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.

Showing 1-4 of 4 results.

A088841 Numerator of the quotient sigma(7*n)/sigma(n).

Original entry on oeis.org

8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 400, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8
Offset: 1

Views

Author

Labos Elemer, Nov 04 2003

Keywords

Links

Crossrefs

Cf. A088837, A088838, A088839, A088840, A088841, A088842 (denominators).
Cf. A000203, A038712, A080278, A214411, A268354, A283078.

Programs

  • Mathematica
    Table[Numerator[DivisorSigma[1, 7*n]/DivisorSigma[1, n]], {n, 1, 128}]
  • PARI
    a(n) = numerator(sigma(7*n)/sigma(n)); \\ Amiram Eldar, Mar 22 2024

Formula

From Amiram Eldar, Mar 22 2024: (Start)
a(n) = numerator(A283078(n)/A000203(n)).
a(n) = (7^(A214411(n)+2)-1)/6 = (49*A268354(n)-1)/6.
Sum_{k=1..n} a(k) ~ (7/log(7))*n*log(n) + (9/2 + 7*(gamma-1)/log(7))*n, where gamma is Euler's constant (A001620).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A088842(k) = 1 + 36 * Sum_{k>=1} 1/(7^k-1) = 7.87276224676... . (End)

A323921 a(n) = (4^(valuation(n, 4) + 1) - 1) / 3.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 85, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 15 2020

Keywords

Comments

Sum of powers of 4 dividing n.

Links

Crossrefs

Cf. A001620, A038712, A080278, A088842, A234957, A235127, A339747, A339748.

Programs

  • Mathematica
    Table[(4^(IntegerExponent[n, 4] + 1) - 1)/3, {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[4^k x^(4^k)/(1 - x^(4^k)), {k, 0, Floor[Log[4, nmax]] + 1}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = (4^(valuation(n, 4) + 1) - 1) / 3; \\ Michel Marcus, Jul 09 2022
  • Python
    def A323921(n): return ((1<<((~n&n-1).bit_length()&-2)+2)-1)//3 # Chai Wah Wu, Jul 09 2022

Formula

G.f.: Sum_{k>=0} 4^k * x^(4^k) / (1 - x^(4^k)).
L.g.f.: -log(Product_{k>=0} (1 - x^(4^k))).
Dirichlet g.f.: zeta(s) / (1 - 4^(1 - s)).
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(2^e) = (4^floor((e+2)/2)-1)/3, and a(p^e) = 1 for p != 2.
Sum_{k=1..n} a(k) ~ n*log_4(n) + (1/2 + (gamma - 1)/log(4))*n, where gamma is Euler's constant (A001620). (End)

A339747 a(n) = (5^(valuation(n, 5) + 1) - 1) / 4.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 15 2020

Keywords

Comments

Sum of powers of 5 dividing n.
Denominator of the quotient sigma(5*n) / sigma(n).

Links

Crossrefs

Cf. A000203, A001620, A038712, A060904, A080278, A088842, A112765, A323921, A339748.

Programs

  • Mathematica
    Table[(5^(IntegerExponent[n, 5] + 1) - 1)/4, {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[5^k x^(5^k)/(1 - x^(5^k)), {k, 0, Floor[Log[5, nmax]] + 1}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = (5^(valuation(n, 5) + 1) - 1)/4; \\ Amiram Eldar, Nov 27 2022

Formula

G.f.: Sum_{k>=0} 5^k * x^(5^k) / (1 - x^(5^k)).
L.g.f.: -log(Product_{k>=0} (1 - x^(5^k))).
Dirichlet g.f.: zeta(s) / (1 - 5^(1 - s)).
a(n) = sigma(n)/(sigma(5*n) - 5*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(5^e) = (5^(e+1)-1)/4, and a(p^e) = 1 for p != 5.
Sum_{k=1..n} a(k) ~ n*log_5(n) + (1/2 + (gamma - 1)/log(5))*n, where gamma is Euler's constant (A001620). (End)

A339748 a(n) = (6^(valuation(n, 6) + 1) - 1) / 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 43, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 43, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 15 2020

Keywords

Comments

Sum of powers of 6 dividing n.

Links

Crossrefs

Cf. A038712, A080278, A088842, A122841, A234959, A323921, A339747.

Programs

  • Mathematica
    Table[(6^(IntegerExponent[n, 6] + 1) - 1)/5, {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[6^k x^(6^k)/(1 - x^(6^k)), {k, 0, Floor[Log[6, nmax]] + 1}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=0} 6^k * x^(6^k) / (1 - x^(6^k)).
L.g.f.: -log(Product_{k>=0} (1 - x^(6^k))).
Dirichlet g.f.: zeta(s) / (1 - 6^(1 - s)).
Showing 1-4 of 4 results.

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