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Exponential harmonic (or e-harmonic) numbers of type 2: numbers k such that the harmonic mean of the exponential divisors of k is an integer.
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9
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
COMMENTS
Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 2.
Apparently, most exponential harmonic numbers of type 1 ( A348961) are also terms of this sequence. Those that are not exponential harmonic numbers of type 2 are 1936, 5808, 9680, 13552, 17424, 29040, ...
EXAMPLE
The squarefree numbers are trivial terms. If k is squarefree, then it has a single exponential divisor, k itself, and thus the harmonic mean of its exponential divisors is also k, which is an integer.
12 is a term since its exponential divisors are 6 and 12, and their harmonic mean, 8, is an integer.
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[100], ehQ]
Exponential unitary harmonic numbers: numbers k such that the harmonic mean of the exponential unitary divisors of k is an integer.
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4
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
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).
EXAMPLE
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.
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]
Numbers k such that the set of divisors {d | k, BitOr(k, d) = k} has an integer harmonic mean.
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3
1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 45, 48, 56, 60, 64, 90, 96, 112, 120, 128, 180, 192, 224, 240, 256, 360, 384, 448, 480, 496, 512, 720, 768, 896, 960, 992, 1024, 1440, 1536, 1792, 1920, 1984, 2048, 2880, 3072, 3584, 3840, 3968, 4096, 5760, 6144, 7168, 7680
COMMENTS
Equivalently, the set of divisors can be defined by {d | k, BitAnd(k, d) = d}.
Analogous to harmonic (or Ore) numbers ( A001599) where the divisors d of k are restricted by BitOr(k, d) = k or BitAnd(k, d) = d. If k is a term then so is 2*k. The primitive terms are in A362805. Thus, this sequence includes all the powers of 2 ( A000079), all the numbers of the form 3*2^m and 15*2^m for m >= 1, and all the numbers of the form 7*2^m for m >= 2. All the even perfect numbers ( A000396) are terms: if k = 2^(p-1)*(2^p-1) is a perfect number (where p is a Mersenne exponent, A000043), then the only divisors of k such that BitOr(k, d) = k are 2^(p-1) and k itself, and the harmonic mean of 2^(p-1) and 2^(p-1)*(2^p-1) is 2^p - 1. Are 1 and 45 the only odd terms in this sequence?
MATHEMATICA
q[n_] := IntegerQ[HarmonicMean[Select[Divisors[n], BitAnd[n, #] == # &]]]; Select[Range[10^4], q]
PROG
(PARI) div(n) = select(x->(bitor(x, n) == n), divisors(n));
is(n) = {my(d = div(n)); denominator(#d/sum(i = 1, #d, 1/d[i])) == 1; }
Tri-unitary harmonic numbers: numbers k such that the harmonic mean of the tri-unitary divisors of k is an integer.
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2
1, 6, 45, 60, 90, 270, 420, 630, 2970, 5460, 8190, 9100, 15925, 27300, 36720, 40950, 46494, 47520, 54600, 81900, 95550, 136500, 163800, 172900, 204750, 232470, 245700, 257040, 332640, 409500, 464940, 491400, 646425, 716625, 790398, 791700, 819000, 900900, 929880
EXAMPLE
45 is a term since its tri-unitary divisors are {1, 5, 9, 45} and their harmonic mean, 3, in an integer.
MATHEMATICA
f1[p_, e_] := If[e == 3 || e == 6, 4, 2]; f2[p_, e_] := If[e == 3, (p^4 - 1)/(p - 1), If[e == 6, (p^8 - 1)/(p^2 - 1), p^e + 1]]; f[p_, e_] := p^e * f1[p, e]/f2[p, e]; tuhQ[1] = True; tuhQ[n_] := IntegerQ[Times @@ (f @@@ FactorInteger[n])]; Select[Range[10^4], tuhQ]
Noninfinitary harmonic numbers: numbers such that the harmonic mean of their noninfinitary divisors is an integer.
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2
4, 9, 12, 18, 25, 45, 49, 60, 96, 112, 121, 126, 150, 169, 289, 294, 336, 361, 448, 486, 529, 540, 560, 600, 637, 672, 726, 841, 961, 1014, 1232, 1344, 1350, 1369, 1638, 1680, 1681, 1734, 1849, 2166, 2209, 2430, 2809, 2850, 3174, 3481, 3721, 3822, 4200, 4320, 4489
COMMENTS
Includes all the squares of primes ( A001248), since they are the numbers with a single noninfinitary divisor.
EXAMPLE
12 is a term since its noninfinitary divisors are {2, 6}, and their harmonic mean, 3, is an integer.
MATHEMATICA
nidiv[1] = {}; nidiv[n_] := Complement[Divisors[n], Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; Select[Range[5000], (d = nidiv[#]) != {} && IntegerQ@ HarmonicMean[d] &]
Numbers that are both unitary and nonunitary harmonic numbers.
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1
45, 60, 3780, 64260, 3112200, 6320160
COMMENTS
a(7) > 10^12, if it exists.
For each term the two sets of unitary and nonunitary divisors both contain more than one element. The only number with a single unitary divisor is 1 which does not have nonunitary divisors. Numbers with a single nonunitary divisor are the squares of primes which are not unitary harmonic numbers. Therefore, this sequence is a subsequence of A348715.
EXAMPLE
45 is a term since the unitary divisors of 45 are 1, 5, 9 and 45, and their harmonic mean is 3, and the nonunitary divisors of 45 are 3 and 15, and their harmonic mean is 5.
MATHEMATICA
usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[65000], !SquareFreeQ[#] && IntegerQ[# * (d = 2^PrimeNu[#])/ (s = usigma[#])] && IntegerQ[# * (DivisorSigma[0, #] - d)/(DivisorSigma[1, #] - s)] &]
Coreful harmonic numbers: nonsquarefree numbers k such that the harmonic mean of the coreful divisors of k is an integer.
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1
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
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.
EXAMPLE
12 is a term since its coreful divisors are 6 and 12 and their harmonic mean, 8, is an integer.
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[#] &]
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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1
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
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.
EXAMPLE
100 is a term since its powerful divisors > 1 are 4, 25 and 100 and their harmonic mean, 10, is an integer.
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]
Nonsquarefree numbers whose harmonic mean of nonsquarefree divisors in an integer.
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1
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
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.
EXAMPLE
12 is a term since its nonsquarefree divisors are 4 and 12 and their harmonic mean is 6 which is an integer.
MATHEMATICA
q[n_] := Length[d = Select[Divisors[n], ! SquareFreeQ[#] &]] > 0 && IntegerQ[HarmonicMean[d]]; Select[Range[2000], q]
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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0
1645, 5742, 6336, 8925, 9450, 88473
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.
EXAMPLE
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.
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] &]
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