A001599 -id:A001599 - cp's OEIS frontend

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.

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]

A046987 Multiply perfect numbers that are neither harmonic numbers nor arithmetic numbers.

Original entry on oeis.org

120, 523776, 1476304896, 31998395520, 30823866178560, 69357059049509038080, 4010059765937523916800, 27099073228001299660800, 686498980761986918441287680, 2827987212986831882236723200, 115131961034430181728489308160, 13361233986454282110797768294400, 32789312424503984621373515366400

Offset: 1

Labos Elemer

nonn

			k = 523776 is a term since s0 = d(k) = 80, s1 = sigma(k) = 1571328, s1/k = 1571328/523776 = 3 is an integer, but (k * s0)/s1 = 80/3 and s1/s0 = 98208/5 are not integers.

Amiram Eldar, Table of n, a(n) for n = 1..100

In A007691 but neither in A003601 nor in A001599.

Cf. A000005, A000203, A046985, A046986.

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && !Divisible[n * d, s] && !Divisible[s, d]]; Select[Range[6*10^5], q] (* Amiram Eldar, May 09 2024 *)

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && ((k * d) % s) && (s % d);} \\ Amiram Eldar, May 09 2024

Let s1 be the sum of divisors of k and s0 be the number of divisors of k. Then, k is a term if k | s1, but (k * s0) is not divisible by s1, and s1 is not divisible by s0.

a(5)-a(10) from Donovan Johnson, Nov 30 2008

Edited and a(11)-a(13) added by Amiram Eldar, May 09 2024

A090240 Numbers which occur as the harmonic mean of the divisors of m for some m.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 24, 25, 26, 27, 29, 31, 35, 37, 39, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 60, 61, 70, 73, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 94, 96, 97, 99, 101, 102, 105, 106, 107, 108, 110, 114, 115

Offset: 1

R. K. Guy, Feb 08 2004

nonn

The equation n = m*tau(m)/sigma(m) has an integer solution m.
Here tau(n) (A000005) is the number of divisors of n and sigma(n) is the sum of the divisors of n (A000203).
A001600 sorted in order.
The Mersenne exponents (A000043) are in this sequence because the even perfect numbers, 2^(p-1)*(2^p-1) where p is in A000043, are all harmonic numbers (A001599) with harmonic mean of divisors p. - Amiram Eldar, Apr 15 2024

For further references see A001599.

T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.

Values of m are in A091911.
Complement of A157849.
Cf. A000043, A001599, A001600, A090758, A090759, A090760, A090761, A090762.

f[n_] := (n*DivisorSigma[0, n]/DivisorSigma[1, n]); a = Table[ 0, {120}]; Do[ b = f[n]; If[ IntegerQ[b] && b < 121 && a[[b]] == 0, a[[b]] = n], {n, 1, 560000000}]; Select[ Range[120], a[[ # ]] > 0 &] (* Robert G. Wilson v, Feb 14 2004 *)

More terms from Robert G. Wilson v, Feb 14 2004

A046986 Multiply perfect numbers that are also harmonic numbers but are not arithmetic numbers.

Original entry on oeis.org

28, 496, 8128, 2178540, 33550336, 142990848, 459818240, 1379454720, 8589869056, 43861478400, 66433720320, 137438691328, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392, 2305843008139952128

Offset: 1

Labos Elemer

nonn

			k = 2178540 is a term since s0 = d(k) = 216 and s1 = sigma(k) = 8714160, s1/s0 = 8714160/216 = 121030/3 is not an integer, and (k * s0)/s1 = (2178540 * 216)/8714160 = 54 and s1/k = 8714160/2178540 = 4 are integers.

Amiram Eldar, Table of n, a(n) for n = 1..207

In A007691 and A001599 but not in A003601.

Cf. A000005, A000203, A046985, A046987.

