A063947 -id:A063947 - cp's OEIS frontend

A319745 Nonunitary harmonic numbers: numbers such that the harmonic mean of their nonunitary divisors is an integer.

4, 9, 12, 18, 24, 25, 45, 49, 54, 60, 112, 121, 126, 150, 168, 169, 270, 289, 294, 336, 361, 529, 560, 594, 637, 726, 841, 961, 1014, 1232, 1369, 1638, 1680, 1681, 1734, 1849, 1984, 2166, 2184, 2209, 2430, 2520, 2688, 2700, 2809, 2850, 3174, 3481, 3721, 3780

Offset: 1

Amiram Eldar, Sep 27 2018

nonn

Includes all the numbers with a single nonunitary divisor. Those with more than one: 12, 18, 24, 45, 54, 60, 112, ...
Supersequence of A064591 (nonunitary perfect numbers).
Ligh & Wall showed that if p, 2p-1 and 2^p-1 are distinct primes (A172461, except for 2), then the following numbers are in the sequence: 6*p^2, p^2*(2p-1), 6*p^2*(2p-1), 2^(p+1)*3*(2^p-1), 2^(p+1)*15*(2^p-1) and 2^(p+1)*(2p-1)*(2^p-1).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.

Cf. A001599, A006086, A063947, A064591, A172461, A286325.

nudiv[n_] := Block[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; nhQ[n_]:= Module[ {divs=nudiv[n]}, Length[divs] > 0 && IntegerQ[HarmonicMean[divs]]]; Select[Range[30000], nhQ]

hm(v) = #v/sum(k=1, #v, 1/v[k]);
vnud(n) = select(x->(gcd(x, n/x)!=1), divisors(n));
isok(n) = iferr(denominator(hm(vnud(n))) == 1, E, 0); \\ Michel Marcus, Oct 28 2018

A348964 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.

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

Amiram Eldar, Nov 05 2021

nonn

Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 2.
Equivalently, numbers k such that A348963(k) | k * A049419(k).
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, ...

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicuşor Minculete, Contribuţii la studiul proprietăţilor analitice ale funcţiilor aritmetice - Utilizarea e-divizorilor, Ph.D. thesis, Academia Română, 2012. See section 4.3, pp. 90-94.
József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

A005117 and A348965 are subsequences.
Cf. A049419, A322791, A348961, A348963.
Similar sequences: A001599, A006086, A063947, A286325, A319745.

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]

A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.

Original entry on oeis.org

1, 4, 3, 8, 5, 2, 7, 32, 9, 20, 11, 12, 13, 7, 5, 32, 17, 12, 19, 8, 21, 22, 23, 16, 25, 52, 27, 14, 29, 10, 31, 128, 11, 68, 35, 72, 37, 38, 39, 32, 41, 7, 43, 44, 3, 23, 47, 48, 49, 100, 17, 104, 53, 18, 55, 56, 57, 116, 59, 4, 61, 31, 63, 256, 65, 11, 67, 136

Offset: 1

Amiram Eldar, Mar 09 2023

			Fractions begin with 1, 4/3, 3/2, 8/5, 5/3, 2, 7/4, 32/15, 9/5, 20/9, 11/6, 12/5, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian Math. Soc., Vol. 41, No. 1 (1990), pp. 151-158.

Cf. A036537, A037445, A049417, A077609, A063947, A138302, A361317 (denominators).

Similar sequences: A099377, A103339.

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}]]; a[1] = 1; a[n_] := Numerator[n * Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), b); numerator(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)))); }

a(n) = numerator(n*A037445(n)/A049417(n)).
a(n)/A361317(n) <= A099377(n)/A099378(n), with equality if and only if n is in A036537.
a(n)/A361317(n) >= A103339(n)/A103340(n), with equality if and only if n is in A138302.

A361317 Denominators of the harmonic means of the infinitary divisors of the positive integers.

Original entry on oeis.org

1, 3, 2, 5, 3, 1, 4, 15, 5, 9, 6, 5, 7, 3, 2, 17, 9, 5, 10, 3, 8, 9, 12, 5, 13, 21, 10, 5, 15, 3, 16, 51, 4, 27, 12, 25, 19, 15, 14, 9, 21, 2, 22, 15, 1, 9, 24, 17, 25, 39, 6, 35, 27, 5, 18, 15, 20, 45, 30, 1, 31, 12, 20, 85, 21, 3, 34, 45, 8, 9, 36, 25, 37, 57

Offset: 1

Amiram Eldar, Mar 09 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian Math. Soc., Vol. 41, No. 1 (1990), pp. 151-158.

Cf. A037445, A049417, A077609, A063947 (positions of 1's), A361316 (numerators).

Similar sequences: A099378, A103340.

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}]]; a[1] = 1; a[n_] := Denominator[n * Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), b); denominator(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)))); }

a(n) = denominator(n*A037445(n)/A049417(n)).

