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.

A349027 Exponential unitary harmonic numbers (A349026) that are not squarefree.

Original entry on oeis.org

12, 18, 36, 40, 60, 75, 84, 90, 120, 126, 132, 135, 144, 150, 156, 180, 198, 204, 228, 234, 252, 270, 276, 280, 306, 342, 348, 360, 372, 396, 414, 420, 440, 444, 450, 468, 492, 516, 520, 522, 525, 540, 544, 558, 564, 588, 600, 612, 630, 636, 660, 666, 675, 680
Offset: 1

Views

Author

Amiram Eldar, Nov 06 2021

Keywords

Comments

First differs from A348965 at n = 13.
All squarefree numbers are exponential unitary harmonic numbers.

Examples

			12 = 2^2 * 3 is a term since it is not squarefree, its exponential unitary divisors are 6 and 12, and their harmonic mean, 8, is an integer.

Links

Crossrefs

Intersection of A013929 and A349026.
Cf. A005117, A348965.

Programs

  • Mathematica
    f[p_, e_] := p^e * 2^PrimeNu[e] / DivisorSum[e, p^(e - #) &, CoprimeQ[#, e/#] &]; euhQ[1] = True; euhQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[1000], ! SquareFreeQ[#] && euhQ[#] &]

A349025 a(n) is multiplicative with a(p^e) = Sum_{d||e} p^(e-d), where d||e are the unitary divisors of e.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 1, 9, 1, 4, 1, 3, 1, 1, 1, 5, 6, 1, 10, 3, 1, 1, 1, 17, 1, 1, 1, 12, 1, 1, 1, 5, 1, 1, 1, 3, 4, 1, 1, 9, 8, 6, 1, 3, 1, 10, 1, 5, 1, 1, 1, 3, 1, 1, 4, 57, 1, 1, 1, 3, 1, 1, 1, 20, 1, 1, 6, 3, 1, 1, 1, 9, 28, 1, 1, 3, 1
Offset: 1

Views

Author

Amiram Eldar, Nov 06 2021

Keywords

Comments

First differs from A348963 at n = 16.
A number k is an exponential unitary harmonic number (A349026) if and only if a(k) | k * A278908(k).

Links

Crossrefs

The unitary version of A348963.
Cf. A005117, A278908, A349026.

Programs

  • Mathematica
    f[p_, e_] := DivisorSum[e, p^(e - #) &, CoprimeQ[#, e/#] &]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Formula

a(n) = 1 if and only if n is squarefree (A005117).

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] &]
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.