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 1-5 of 5 results.

A360526 Odd numbers k such that A360522(k) > 2*k.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035, 116655
Offset: 1

Views

Author

Amiram Eldar, Feb 10 2023

Keywords

Comments

First differs from A112643, A129485, A249263 at n = 46: a(46) = 165165 is not a term of these sequences.

Examples

			15015 is a term since A360522(15015) = 32256 > 2*15015.

Links

Crossrefs

Cf. A360522.
Subsequence of A005101, A005231 and A360525.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A348275, A348605.

Programs

  • Mathematica
    f[p_, e_] := p^e + e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[1, 10^5, 2], q]
  • PARI
    isab(n) = { my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]) > 2*n;}
is(n) = n%2 && isab(n);

A360525 Numbers k such that A360522(k) > 2*k.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 140, 150, 156, 168, 174, 180, 186, 204, 210, 222, 228, 246, 252, 258, 276, 282, 294, 300, 318, 330, 348, 354, 360, 366, 372, 390, 402, 420, 426, 438, 444, 462, 474, 492, 498, 510, 516, 534, 546, 564
Offset: 1

Views

Author

Amiram Eldar, Feb 10 2023

Keywords

Comments

First differs from A308127 at n = 15.
Analogous to abundant numbers (A005101) with A360522 instead of A000203.
Subsequence of A005101 because A360522(n) <= A000203(n) for all n.
The least odd term is a(1698) = A360526(1) = 15015.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 8, 95, 1135, 10890, 110867, 1104596, 11048123, 110534517, 1105167384, 11051009278, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1105...

Examples

			30 is a term since A360522(30) = 72 > 2*30.

Links

Crossrefs

Cf. A000203, A308127, A360522, A360524, A360526.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

Programs

  • Mathematica
    f[p_, e_] := p^e + e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[1000], q]
  • PARI
    is(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]) > 2*n;}

A360524 Numbers k such that A360522(k) = 2*k.

Original entry on oeis.org

6, 12, 198, 240, 264, 270, 396, 540, 6720, 7920, 11880, 13770, 27540, 221760, 337440, 605880, 2500344, 6072570, 11135520, 12145140, 267193080, 441692160, 1112629770, 2225259540, 14575841280, 48955709880
Offset: 1

Views

Author

Amiram Eldar, Feb 10 2023

Keywords

Comments

Analogous to perfect numbers (A000396) with A360522 instead of A000203.
a(27) > 10^11, if it exists.

Examples

			6 is a term since A360522(6) = 12 = 2 * 6.

Crossrefs

Cf. A000203, A360522.
Similar sequences: A000396, A002827, A007357, A054979, A322486, A324707.

Programs

  • Mathematica
    f[p_, e_] := p^e + e; q[n_] := Times @@ f @@@ FactorInteger[n] == 2*n; Select[Range[10^6], q]
  • PARI
    is(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]) == 2*n;}

A360527 Numbers k such that A360522(k) = A360522(k+1).

Original entry on oeis.org

4, 8, 14, 176, 895, 956, 957, 1334, 1634, 1724, 1725, 1844, 1934, 2685, 2871, 3404, 3759, 4047, 4136, 5175, 7004, 7315, 7599, 8055, 12104, 13760, 18415, 20145, 29392, 32944, 33998, 42818, 44095, 44516, 49599, 60356, 74918, 79826, 79833, 84134, 85172, 85744, 86343
Offset: 1

Views

Author

Amiram Eldar, Feb 10 2023

Keywords

Comments

Numbers k such that A360522(k) = A360522(k+1) = A360522(k+2) exist: 956 and 1724. Are there any other terms like these? There are none below 1.8*10^10.

Examples

			4 is a term since A360522(4) = A360522(5) = 6.

Links

Crossrefs

Cf. A360522.
Similar sequences: A002961, A064115, A064125, A293183, A306985.

Programs

  • Mathematica
    f[p_, e_] := p^e + e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Position[Partition[Array[s, 10^5], 2, 1], _?(SameQ @@ # &)] // Flatten
  • PARI
    s(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]);}
lista(nmax) = {my(s1 = s(1), s2); for(n=2, nmax, s2=s(n); if(s1 == s2, print1(n-1, ", ")); s1 = s2); }

A360523 a(n) = Sum_{d|n} mu(rad(d)) * delta_d(n/d), where rad(n) = A007947(n) and delta_d(n) is the greatest divisor of n that is relatively prime to d.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 5, 7, 4, 10, 4, 12, 6, 8, 12, 16, 7, 18, 8, 12, 10, 22, 10, 23, 12, 24, 12, 28, 8, 30, 27, 20, 16, 24, 14, 36, 18, 24, 20, 40, 12, 42, 20, 28, 22, 46, 24, 47, 23, 32, 24, 52, 24, 40, 30, 36, 28, 58, 16, 60, 30, 42, 58, 48, 20, 66, 32, 44, 24
Offset: 1

Views

Author

Amiram Eldar, Feb 10 2023

Keywords

Comments

Analogous to the Euler totient function (A000010) as A360522 is analogous to A000203.

Links

Crossrefs

Cf. A000010, A000203, A005117, A007947 (rad), A008683 (mu), A047994, A078779, A360522.

Programs

  • Mathematica
    f[p_, e_] := p^e - e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - f[i,2]);}

Formula

Multiplicative with a(p^e) = p^e - e.
Dirichlet g.f.: zeta(s-1)*zeta(s)^2 * Product_{p prime} (1 - 3/p^s + 1/p^(2*s-1) + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p/((p-1)*(p+1)^2)) = 0.3243742337... .
A000010(n) <= a(n) <= A047994(n) (Khan, 2005).
a(n) = A000010(n) if and only if n is in A078779 (i.e., n is either squarefree or twice a squarefree number).
a(n) = A047994(n) if and only if n is in A005117 (i.e., n is squarefree).
Showing 1-5 of 5 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.