Let s1 be the sum of divisors of k and s0 be the number of divisors of k. Then, k is a term if k | s1, s1 | (k * s0), but s1 is not divisible by s0.

a(12)-a(17) from Donovan Johnson, Nov 30 2008

Edited and a(18)-a(21) added by Amiram Eldar, May 09 2024

A324051 a(n) = A106315(A156552(n)).

Original entry on oeis.org

0, 1, 2, 5, 4, 2, 6, 0, 1, 18, 10, 3, 16, 4, 12, 67, 12, 4, 18, 30, 36, 260, 22, 16, 8, 8, 44, 5, 20, 1029, 30, 28, 164, 36, 28, 6, 256, 96, 44, 4102, 36, 7, 66, 16, 104, 16391, 46, 12, 13, 32, 130, 8, 28, 51, 70, 480, 942, 65544, 42, 9, 2724, 32, 66, 30, 84, 262153, 124, 508, 40, 10, 4, 1048586, 3320, 20, 188, 50, 52, 11, 78, 24

Offset: 2

Antti Karttunen, Feb 19 2019

nonn

Positions of zeros, which is sequence A005940(1+A001599(n)) sorted into ascending order: 2, 9, 125, 325, 351, 4199, ..., has A324201 as its subsequence.

Antti Karttunen, Table of n, a(n) for n = 2..4473
Index entries for sequences computed from indices in prime factorization

Cf. A001599, A005940, A106315, A156552, A158879, A323243, A323244, A324049, A324105, A324201.

Cf. also A324057, A324187.

A106315(n) = (n*numdiv(n) % sigma(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A324051(n) = A106315(A156552(n));

a(n) = A106315(A156552(n)).

a(n) = (A156552(n)*A324105(n)) mod A323243(n).

A349474 a(n) is the length of the continued fraction of the harmonic mean of the divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 3, 1, 3, 3, 2, 3, 3, 4, 3, 2, 2, 7, 3, 4, 3, 3, 5, 3, 3, 2, 6, 3, 4, 1, 3, 2, 3, 2, 3, 4, 3, 8, 3, 4, 5, 4, 3, 2, 3, 2, 3, 4, 3, 5, 6, 4, 3, 4, 3, 4, 2, 5, 5, 7, 3, 3, 3, 5, 7, 7, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 7, 4, 4, 4, 3, 4, 6, 6, 3, 2, 4, 6, 3

Offset: 1

Amiram Eldar, Nov 19 2021

nonn

a(n) = 1 if and only if n is a harmonic number (A001599).

a(n) <= 2 if and only if n is in A348865.

			a(1) = 1 since the harmonic mean of the divisors of 1 is 1 and its continued fraction has 1 element, {1}.
a(2) = 2 since the harmonic mean of the divisors of 2 is 4/3 = 1 + 1/3 and its continued fraction has 2 elements, {1, 3}.
a(4) = 4 since the harmonic mean of the divisors of 4 is 12/7 = 1 + 1/(1 + 1/(2 + 1/2)) and its continued fraction has 4 elements, {1, 1, 2, 2}.

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

Row length of A349473.

Cf. A001599, A071862, A099377, A099378, A348865.

a[n_] := Length @ ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]; Array[a, 100]

A047728 Intersection of A046985 and A033950: multiply perfect, refactorable numbers with integer average divisor dividing the number.

Original entry on oeis.org

1, 672, 30240, 23569920, 45532800, 14182439040, 153003540480, 403031236608, 13661860101120, 154345556085770649600, 143573364313605309726720, 352338107624535891640320, 680489641226538823680000, 34384125938411324962897920, 156036748944739017459105792, 3638193973609385308194865152

Offset: 1

Labos Elemer

nonn

Colton proves that perfect numbers (A000396) cannot be refactorable.

			k = 45532800 is a term since s0 = d(k) = 384, s1 = sigma(k) = 571963392, and the four quotients s1/s0 = 474300, (k * s0)/s1 = 96, s1/k = 4 and k/s0 = 118580 are all integers.

Amiram Eldar, Table of n, a(n) for n = 1..305
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.

Cf. A000005, A000203, A000396, A046985, A001599, A007691.

