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 1-8 of 8 results.

A348963 a(n) is multiplicative with a(p^e) = Sum_{d|e} p^(e-d).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 1, 13, 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, 13, 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, 13, 37, 1, 1, 3
Offset: 1

Views

Author

Amiram Eldar, Nov 05 2021

Keywords

Comments

The function S_e(n) in Sándor (2006).
A number k is an exponential harmonic of type 2 (A348964) if and only if a(k) | k * A049419(k).

Links

Crossrefs

Cf. A005117, A049419, A348964.

Programs

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

Formula

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

A348965 Exponential harmonic numbers of type 2 that are not squarefree.

Original entry on oeis.org

12, 18, 36, 40, 60, 75, 84, 90, 120, 126, 132, 135, 150, 156, 180, 198, 204, 208, 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, 624, 630, 636, 660, 666, 675
Offset: 1

Views

Author

Amiram Eldar, Nov 05 2021

Keywords

Comments

Sándor (2006) proved that all squarefree numbers are exponential harmonic numbers of type 2.

Examples

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

Links

Crossrefs

Intersection of A013929 and A348964.
Cf. A005117, A348962.

Programs

  • Mathematica
    f[p_, e_] := p^e * DivisorSigma[0, e] / DivisorSum[e, p^(e-#) &]; ehQ[1] = True; ehQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[1000], ! SquareFreeQ[#] && ehQ[#] &]

A349026 Exponential unitary harmonic numbers: numbers k such that the harmonic mean of the exponential unitary divisors of k is an integer.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94
Offset: 1

Views

Author

Amiram Eldar, Nov 06 2021

Keywords

Comments

First differs from A348964 at n = 102. a(102) = 144 is not an exponential harmonic number of type 2.
The exponential unitary divisors of n = Product p(i)^e(i) are all the numbers of the form Product p(i)^b(i) where b(i) is a unitary divisor of e(i) (see A278908).
Equivalently, numbers k such that A349025(k) | k * A278908(k).

Examples

			The squarefree numbers are trivial terms. If k is squarefree, then it has a single exponential unitary divisor, k itself, and thus the harmonic mean of its exponential unitary divisors is also k, which is an integer.
144 is a term since its exponential unitary divisors are 6, 18, 48 and 144, and their harmonic mean, 16, is an integer.

Links

Crossrefs

Cf. A278908 (number of exponential unitary divisors), A322857, A322858, A323310, A349025, A349027.
Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.

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[100], euhQ]

A348961 Exponential harmonic (or e-harmonic) numbers of type 1: numbers k such that esigma(k) | k * d_e(k), where d_e(k) is the number of exponential divisors of k (A049419) and esigma(k) is their sum (A051377).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105
Offset: 1

Views

Author

Amiram Eldar, Nov 05 2021

Keywords

Comments

First differs from A005117 at n = 24, from A333634 and A348499 at n = 47, and from A336223 at n = 63.
Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 1, and that an e-perfect number (A054979) is a term of this sequence if and only if at least one of the exponents in its prime factorization is not a perfect square.
Since all the e-perfect numbers are products of a primitive e-perfect number (A054980) and a coprime squarefree number, and all the known primitive e-perfect numbers have a nonsquare exponent in their prime factorizations, there is no known e-perfect number that is not in this sequence.

Examples

			3 is a term since esigma(3) = 3, 3 * d_e(3) = 3 * 1, so esigma(3) | 3 * d_e(3).

Links

Crossrefs

A005117 and A348962 are subsequences.
Cf. A049419, A051377, A322791, A333634, A336223, A348499, A348964.

Programs

  • Mathematica
    f[p_, e_] := p^e * DivisorSigma[0, e] / DivisorSum[e, p^# &]; ehQ[1] = True; ehQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], ehQ]

A349019 Modified e-perfect numbers: numbers k such that A348963(k) | k.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101
Offset: 1

Views

Author

Amiram Eldar, Nov 06 2021

Keywords

Comments

First differs from A225354 at n = 25.
Not to be confused with modified exponential perfect numbers (A323757).
Sándor (2006) showed that the exponential harmonic numbers of type 2 (A348964) are terms in this sequence.
All the squarefree numbers are terms (A005117), since A348963(k) = 1 if k is squarefree.

Examples

			12 is a term since A348963(12) = 3 is a divisor of 12.

Links

Crossrefs

A005117, A348964 and A349020 are subsequences.
Cf. A225354, A323757, A348963.

Programs

  • Mathematica
    f[p_, e_] := p^e/DivisorSum[e, p^(e - #) &]; modEPerfQ[1] = True; modEPerfQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], modEPerfQ]

A349180 Coreful harmonic numbers: nonsquarefree numbers k such that the harmonic mean of the coreful divisors of k is an integer.

Original entry on oeis.org

12, 18, 36, 56, 60, 75, 84, 90, 126, 132, 150, 156, 168, 180, 198, 204, 228, 234, 240, 252, 276, 280, 306, 342, 348, 351, 372, 392, 396, 414, 420, 444, 450, 468, 492, 504, 516, 522, 525, 558, 564, 588, 612, 616, 630, 636, 660, 666, 684, 702, 708, 720, 726, 728
Offset: 1

Views

Author

Amiram Eldar, Nov 09 2021

Keywords

Comments

A divisor of a number k is coreful if it is divisible by every prime that divides k.
The sequence is restricted to nonsquarefree numbers since the squarefree numbers have a single coreful divisor and thus they trivially have an integer harmonic mean.

Examples

			12 is a term since its coreful divisors are 6 and 12 and their harmonic mean, 8, is an integer.

Links

Crossrefs

Cf. A005117, A013929, A057723, A307958, A307959.
Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.

Programs

  • Mathematica
    rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; corHarmQ[n_] := Module[{r = rad[n], d}, d = Select[Divisors[n], rad[#] == r &]; IntegerQ[HarmonicMean[d]]]; Select[Range[10^3], !SquareFreeQ[#] && corHarmQ[#] &]

A349181 Powerful harmonic numbers: numbers k such that the set of powerful divisors of k that are larger than 1 has more than one element and that the harmonic mean of this set is an integer.

Original entry on oeis.org

100, 300, 700, 1100, 1225, 1300, 1700, 1900, 2100, 2300, 2450, 2900, 3100, 3300, 3675, 3700, 3900, 4100, 4225, 4300, 4700, 5100, 5300, 5700, 5900, 6100, 6700, 6900, 7100, 7300, 7350, 7700, 7900, 8300, 8450, 8700, 8900, 9100, 9300, 9700, 10100, 10300, 10700, 10900
Offset: 1

Views

Author

Amiram Eldar, Nov 09 2021

Keywords

Comments

Numbers with a single powerful divisor > 1 are A060687 and trivially have an integer harmonic mean.
The least term that is not divisible by 5 (or 25) is a(5446) = 1413721.

Examples

			100 is a term since its powerful divisors > 1 are 4, 25 and 100 and their harmonic mean, 10, is an integer.

Links

Crossrefs

Cf. A001694, A060687.
Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.

Programs

  • Mathematica
    powQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 1; powHarmQ[n_] := Module[{d = Select[Divisors[n], powQ]}, Length[d] > 1 && IntegerQ[HarmonicMean[d]]]; Select[Range[10^4], powHarmQ]

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 1-8 of 8 results.

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