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.

A383693 Exponential unitary abundant numbers: numbers k such that A322857(k) > 2*k.

Original entry on oeis.org

900, 1764, 4356, 4500, 4900, 6084, 6300, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22500, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 36900, 38700, 40572, 42300, 42588, 44100, 47700, 47916, 49284, 49500
Offset: 1

Views

Author

Amiram Eldar, May 05 2025

Keywords

Comments

First differs from its subsequence A383697 at n = 21.
All the terms are nonsquarefree numbers (A013929), since A322857(k) = k if k is a squarefree number (A005117).
If an exponential abundant number (A129575) is exponentially squarefree (A209061), then it is in this sequence. Terms of this sequence that are not exponentially squarefree are a(21) = 22500, a(77) = 86436, a(140) = 157500, etc..
The least odd term is a(202273) = 225450225, and the least term that is coprime to 6 is a(1.002..*10^18) = 1117347505588495206025.
The asymptotic density of this sequence is Sum_{n>=1} f(A383694(n)) = 0.00089722..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)).

Examples

			900 is a term since A322857(900) = 2160 > 2*900 = 1800.

Links

Crossrefs

Cf. A005117, A209061, A322857, A322858, A361255.
Subsequence of A013929 and A129575.
Subsequences: A383694, A383697, A383698.

Programs

  • Mathematica
    f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Times @@ f @@@ FactorInteger[n] > 2 n; Select[Range[50000], q]
  • PARI
    fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
isok(k) = {my(f = factor(k)); prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}

A383697 Exponential squarefree exponential abundant numbers: numbers k such that A361174(k) > 2*k.

Original entry on oeis.org

900, 1764, 4356, 4500, 4900, 6084, 6300, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 36900, 38700, 40572, 42300, 42588, 44100, 47700, 47916, 49284, 49500
Offset: 1

Views

Author

Amiram Eldar, May 06 2025

Keywords

Comments

Subsequence of A383693 and first differs from it at n = 21.
All the terms are nonsquarefree numbers (A013929), since A361174(k) = k if k is a squarefree number (A005117).
The least odd term is a(198045) = 225450225, and the least term that is coprime to 6 is a(9.815...*10^17) = 1117347505588495206025.
The least term that is not an exponentially squarefree number (A209061) is a(8.85...*10^1324) = 2^4 * Product_{k=2..248} prime(k)^2 = 1.00786...*10^1328.
The asymptotic density of this sequence is Sum_{n>=1} f(A383698(n)) = 0.000878475..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)).

Links

Crossrefs

Subsequence of A013929, A129575 and A383693.
A383698 is a subsequence.
Cf. A005117, A209061, A361174.

Programs

  • Mathematica
    f[p_, e_] := DivisorSum[e, p^# &, SquareFreeQ[#] &]; q[k_] := Times @@ f @@@ FactorInteger[k] > 2*k; Select[Range[1000], q]
  • PARI
    ff(p, e) = sumdiv(e, d, if(issquarefree(d), p^d, 0));
isok(k) = {my(f = factor(k)); prod(i=1, #f~, ff(f[i, 1], f[i, 2])) > 2*k; }

A383694 Primitive exponential unitary abundant numbers: the powerful terms of A383693.

Original entry on oeis.org

900, 1764, 4356, 4500, 4900, 6084, 10404, 12348, 12996, 19044, 22500, 30276, 34596, 44100, 47916, 49284, 60516, 66564, 79092, 79524, 86436, 88200, 101124, 108900, 112500, 125316, 132300, 133956, 152100, 161604, 176400, 176868, 181476, 191844, 213444, 217800, 220500
Offset: 1

Views

Author

Amiram Eldar, May 05 2025

Keywords

Comments

First differs from its subsequence A383698 at n = 11.
For squarefree numbers k, eusigma(k) = k, where eusigma is the sum of exponential unitary divisors function (A322857). Thus, if m is a term (eusigma(m) > 2*m) and k is a squarefree number coprime to m, then eusigma(k*m) = eusigma(k) * eusigma(m) = k * eusigma(m) > 2*k*m, so k*m is an exponential unitary abundant number. Therefore, the sequence of exponential unitary abundant numbers (A383693) can be generated from this sequence by multiplying with coprime squarefree numbers.
The least odd term is a(1455) = 225450225, and the least term that is coprime to 6 is 1117347505588495206025.

Examples

			900 is a term since eusigma(900) = 2160 > 2 * 900, and 900 = 2^2 * 3^2 * 5^2 is a powerful number.
6300 is exponential unitary abundant, since eusigma(6300) = 15120 > 2 * 6300, but it is not a powerful number: 6300 = 2^2 * 3^2 * 5^2 * 7. Thus it is not in this sequence. It can be generated as a term of A383693 from a(1) = 900 by 7 * 900 = 6300, since 7 is squarefree and gcd(7, 900) = 1.

Links

Crossrefs

Intersection of A001694 and A383693.
A383698 is a subsequence.
Cf. A005117, A322857, A328136.

Programs

  • Mathematica
    fun[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Min[(f = FactorInteger[n])[[;; , 2]]] > 1 && Times @@ fun @@@ f > 2*n; Select[Range[250000], q]
  • PARI
    fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
isok(k) = {my(f = factor(k)); ispowerful(f) && prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.