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 1-9 of 9 results.

A234516 Composite numbers n sorted by decreasing values of alpha(n) = log_n(sigma(n)) - log_n(n+1), where sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

12, 6, 24, 36, 18, 30, 60, 8, 4, 48, 20, 72, 120, 84, 16, 42, 10, 40, 180, 90, 96, 144, 240, 168, 108, 360, 28, 54, 420, 252, 132, 80, 216, 210, 32, 126, 300, 336, 480, 56, 192, 288, 720, 840, 66, 504, 156, 540, 150, 264, 14, 600, 140, 270, 1260, 432, 78, 1080
Offset: 1

Views

Author

Jaroslav Krizek, Jan 03 2014

Keywords

Comments

The number alpha(n) = log_n(sigma(n)) - log_n(n+1) = log_n[sigma(n) / (n+1)] is called the alpha-deviation from primality of number n; alpha(p) = 0 for p = prime. See A234520 for definition of beta(n).
Lim_n->infinity alpha(n) = 0.
Conjecture: Every composite number n has a unique value of alpha(n).
Conjecture: sequence A234517 is not the sequence of numbers from a(n) such that a(n) > a(k) for all k < n.

Examples

			For the number 12; alpha(12) = log_12(sigma(12)) - log_12(12+1) = log_12(28) - log_12(13) = 0.308766187… = A234518 (maximal value of function alpha(n)).

Links

Crossrefs

Cf. A234515, A234517, A234518, A234519, A234520, A234521, A234522, A234523, A234524.

Programs

  • PARI
    lista(nn) = {v = vector(nn, n, if ((n==1) || isprime(n), 0, log(sigma(n)/(n+1))/log(n))); v = vecsort(v,,5); for (i=1, 80, print1(v[i], ", "));} \\ Michel Marcus, Dec 10 2014

A234520 Composite numbers n sorted by decreasing values of beta(n) = sigma(n)^(1/n) - (n+1)^(1/n), where sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

4, 6, 8, 12, 10, 18, 16, 24, 14, 20, 9, 15, 30, 36, 28, 22, 32, 40, 48, 42, 21, 26, 60, 54, 44, 27, 72, 56, 34, 50, 45, 52, 38, 66, 84, 33, 64, 90, 80, 70, 96, 78, 46, 39, 120, 68, 108, 35, 88, 76, 63, 25, 100, 58, 102, 126, 144, 112, 132, 62, 104, 75, 51, 92
Offset: 1

Views

Author

Jaroslav Krizek, Jan 14 2014

Keywords

Comments

The number beta(n) = sigma(n)^(1/n) - (n+1)^(1/n) is called the beta-deviation from primality of the number n; beta(p) = 0 for p = prime. See A234516 for definition of alpha(n).
For number 4; beta(4) = sigma(4)^(1/4) - (4+1)^(1/4), = 7^(1/4) - 5^(1/4) = 0,131227780… = A234522 (maximal value of function beta(n)).
Lim_n->infinity beta(n) = 0.
Conjecture: Every composite number n has a unique value of number beta(n).
See A234523 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

Links

Crossrefs

Cf. A234515, A234516, A234517, A234518, A234519, A234521, A234522, A234523, A234524.

A234515 Natural numbers n sorted by decreasing values of number k(n) = log_n(sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

2, 4, 6, 12, 8, 24, 18, 3, 36, 30, 10, 60, 20, 48, 16, 72, 120, 84, 42, 40, 180, 90, 96, 28, 144, 240, 168, 14, 108, 360, 54, 32, 420, 80, 252, 132, 216, 56, 210, 126, 300, 66, 336, 480, 192, 288, 720, 840, 156, 504, 150, 540, 264, 140, 600, 78, 270, 1260, 432
Offset: 1

Views

Author

Jaroslav Krizek, Jan 03 2014

Keywords

Comments

Number k(n) = log_n(sigma(n)) = log(sigma(n)) / log(n) is number such that n^k(n) = sigma(n).
The last term of this infinite sequence is number 1, k(1) = 1 (minimal value of function k(n)).
Conjecture: Every natural number n has a unique value of number k(n).
See A234517 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

Examples

			For number 2; k(2) = log_2(sigma(2)) = log_2(3) = 1,5849625007… = A020857 (maximal value of function k(n)).

Links

Crossrefs

Cf. A234516, A234517, A234518, A234519, A234520, A234521, A234522, A234523, A234524.

Programs

  • PARI
    lista(nn=100000) = {v = vector(nn, n, if (n==1, 0, log(sigma(n))/log(n))); v = vecsort(v,,5); for (i=1, 80, print1(v[i], ", "));} \\ Michel Marcus, Dec 11 2014

A234519 Natural numbers n sorted by decreasing values of number k(n) = sigma(n)^(1/n), where sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

2, 4, 3, 6, 5, 8, 7, 10, 9, 12, 14, 11, 16, 15, 18, 13, 20, 24, 17, 21, 22, 19, 28, 26, 30, 23, 25, 27, 32, 36, 34, 33, 29, 40, 31, 35, 42, 38, 39, 44, 48, 37, 45, 46, 41, 50, 54, 52, 43, 56, 60, 51, 49, 47, 55, 58, 57, 64, 66, 53, 63, 62, 72, 68, 70, 59, 65
Offset: 1

Views

Author

Jaroslav Krizek, Jan 04 2014

Keywords

Comments

Number k(n) = sigma(n)^(1/n) is number such that k(n)^n = sigma(n).
For number 2; k(2) = sigma(2)^(1/2) = sqrt(3) = 1,732050807568… = A002194 (maximal value of function k(n)).
The last term of this infinite sequence is number 1, k(1) = 1 (minimal value of function k(n)).
Conjecture: Every natural number n has a unique value of number k(n).
See A234521 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

Links

Crossrefs

Cf. A234515, A234516, A234517, A234518, A234520, A234521, A234522, A234523, A234524.

