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 1-10 of 10 results.

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

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.

A340637 Integers whose number of divisors that are Niven numbers sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 60, 72, 120, 180, 240, 360, 720, 1080, 1800, 2160, 2520, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 30240, 45360, 50400, 60480, 75600, 90720, 100800, 110880, 120960, 151200, 166320, 221760, 277200, 302400, 332640, 453600, 498960, 554400
Offset: 1

Views

Author

Bernard Schott, Jan 14 2021

Keywords

Comments

A Niven number (A005349) is a number that is divisible by the sum of its digits.
The first 13 terms are the first 13 terms of A236021, then A236021(14) = 420 while a(14) = 720.

Examples

			The 8 divisors of 24 are all Niven numbers, and also, 24 is the smallest integer that has at least 8 divisors that are Niven numbers, hence 24 is a term.

Crossrefs

Cf. A005349, A236021, A332268, A337741.
Subsequence of A333456.
Similar for palindromes (A093036), repdigits (A340548), repunits (A340549), Zuckerman numbers (A340638).

Programs

  • Mathematica
    s[n_] := DivisorSum[n, 1 &, Divisible[#, Plus @@ IntegerDigits[#]] &]; smax = 0; seq = {}; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^6}]; seq (* Amiram Eldar, Jan 14 2021 *)
  • PARI
    f(n) = sumdiv(n, d, !(d % sumdigits(d))); \\ A332268
lista(nn) = {my(m=0); for (n=1, nn, my(x = f(n)); if (x > m, m = x; print1(n, ", ")););} \\ Michel Marcus, Jan 14 2021

Extensions

More terms from Amiram Eldar, Jan 14 2021

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.

A275026 a(n) is the largest number k such that the sum of divisors of k does not exceed the n-th power of the number of divisors of k.

Original entry on oeis.org

1, 24, 122522400, 41936006482988380963200, 2818633727625754852693848168481445291030176361088000
Offset: 1

Views

Author

Jon E. Schoenfield, Nov 12 2016

Keywords

Comments

Largest number k such that sigma(k) <= tau(k)^n.
a(4) >= 41936006482988380963200.
From Jon E. Schoenfield, Nov 01 2017: (Start)
a(5) >= 2812833572480164685801568964499317649172616193664000;
a(6) >= A002110(49)*2321816378289408000 = 1.934333...*10^107.
(End)
a(6) >= A002110(47) * 117664981274811979008000 = 1.9365109... * 10^107. - Max Alekseyev, Mar 21 2023

Examples

			24 has 8 divisors (1, 2, 3, 4, 6, 8, 12, and 24), and their sum is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60, which does not exceed 8^2 = 64. Every number k > 24 has sigma(k) > tau(k)^2, so a(2) = 24.

Crossrefs

Cf. A000005, A000203, A236021.

Extensions

a(4)-a(5) from Max Alekseyev, Mar 21 2023

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 1-10 of 10 results.

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

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