A006086 -id:A006086 - cp's OEIS frontend

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

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

Amiram Eldar, Nov 09 2021

nonn

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.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A013929, A057723, A307958, A307959.

Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.

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

Amiram Eldar, Nov 09 2021

nonn

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.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001694, A060687.

Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.

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]

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

Amiram Eldar, Oct 01 2022

nonn

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.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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).

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

A361385 a(n) is the number of "Fermi-Dirac prime" factors (or I-components) of the n-th infinitary harmonic number.

Original entry on oeis.org

0, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 4, 3, 5, 5, 5, 4, 6, 5, 5, 6, 6, 5, 6, 5, 6, 6, 6, 5, 7, 4, 5, 5, 6, 7, 6, 6, 6, 7, 6, 6, 7, 6, 6, 6, 7, 6, 8, 7, 7, 7, 6, 7, 7, 7, 6, 8, 6, 5, 6, 7, 6, 7, 7, 6, 8, 7, 7, 8, 7, 6, 7, 8, 7, 6, 8, 7, 7, 7, 7, 9, 6, 8, 6, 8, 8, 7

Offset: 1

Amiram Eldar, Mar 10 2023

nonn

Each term appears a finite number of times in the sequence (Hagis and Cohen, 1990).

Amiram Eldar, Table of n, a(n) for n = 1..239
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian math. Soc., Vol. 41 (1990), pp. 151-158.

Cf. A006086, A006087, A361384 (analogous unitary sequence).

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; ih[1] = 1; ih[n_] := n*Times @@ f @@@ FactorInteger[n]; ic[n_] := Plus @@ (DigitCount[Last /@ FactorInteger[n], 2, 1]); ic[1] = 0; ic /@ Select[Range[10^5], IntegerQ[ih[#]] &]

A064547(n) = {my(f = factor(n)[, 2]); sum(k=1, #f, hammingweight(f[k])); } \\ Michel Marcus at A064547
ihmean(n) = {my(f = factor(n), b); n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1))); };
lista(kmax) = {my(ih); for(k = 1, kmax, ih = ihmean(k); if(denominator(ih) == 1, print1(A064547(k), ", ")));}

a(n) = A064547(A063947(n)).

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

Amiram Eldar, Nov 09 2021

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.

			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.

Cf. A160097, A160135.

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

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] &]

cp's OEIS Frontend

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

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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.

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

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A361385 a(n) is the number of "Fermi-Dirac prime" factors (or I-components) of the n-th infinitary harmonic number.

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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.

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