Programs

A234517 Sequence of numbers from A234515 such that A234515(n) > A234515(k) for all k < n.

Original entry on oeis.org

2, 4, 6, 12, 24, 36, 60, 72, 120, 180, 240, 360, 420, 480, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 12600, 15120, 20160, 27720, 30240, 32760, 55440, 65520, 83160, 110880, 131040, 166320, 196560, 221760, 277200, 332640, 360360, 393120, 415800, 443520
Offset: 1

Views

Author

Jaroslav Krizek, Jan 03 2014

Keywords

Comments

A234515 = natural numbers n sorted by decreasing values of number k(n) = log_n(sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n.
Conjecture: sequence a(n) for n >= 4 is not the same as sequence of numbers from A234516 such that A234516(n) > A234516(k) for all k < n.

Crossrefs

Cf. A002182 (numbers n such that tau(n) > tau(k) for all k < n), A002473 (numbers whose prime divisors are all <= 7).
Cf. A234515, A234516, A234518, A234519, A234520, A234521, A234522, A234523, A234524.

Programs

  • PARI
    lista(nn) = {v = vector(nn, n, if (n==1, 0, log(sigma(n))/log(n))); v = vecsort(v,,5); m = 0; for (n=1, #v, if (v[n] > m, m = v[n]; print1(m, ", ")););} \\ Michel Marcus, Dec 11 2014

Extensions

480 inserted and more terms from Michel Marcus, Dec 11 2014

A234518 Decimal expansion of log_12 (28/13).

Original entry on oeis.org

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

Views

Author

Jaroslav Krizek, Jan 03 2014

Keywords

Comments

Decimal expansion of maximal value of function alpha(n) = alpha-deviation from primality of number n = log_n(sigma(n)) - log_n(n+1) = log_n[sigma(n) / (n+1)] for n = 12, when alpha(12) = log_12(sigma(12)) - log_12(12+1) = log_12(28) - log_12(13) = log_12 (28/13) = 0,308766187…; alpha(p) = 0 for p = prime.

Examples

			0,3087661875664928997884010546628878661481631771557148…

Links

Crossrefs

Cf. A234515, A234516, A234517, A234519, A234520, A234521, A234522, A234523, A234524.

Programs

  • Mathematica
    RealDigits[Log[12,28/13],10,120][[1]] (* Harvey P. Dale, Apr 17 2022 *)
  • PARI
    log(28/13)/log(12) \\ Michel Marcus, Dec 11 2014

Formula

Decimal expansion of (A016651-A016636) / A016635.

A234521 Sequence of numbers from A234519 such that A234519(n) > A234519(k) for all k < n.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 66, 72, 78, 80, 84, 90, 96, 100, 102, 108, 112, 120, 126, 132, 144, 150, 156, 160, 168, 180, 192, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 276, 280, 288, 300
Offset: 1

Views

Author

Jaroslav Krizek, Jan 13 2014

Keywords

Comments

A234519 = natural numbers n sorted by decreasing values of number k(n) = sigma(n)^(1/n), where sigma(n) = A000203(n) = the sum of divisors of n.
Conjecture: a(n) = supersequence of A002093 - highly abundant numbers - numbers n such that sigma(n) > sigma(m) for all m < n.

Crossrefs

Cf. A234515, A234516, A234517, A234518, A234519, A234520, A234522, A234523, A234524.

A234522 Decimal expansion of 7^(1/4) - 5^(1/4).

Original entry on oeis.org

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

Views

Author

Jaroslav Krizek, Jan 14 2014

Keywords

Comments

Decimal expansion of maximal value of function beta(n) = sigma(n)^(1/n) - (n+1)^(1/n) for n = 4, where beta(n) is called the beta-deviation from primality of number n (see A234520). Lim_n->infinity beta(n) = 0.
An algebraic integer with degree 16 and minimal polynomial x^16 - 48x^12 - 3896x^8 - 53952x^4 + 16. - Charles R Greathouse IV, Apr 25 2016

Examples

			0.13122778047656520129933335...

Links

Crossrefs

Cf. A234515, A234516, A234517, A234518, A234519, A234520, A234521, A234523, A234524.

Programs

Formula

Equals A011005-A011003.

Extensions

a(97) corrected by Georg Fischer, Apr 04 2020

A234524 Numbers n such that A234519(n) = n.

Original entry on oeis.org

3, 5, 7, 9, 38, 39, 55, 57, 62, 69, 82, 99, 122, 146, 147, 207, 254, 274, 278, 429, 454, 513, 554, 561, 634, 694, 783, 794, 987, 1539, 1682, 2373, 5846, 6106, 6758, 6806, 7011, 7102, 7138, 7426, 8066, 8494, 9106, 9686, 10106, 16306, 19654, 22287, 23722, 28749
Offset: 1

Views

Author

Jaroslav Krizek, Jan 18 2014

Keywords

Comments

A234519 = natural numbers n sorted by decreasing values of number k(n) = sigma(n)^(1/n), where sigma(n) = A000203(n) = the sum of divisors of n.

Crossrefs

Cf. A234515, A234516, A234517, A234518, A234519, A234520, A234521, A234522, A234523.
Showing 1-9 of 9 results.

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