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.

Previous Showing 11-13 of 13 results.

A357497 Nonsquarefree numbers whose harmonic mean of nonsquarefree divisors in an integer.

Original entry on oeis.org

4, 9, 12, 18, 24, 25, 28, 45, 49, 54, 60, 90, 112, 121, 126, 132, 150, 153, 168, 169, 198, 270, 289, 294, 336, 361, 364, 414, 529, 560, 594, 630, 637, 684, 726, 841, 918, 961, 1014, 1140, 1232, 1305, 1350, 1369, 1512, 1521, 1638, 1680, 1681, 1710, 1734, 1849, 1984
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2022

Keywords

Comments

Analogous to harmonic numbers (A001599) with nonsquarefree divisors.
The squares of primes (A001248) are terms since they have a single nonsquarefree divisor.
If p is a prime then 6*p^2 is a term.

Examples

			12 is a term since its nonsquarefree divisors are 4 and 12 and their harmonic mean is 6 which is an integer.

Links

Crossrefs

Cf. A162296, A322609, A357493, A357494, A357495, A357496.
Subsequence of A013929.
Subsequence: A001248.
Similar sequences: A001599 (harmonic numbers), A006086 (unitary), A063947 (infinitary), A286325 (bi-unitary), A319745 (nonunitary), A335387 (tri-unitary).

Programs

  • Mathematica
    q[n_] := Length[d = Select[Divisors[n], ! SquareFreeQ[#] &]] > 0 && IntegerQ[HarmonicMean[d]]; Select[Range[2000], q]

A361787 Bi-unitary arithmetic numbers k whose mean bi-unitary divisor is a bi-unitary divisor of k.

Original entry on oeis.org

1, 6, 60, 270, 420, 630, 672, 2970, 5460, 8190, 10080, 22848, 30240, 99792, 136500, 172900, 204750, 208656, 245700, 249480, 312480, 332640, 342720, 385560, 491400, 695520, 708288, 791700, 819000, 861840, 1028160, 1037400, 1187550, 1228500, 1421280, 1528800, 1571328
Offset: 1

Views

Author

Amiram Eldar, Mar 24 2023

Keywords

Comments

Also, bi-unitary harmonic numbers k whose harmonic mean of the bi-unitary divisors of k is a bi-unitary divisor of k.

Examples

			6 is a term since the arithmetic mean of its bi-unitary divisors, {1, 2, 3, 6}, is 3, and 3 is also a bi-unitary divisor of 6.
60 is a term since the arithmetic mean of its bi-unitary divisors, {1, 3, 4, 5, 12, 15, 20, 60}, is 15, and 15 is also a bi-unitary divisor of 60.

Links

Crossrefs

Subsequence of A286325 and A361786.
Similar sequence: A007340, A353039, A361387.
Cf. A188999, A222266, A286324.

Programs

  • Mathematica
    biudivQ[f_, d_] := AllTrue[f, OddQ[Last[#]] || IntegerExponent[d, First[#]] != Last[#]/2 &]; biuDivs[n_] := Module[{d = Divisors[n], f = FactorInteger[n]}, Select[d, biudivQ[f, #] &]]; Select[Range[10^5], IntegerQ[(r = Mean[(i = biuDivs[#])])] && MemberQ[i, r] &]
  • PARI
    isbdiv(f, d) = {for (i=1, #f~, if(f[i, 2]%2 == 0 && valuation(d, f[i, 1]) == f[i, 2]/2, return(0))); 1;}
is(n) = {my(f = factor(n), r, p, e); r = prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (p^(e+1)-1)/((e + 1)*(p-1)), ((p^(e+1)-1)/(p-1)-p^(e/2))/e)); denominator(r) == 1 && n%r==0 && isbdiv(f, r); }

A349178 Nonexponential harmonic numbers: numbers k that are not prime powers such that the harmonic mean of the nonexponential divisors of k is an integer.

Original entry on oeis.org

1645, 5742, 6336, 8925, 9450, 88473
Offset: 1

Views

Author

Amiram Eldar, Nov 09 2021

Keywords

Comments

The prime powers are excluded since the primes and the squares of primes have a single nonexponential divisor (the number 1).
a(7) > 6.6*10^10, if it exists.

Examples

			1645 is a term since the set of its nonexponential divisors is {1, 5, 7, 35, 47, 235, 329} and the harmonic mean of this set, 5, is an integer.

Crossrefs

Cf. A160097, A160135.
Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964, A349026.

Programs

  • Mathematica
    dQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m]); expDivQ[n_, d_] := Module[{ft = FactorInteger[n]}, And @@ MapThread[dQ, {ft[[;; , 2]], IntegerExponent[d, ft[[;; , 1]]]}]]; neDivs[1] = {0}; neDivs[n_] := Module[{d = Divisors[n]}, Select[d, ! expDivQ[n, #] &]]; Select[Range[10^4], Length[(d = neDivs[#])] > 1 && IntegerQ @ HarmonicMean[d] &]
Previous Showing 11-13 of 13 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.