 896
Offset: 1

Views

Author

Werner Schulte, Nov 09 2022

Keywords

Crossrefs

Cf. A000010, A005361, A018804, A112526.
Cf. A001620, A002117, A013664, A244115, A157289, A306016.

Programs

  • Mathematica
    f[p_, e_] := ((p - 1)*(1 + e*(e + 1)/2) + e)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2022 *)
  • PARI
    a(n) = { my (f=factor(n), p, e, v=1); for (k=1, #f~, p=f[k,1]; e=f[k,2]; v *= ((p-1) * (1 + e*(e+1)/2) + e) * p^(e-1)); return (v) } \\ Rémy Sigrist, Jan 18 2023

Formula

a(n) = Sum_{k=1..n} gcd(k, n) * A005361(gcd(k, n)) for n > 0.
Equals Dirichlet convolution of A000010 and n * A005361.
Dirichlet g.f.: (zeta(s-1)^2 * zeta(2*s-2) * zeta(3*s-3)) / (zeta(s) * zeta(6*s-6)).
Equals Dirichlet convolution of A018804 and A112526.
Sum_{k=1..n} a(k) ~ (zeta(3)/(2*zeta(6))) * n^2 * (log(n) + 2*gamma - 1/2 + zeta'(2)/zeta(2) + 3*zeta'(3)/zeta(3) + 6*zeta'(6)/zeta(6)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 13 2024

A386738 Decimal expansion of Integral_{x=0..1} {1/x}^4 dx, where {} denotes fractional part.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Aug 01 2025

Keywords

Examples

			0.14553289487913287197745596494722440166566646379514...

Links

Crossrefs

Cf. A001620 (gamma), A002117, A061444, A073002, A306016.
Cf. A153810 (m=1), A345208 (m=2), A345208 (m=3), this constant (m=4).

Programs

  • Mathematica
    RealDigits[Log[2*Pi] - 2*EulerGamma - 1/3 + (Zeta[3]/2 + Zeta'[2])/Zeta[2], 10, 120][[1]]
  • PARI
    log(2*Pi) - 2*Euler - 1/3 + (zeta(3)/2 + zeta'(2))/zeta(2)

Formula

Equals log(2*Pi) - 2*gamma - 1/3 + 3*zeta(3)/Pi^2 + 6*zeta'(2)/Pi^2.
In general, for m >= 2, Integral_{x=0..1} {1/x}^m dx = log(2*Pi) - m*gamma/2 - 1/(m-1) - Sum_{k=1..floor((m-2)/2)} (-1)^k * (m!/(m-2*k-1)!) * zeta(2*k+1) / (2^(2*k+1) * Pi^(2*k)) + 2 * Sum_{k=1..floor((m-1)/2)} (-1)^(k-1) * (m!/(m-2*k)!) * zeta'(2*k) / (2*Pi)^(2*k).
Previous Showing 41-44 of 44 results.

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