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

Amiram Eldar, Nov 06 2021

nonn

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).

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicuşor Minculete, Contribuţii la studiul proprietăţilor analitice ale funcţiilor aritmetice - Utilizarea e-divizorilor, Ph.D. thesis, Academia Română, 2012. See section 4.3, pp. 90-94.

Cf. A278908 (number of exponential unitary divisors), A322857, A322858, A323310, A349025, A349027.

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

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]

A362804 Numbers k such that the set of divisors {d | k, BitOr(k, d) = k} has an integer harmonic mean.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 04 2023

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?

Amiram Eldar, Table of n, a(n) for n = 1..406

Cf. A000043, A000396, A246600, A246601.
Subsequences: A000079, A007283 \ {3}, A005009 \ {7, 14}, A110286 \ {15}, A362805.
Similar sequences: A001599, A006086, A063947, A286325, A319745.

q[n_] := IntegerQ[HarmonicMean[Select[Divisors[n], BitAnd[n, #] == # &]]]; Select[Range[10^4], q]

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;}

A335387 Tri-unitary harmonic numbers: numbers k such that the harmonic mean of the tri-unitary divisors of k is an integer.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 04 2020

nonn

Equivalently, numbers k such that A324706(k) | (k * A335385(k)).

Differs from A063947 from n >= 18.

			45 is a term since its tri-unitary divisors are {1, 5, 9, 45} and their harmonic mean, 3, in an integer.

Amiram Eldar, Table of n, a(n) for n = 1..200

A324707 is a subsequence.
Analogous sequences: A001599 (harmonic numbers), A006086 (unitary), A063947 (infinitary), A286325 (bi-unitary), A319745 (nonunitary).
Cf. A324706, A335385.

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]

A348918 Noninfinitary harmonic numbers: numbers such that the harmonic mean of their noninfinitary divisors is an integer.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 04 2021

nonn

Includes all the squares of primes (A001248), since they are the numbers with a single noninfinitary divisor.

			12 is a term since its noninfinitary divisors are {2, 6}, and their harmonic mean, 3, is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A001248, A001599, A006086, A063947, A286325, A319745.

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] &]

A361318 Harmonic means of the infinitary divisors of the infinitary harmonic numbers.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 7, 7, 11, 13, 13, 10, 7, 15, 16, 15, 9, 20, 18, 14, 25, 24, 19, 25, 15, 27, 28, 30, 18, 36, 13, 21, 17, 29, 40, 33, 24, 28, 38, 31, 29, 45, 34, 27, 28, 44, 27, 60, 36, 52, 46, 26, 51, 42, 55, 33, 66, 40, 24, 37, 49, 29, 47, 57, 34, 68, 49, 44

Offset: 1

Amiram Eldar, Mar 09 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, No. 1 (1990), pp. 151-158.

Cf. A063947, A361316, A361317.

Similar sequences: A001600, A006087.

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}]]; s[1] = 1; s[n_] := n * Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10^4], IntegerQ]

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(ih, ", ")));}

a(n) = A361316(A063947(n)).

A361387 Infinitary arithmetic numbers k whose mean infinitary divisor is an infinitary divisor of k.

Original entry on oeis.org

1, 6, 60, 270, 420, 630, 2970, 5460, 8190, 36720, 136500, 172900, 204750, 245700, 491400, 790398, 791700, 819000, 1037400, 1138320, 1187550, 1228500, 1801800, 2457000, 3767400, 4176900, 4504500, 5405400, 6397300, 6688500, 6741630, 7698600, 8353800, 10032750, 10228680

Offset: 1

Amiram Eldar, Mar 10 2023

nonn

Also, infinitary harmonic numbers k whose harmonic mean of the infinitary divisors of k is an infinitary divisor of k.

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

Amiram Eldar, Table of n, a(n) for n = 1..135

Subsequence of A063947 and A361386.
Similar sequence: A007340, A353039.
Cf. A037445, A049417, A077609.

idivs[1] = {1}; idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])]; Select[Range[10^5], IntegerQ[(r = Mean[(i = idivs[#])])] && MemberQ[i, r] &]

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); } \\ Michel Marcus at A077609
is(n) = {my(f = factor(n), b, r); r = prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], (f[i, 1]^(2^(#b-k))+1)/2, 1))); denominator(r) == 1 && n%r==0 && isidiv(r, f); }

cp's OEIS Frontend

A319745 Nonunitary harmonic numbers: numbers such that the harmonic mean of their nonunitary divisors is an integer.

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A348964 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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A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.

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A361317 Denominators of the harmonic means of the infinitary divisors of the positive integers.

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A349026 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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A362804 Numbers k such that the set of divisors {d | k, BitOr(k, d) = k} has an integer harmonic mean.

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A335387 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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A348918 Noninfinitary harmonic numbers: numbers such that the harmonic mean of their noninfinitary divisors is an integer.

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