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 1-3 of 3 results.

A245214 Numbers k such that A245212(k) < 0.

Original entry on oeis.org

144, 192, 216, 240, 288, 336, 360, 384, 432, 480, 504, 540, 576, 600, 648, 672, 720, 768, 792, 840, 864, 900, 936, 960, 1008, 1056, 1080, 1152, 1200, 1248, 1260, 1296, 1320, 1344, 1440, 1512, 1536, 1560, 1584, 1620, 1632, 1680, 1728, 1800, 1824, 1848, 1872, 1920, 1944, 1980, 2016, 2040, 2100, 2112, 2160, 2240
Offset: 1

Views

Author

Jaroslav Krizek, Jul 23 2014

Keywords

Comments

If d are divisors of k then values of sequence A245212(k) are by bending moments in point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length k with bracket in point 0 by the schema: A245212(k) = (k * tau(k)) - Sum_{(d
Numbers k such that A038040(k) = k * tau(k) < A245211(k) = Sum_{(d
From Amiram Eldar, Jul 19 2024: (Start)
Numbers whose divisors have a mean abundancy index that is larger than 2.
The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 24, 243, 2571, 25583, 254794, 2551559, 25514104, 255112225, ... . Apparently, the asymptotic density of this sequence exists and equals 0.02551... .
The least odd term in this sequence is a(276918705) = 10854718875. (End)

Examples

			Number 144 is in sequence because 144 * tau(144) = 2160  < Sum_{(d<144) | 144} (d * tau(d)) = 2226.

Links

Crossrefs

Cf. A000005, A038040, A245211, A245212, A245213.

Programs

  • Magma
    [n: n in [1..100000] | (2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])) lt 0]
  • Mathematica
    f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2500], s[#] > 2 &]  (* Amiram Eldar, Jul 19 2024 *)
  • PARI
    isok(n) = (n*numdiv(n) - sumdiv(n, d, (dMichel Marcus, Aug 06 2014
  • PARI
    is(n) = {my(f = factor(n)); prod(i = 1, #f~, p=f[i,1]; e=f[i,2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2)) > 2;} \\ Amiram Eldar, Jul 19 2024

A245213 Numbers n such that A245212(n) < n.

Original entry on oeis.org

72, 96, 120, 144, 180, 192, 216, 240, 288, 336, 360, 384, 432, 480, 504, 528, 540, 576, 600, 624, 648, 672, 720, 756, 768, 792, 840, 864, 900, 936, 960, 972, 1008, 1056, 1080, 1120, 1152, 1176, 1200, 1224, 1248, 1260, 1280, 1296, 1320, 1344, 1368, 1440, 1512
Offset: 1

Views

Author

Jaroslav Krizek, Jul 23 2014

Keywords

Comments

Numbers n such that A245212(n) = (n * tau(n)) - Sum_((d
If d are divisors of n then values of sequence A245212(n) are the bending moments at point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length n with support at point 0 by the schema: A245212(n) = (n * tau(n)) - Sum_((d

Examples

			Number 72 is in sequence because A245212(72) = 62 < 72.

Links

Crossrefs

Cf. A000005, A100484, A245211, A245212, A245214.

Programs

  • Magma
    [n: n in [1..100000] | (2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])) lt n]

A245211 a(n) = Sum_{(d

Original entry on oeis.org

0, 1, 1, 5, 1, 11, 1, 17, 7, 15, 1, 47, 1, 19, 17, 49, 1, 62, 1, 67, 21, 27, 1, 151, 11, 31, 34, 87, 1, 145, 1, 129, 29, 39, 25, 254, 1, 43, 33, 219, 1, 189, 1, 127, 104, 51, 1, 423, 15, 130, 41, 147, 1, 278, 33, 287, 45, 63, 1, 589, 1, 67, 132, 321, 37, 277
Offset: 1

Views

Author

Jaroslav Krizek, Jul 23 2014

Keywords

Comments

If q are proper divisors of n then values of sequence a(n) are the bending moments at point 0 of static forces of sizes tau(q) operating in places q on the cantilever as the nonnegative number axis of length n with support at point 0 by the schema: a(n) = Sum_{q | n} (q * tau(q)).
Number n = 144 is the smallest number n such that a(n) > n * tau(n) (see A245212 and A245214).
Conjecture: 21 is only number such that a(n) = n.

Examples

			For n = 21 with proper divisors [1, 3, 7] we have: a(21) = 7 * tau(7) + 3 * tau(3) + 1 * tau(1) = 7*2 + 3*2 + 1*1 = 21.

Links

Crossrefs

Cf. A000005, A100484, A245212, A245213, A245214.

Programs

  • Magma
    [(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])-(n*(#[d: d in Divisors(n)])): n in [1..1000]];
  • PARI
    a(n) = sumdiv(n, d, (dJens Kruse Andersen, Aug 13 2014

Formula

a(n) = A060640(n) - A038040(n) = Sum_{d | n} (d * tau(d)) - n*tau(n).
a(n) = A038040(n) - A245212(n).
a(n) = 1 for n = primes.
a(n) = n + 5 for even semiprimes q = 2p > 4 (see A100484) where p = odd prime.
Showing 1-3 of 3 results.

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