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 1-7 of 7 results.

A236024 Record values in A236022.

Original entry on oeis.org

4, 9, 10, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761
Offset: 1

Views

Author

Jaroslav Krizek, Jan 19 2014

Keywords

Comments

Sequence of numbers from A236022 such that A236022(n) > A236022(k) for all k < n.
A236022 = composite numbers n sorted by increasing values of gamma(n) = log_2(n+1) - log_tau(n) (sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n and tau(n) = A000005(n) = the number of divisors of n.
Conjecture: union of 10 and squares of primes A001248.

Crossrefs

Cf. A236020, A236021, A236022, A236023, A236025, A236026.

A236021 Record values in A236020.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 60, 72, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 27720, 30240, 55440, 65520, 83160, 110880, 166320, 196560, 221760, 277200, 332640, 360360, 393120, 720720, 831600, 1441440, 2162160, 2882880
Offset: 1

Views

Author

Jaroslav Krizek, Jan 18 2014

Keywords

Comments

Sequence of numbers from A236020 such that A236020(n) > A236020(k) for all k < n.
A236020 = natural numbers n sorted by increasing values of k(n) = log_tau(n) (sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n and tau(n) = A000005(n) = the number of divisors of n.
Conjecture: subsequence of A094348.

Crossrefs

Cf. A236020, A236022, A236023, A236024, A236025, A236026.

Extensions

More terms from Jon E. Schoenfield, Nov 12 2016

A236020 Natural numbers n sorted by increasing values of k(n) = log_tau(n) (sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n and tau(n) = A000005(n) = the number of divisors of n.

Original entry on oeis.org

1, 2, 4, 6, 12, 8, 24, 3, 18, 36, 30, 60, 10, 20, 48, 72, 120, 16, 40, 84, 180, 42, 90, 240, 144, 360, 96, 168, 28, 420, 108, 80, 252, 720, 14, 15, 210, 840, 54, 56, 336, 480, 216, 126, 32, 504, 288, 9, 540, 1260, 300, 132, 140, 1680, 192, 2520, 1080, 600, 630
Offset: 1

Views

Author

Jaroslav Krizek, Jan 18 2014

Keywords

Comments

The number k(n) = log_tau(n) (sigma(n)) = log(sigma(n)) / log(tau(n)) is such that tau(n)^k(n) = sigma(n).
Conjecture: every natural number n has a unique value of k(n). [The conjecture is wrong: e.g., k(5) = k(22) = log(6)/log(2). - Amiram Eldar, Jan 17 2021]
See A236021 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

Examples

			For number 1; k(1) = 1.
For number 2; k(2) = log_tau(2) (sigma(2)) = log_2 (3) = 1.5849625007... = A020857.

Links

Crossrefs

Cf. A000005, A000203, A236021.
Cf. A236022, A236023, A236024, A236025, A236026.

Programs

  • Mathematica
    A[nn_] := Ordering[ N[ Join[ {1}, Table[ Log[DivisorSigma[0, i], DivisorSigma[1, i]], {i, 2, nn} ] ] ] ];
A236020[nn_] := A[nn^2][[1 ;; nn]];
A236020[59] (* Robert P. P. McKone, Jan 17 2021 *)
  • PARI
    \\ warning: does not generate all the terms up to nn
f(k) = if (k==1, 1, log(sigma(k)) / log(numdiv(k)));
lista(nn) = vecsort(vector(nn, k, f(k)),, 1); \\ Michel Marcus, Jan 16 2021

A236025 Composite numbers n sorted by increasing values of delta(n) = (n+1)^(1/2) - sigma(n)^(1/tau(n)), where sigma(n) = A000203(n) = the sum of divisors of n and tau(n) = A000005(n) = the number of divisors of n.

Original entry on oeis.org

4, 6, 9, 8, 10, 14, 15, 12, 25, 16, 21, 22, 18, 26, 20, 27, 33, 34, 49, 24, 35, 28, 38, 39, 32, 30, 46, 51, 36, 55, 58, 44, 57, 40, 45, 42, 62, 50, 65, 52, 69, 48, 74, 54, 77, 56, 82, 63, 86, 121, 85, 64, 68, 87, 60, 94, 66, 93, 91, 81, 75, 95, 76, 70, 106, 78
Offset: 1

Views

Author

Jaroslav Krizek, Jan 19 2014

Keywords

Comments

The number delta(n) = (n+1)^(1/2) - sigma(n)^(1/tau(n)) is called the delta-deviation from primality of the number n; delta(p) = 0 for p = prime.
For number 4; delta(4) = (4+1)^(1/2) - sigma(4)^(1/tau(4)) = 5^(1/2) - 7^(1/3) = 0.32313679472... = A236027 (minimal value of function delta(n)).
See A234516, A234520 and A236022 for definitions of functions alpha(n), beta(n) and gamma(n).
See A236026 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.
Conjecture: Every natural number n has a unique value of number delta(n).

Links

Crossrefs

Cf. A236020, A236021, A236022, A236023, A236024, A236026, A236027.

A236023 Decimal expansion of log_2 (5) - log_3 (7).

Original entry on oeis.org

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

Views

Author

Jaroslav Krizek, Jan 18 2014

Keywords

Comments

Decimal expansion of minimal value of function gamma(n) = log_2(n+1) - log_tau(n) (sigma(n)) for n = 4, where gamma(n) is called the gamma-deviation from primality of the number n (see A236022).

Examples

			0.550684345725940087802391122406...

Crossrefs

Cf. A236020, A236021, A236022, A236024, A236025, A236026.
Cf. A020858, A152565.

Programs

  • Mathematica
    RealDigits[Log2[5] - Log[3, 7], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Formula

Equals log(5)/log(2) - log(7)/log(3) = A020858 - A152565.

A236026 Sequence of numbers from A236025 such that A236025(n) > A236025(k) for all k < n.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 25, 26, 27, 33, 34, 49, 51, 55, 58, 62, 65, 69, 74, 77, 82, 86, 121, 169, 178, 183, 185, 194, 202, 206, 289, 361, 362, 365, 382, 386, 529, 538, 542, 543, 554, 562, 566, 573, 586, 591, 597, 614, 622, 841, 961, 974, 982, 998, 1006, 1018
Offset: 1

Views

Author

Jaroslav Krizek, Jan 21 2014

Keywords

Comments

A236025 = composite numbers n sorted by increasing values of number delta(n) = (n+1)^(1/2) - sigma(n)^(1/tau(n)), where sigma(n) = A000203(n) = the sum of divisors of n and tau(n) = A000005(n) = the number of divisors of n.

Crossrefs

Cf. A236020, A236021, A236022, A236023, A236024, A236025, A236027.

A236027 Decimal expansion of 5^(1/2) - 7^(1/3).

Original entry on oeis.org

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

Views

Author

Jaroslav Krizek, Jan 19 2014

Keywords

Comments

Decimal expansion of minimal value of the function delta(n) = (n+1)^(1/2) - sigma(n)^(1/tau(n)) for n = 4, where delta(n) is called the delta-deviation from primality of the number n (see A236025).

Examples

			0.32313679472740059521005682...

Crossrefs

Cf. A236020, A236021, A236022, A236023, A236024, A236025, A236026.
Cf. A002163, A005482.

Programs

  • Mathematica
    RealDigits[Sqrt[5] - Surd[7, 3], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Formula

Equals A002163 - A005482.
Showing 1-7 of 7 results.

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