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 1-8 of 8 results.

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

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.

Original entry on oeis.org

4, 9, 6, 8, 10, 25, 14, 15, 12, 22, 16, 21, 49, 26, 27, 18, 34, 33, 20, 38, 35, 39, 46, 121, 28, 51, 58, 169, 24, 62, 57, 55, 32, 74, 69, 65, 82, 30, 86, 289, 94, 77, 87, 44, 85, 93, 106, 361, 45, 91, 95, 50, 118, 36, 52, 122, 111, 40, 42, 134, 123, 115, 142
Offset: 1

Views

Author

Jaroslav Krizek, Jan 18 2014

Keywords

Comments

The number gamma(n) = log_2(n+1) - log_tau(n) (sigma(n)) is called the gamma-deviation from primality of the number n; gamma(p) = 0 for p = prime.
Conjecture: every natural number n has a unique value of gamma(n).
For number 4; gamma(4) = log_2 (4+1) - log_tau(4) (sigma(4)) = log_2 (5) - log_3 (7) = 0,5506843457… = A236023 (minimal value of function gamma(n)).
See A234516, A234520 and A236025 for definitions of functions alpha(n), beta(n) and delta(n).
See A236024 - sequence of numbers from a(n) such that a(n) > a(k) for all k < n.

Links

Crossrefs

Cf. A236020, A236021, A236023, A236024, A236025, A236026.

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.

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.

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.

A354770 Numbers k such that d(k)/log(k) sets a new record, where d(k) is the number-of-divisors function A000005(k).

Original entry on oeis.org

2, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880, 3603600, 4324320, 6486480, 7207200, 8648640
Offset: 1

Views

Author

N. J. A. Sloane, Jun 22 2022

Keywords

Comments

A related sequence, not yet in the OEIS, is "Numbers k such that log(d(k))/log(k) > log(d(m))/log(m) for all m > k". It begins 2, 4, 6, 12, 24, 36, 60, 72, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 5040, 7560, ..., and up to this point it agrees with A236021 (except that it doesn't include 1). Does it continue to agree with A236021?

Examples

			The values of d(k)/log(k) for k = 2, 3, ... are 2.885390082, 1.820478453, 2.164042562, 1.242669869, 2.232442506, 1.027796685, 1.923593388, 1.365358840, 1.737177928, 0.8340647828, ... and reach record highs at k = 2 (2.885390082...), k = 60 (2.930872040...), and so on.

References

  • David desJardins, Posting to Math Fun Mailing List, Jun 22 2022.

Crossrefs

Cf. A000005, A066523, A236020, A236021, A354768, A354369.

Programs

  • Mathematica
    s = {}; rm = 0; Do[If[(r = DivisorSigma[0, n]/Log[n]) > rm, rm = r; AppendTo[s, n]], {n, 2, 10^5}]; s (* Amiram Eldar, Jun 22 2022 *)

Extensions

More terms from Amiram Eldar, Jun 22 2022
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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