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.

A023199 a(n) is the least k with sigma(k) >= n*k.

Original entry on oeis.org

1, 6, 120, 27720, 122522400, 130429015516800, 1970992304700453905270400, 1897544233056092162003806758651798777216000, 4368924363354820808981210203132513655327781713900627249499856876120704000
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Following a suggestion from Ed Pegg Jr, the sequence can be written in a more readable form as: 1!, 3!, 5!, 11# * 3! * 2, 17# * 5! * 2, 29# * 7! * 4, 53# * 7! * 12, 89# * 11! * 2, 157# * 17# * 8! * 6, 271# * 23# * 10!, 487# * 29# * 10!, 857# * 37# * 11! * 42, 1487# * 53# * 15! * 2, ..., where p# = primorial(p) = A034386.
From T. D. Noe, Jul 06 2005: (Start)
Let c(p) be the smallest colossally-abundant number having the prime factor p. See A073751 for info about computing these numbers.
Then the terms of this sequence can be expressed as
a(2) = c(3)
a(3) = c(5) * 2
a(4) = c(11) / 2
a(5) = c(17) / 3
a(6) = c(29) * 14
a(7) = c(53)
a(8) = c(89) * 4
a(9) = c(157) * 34
a(10) = c(271) * 23
a(11) = c(487) / 2
a(12) = c(857) / 2
a(13) = c(1487) * 212
a(14) = c(2621) * 710
a(15) = c(4567) * 2/21
a(16) = c(8011) / 2
a(17) = c(13999) * 1630. (End)
Initially, each term is divisible by the previous one. Is there a reason this should always be true? - Santi Spadaro, Aug 13 2002
The conjecture a(n)|a(n+1) holds out to n=10. - Devin Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003
The conjecture a(n)|a(n+1) fails for n=15. - T. D. Noe, Jul 08 2005
We have a(n) = min{A007539(n), A134716(n)}, and clearly A007539(n) != A134716(n) for every n. For what values of n is the former less than the latter? - Jeppe Stig Nielsen, Jun 16 2015

Links

Crossrefs

A subsequence of A004394.
The dominating primes are in A108402.
Cf. A000203 (sigma), A007539, A034386, A073751, A134716.

Programs

  • PARI
    a(n) = my(k=1); while (sigma(k)/k < n, k++); k; \\ Michel Marcus, Oct 07 2019

Extensions

More terms from Walter Nissen, Apr 15 1997
Further terms from Devin Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003
The term a(10) = 271#23#10! was apparently found independently by Bodo Zinser and Don Reble, circa Jul 05 2005
The next term, a(11) = 487#29#10!, was corrected by Don Reble, Jul 06 2005
a(12) = 857#37#11!42 from Don Reble, Jul 06 2005
a(13) = 1487#53#15!2 found by T. D. Noe and confirmed by Don Reble, Jul 07 2005
a(14)-a(17) found by T. D. Noe and rechecked by him Oct 11 2005
a(15) corrected. The conjecture still fails at n=15. - T. D. Noe, Oct 13 2009

A216756 a(n) = floor(e^e^(n/e^gamma)).

Original entry on oeis.org

2, 5, 21, 219, 12686, 15636041, 4101615513409, 12983864742986180266005, 588668718546098799238432431873920456606, 93859726557327916653208320240553220872245170780539985934198524439788
Offset: 0

Views

Author

Arkadiusz Wesolowski, Sep 15 2012

Keywords

Comments

Assuming the Riemann hypothesis we have a(n) < A134716(n) for n >= 4.

Examples

			a(3) = 219 because exp(exp(n/exp(gamma))) = 219.0062568829....

Links

Crossrefs

Cf. A001620, A134716, A058210.

Programs

  • Magma
    [Floor(Exp(Exp(n/Exp(EulerGamma(RealField(100)))))) : n in [0..9]];
  • Mathematica
    Table[Floor[E^E^(n/E^EulerGamma)], {n, 0, 9}]

A352281 Regular triangle read by rows, T(n,k) is the least integer m such that sigma(m)/m is greater than n + (k-1)/n.

Original entry on oeis.org

2, 12, 36, 180, 720, 2520, 27720, 110880, 720720, 10810800, 122522400, 1396755360, 6983776800, 160626866400, 4497552259200, 130429015516800, 2021649740510400, 74801040398884800, 3066842656354276800, 263748468446467804800, 18594267025475980238400
Offset: 1

Views

Author

Michel Marcus, Apr 25 2022

Keywords

Examples

			Triangle begins:
  2,
  12, 36,
  180, 720, 2520,
  27720, 110880, 720720, 10810800,
  122522400, 1396755360, ...

Crossrefs

Cf. A000203 (sigma), A004394 (supersequence), A134716 (1st column).

Programs

  • PARI
    isok(i,n,k) = sigma(i)/i > n+(k-1)/n;
T(n,k) = my(i=1); while(!isok(i,n,k), i++); i;

Extensions

More terms from David A. Corneth, Apr 25 2022

A368063 a(n) is the least number k such that sigma(sigma(k) * k) > n * sigma(k) * k.

Original entry on oeis.org

1, 2, 3, 10, 160, 12155, 26558675
Offset: 0

Views

Author

Max Z. Scialabba, Dec 10 2023

Keywords

Comments

Application of A134716 (sigma(k) / k > n) to A064987.
From Daniel Suteu, Dec 21 2023: (Start)
a(7) <= 114775357632650.
a(8) <= 272113056574982766111055794421. (End)

Examples

			For n = 4, the divisors of 160 sum to 378. 160 * 378 = 60480, whose divisors sum up to 243840 > 4 * 60480.

Crossrefs

Cf. A064987, A134716.

Programs

  • Java
    public static void main(String[] args)
    {
        long max = 0;
        for (long c = 1; c < Math.pow(10, 8); c = c + 1)
        {
            if (factorSum(factorSum(c) * c) > max * factorSum(c) * c)
            {
                System.out.println(c + ": " + factorSum(c) * c);
                max = max + 1;
            }
        }
    }
    public static long factorSum(long n)
    {
        long sum = 0;
        for (long c = 1; c <= Math.sqrt(n); c = c + 1)
        {
            if (n % c == 0)
            {
                sum = sum + c;
                if (c != Math.sqrt(n))
                {
                    sum = sum + n / c;
                }
            }
        }
        return sum;
    }
  • Mathematica
    a={}; For[n=0, n<=6, n++, k=1; While[DivisorSigma[1,DivisorSigma[1,k]k] <= n DivisorSigma[1,k] k, k++]; AppendTo[a,k]]; a (* Stefano Spezia, Dec 10 2023 *)
  • PARI
    a(n) = my(k=1); while (sigma(sigma(k)*k) <= n * sigma(k) * k, k++); k; \\ Michel Marcus, Dec 10 2023

Extensions

a(6) from Michel Marcus, Dec 10 2023
Showing 1-4 of 4 results.

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