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && Divisible[n * d, s] && Divisible[s, d] && Divisible[n, d]]; Select[Range[31000], q] (* Amiram Eldar, May 09 2024 *)

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && !((k * d) % s) && !(s % d) && !(k % d);} \\ Amiram Eldar, May 09 2024

Let s1 = sigma(k) = A000203(k) be the sum of divisors of k and s0 = d(k) = A000005(k) be the number of divisors of k. Then, k is a term if s1/s0, (k * s0)/s1, s1/k, and k/s0 are all integers.

a(7)-a(13) from Donovan Johnson, Apr 09 2010

Edited and a(14)-a(16) added by Amiram Eldar, May 09 2024

A074266 Numbers k such that the harmonic mean of the divisors of k is the square of a rational number.

Original entry on oeis.org

1, 216, 468, 810, 1550, 1638, 3744, 10880, 11340, 13965, 21700, 23716, 40176, 45847, 50274, 56896, 80262, 90720, 97969, 126360, 128744, 137940, 139159, 161728, 173600, 189728, 224450, 319579, 434511, 482790, 515450, 526500, 555660

Offset: 1

Joseph L. Pe, Sep 20 2002

nonn

			The harmonic mean of the divisors of 468 is 324/49 = (18/7)^2, the square of a rational number, so 468 is a term of the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A001599, A099377, A099378.

H[l_] := Module[{m, s}, m = Length[l]; s = 0; For[i = 1, i <= m, i++, s = s + (1/l[[i]])]; s = s/m; s = 1/s; s] r = {}; Do[d = Divisors[n]; h = H[d]; num = Numerator[h]; den = Denominator[h]; If[IntegerQ[num^(1/2)] && IntegerQ[den^(1/2)], r = Append[r, n]], {n, 1, 10^6}]; r

A247078 Numbers for which the harmonic mean of nontrivial divisors is an integer.

Original entry on oeis.org

4, 9, 25, 49, 121, 169, 289, 345, 361, 529, 841, 961, 1050, 1369, 1645, 1681, 1849, 2209, 2809, 3481, 3721, 4386, 4489, 5041, 5329, 6241, 6489, 6889, 7921, 8041, 9409, 10201, 10609, 11449, 11881, 12769, 13026, 16129, 17161, 18769, 19321, 22201, 22801

Offset: 1

Daniel Lignon, Nov 17 2014

nonn

All the squares of prime numbers (A001248) have this property but there are other numbers (A247079): 345, 1050, 1645, 4386, 6489, 8041, ...

			The divisors of 25 are [1,5,25] and the nontrivial divisors are [5]. The harmonic mean is 1/(1/5)=5. That's the same for all squares of prime numbers.
The nontrivial divisors of 345 are [3,5,15,23,69,115] and their harmonic mean is 6/(1/3+1/5+1/15+1/23+1/69+1/115) = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1269 from Daniel Lignon)

Cf. similar sequences: A001599 (with all divisors), A247077 (with proper divisors).

hm:= S -> nops(S)/convert(map(t->1/t,S),`+`):
filter:= n -> not isprime(n) and type(hm(numtheory:-divisors(n) minus {1,n}),integer):
select(filter, [$2..10^5]); # Robert Israel, Nov 17 2014

Select[Range[2,100000],(Not[PrimeQ[#]] && IntegerQ[HarmonicMean[Rest[Most[Divisors[#]]]]])&]

isok(n) = my(d=divisors(n)); (#d >2) && (denominator((#d-2)/sum(i=2, #d-1, 1/d[i])) == 1);

A324199 Numbers n such that A324187(n) = 0.

Original entry on oeis.org

0, 6, 60, 98, 108, 928, 930, 946, 1874, 3506, 6688, 7496, 7980, 13640, 14476, 15148, 15956, 27168, 30290, 30292, 32752, 59528, 59986, 60556, 60580, 64338, 65432, 108680, 109732, 119972, 121108, 126280, 126872, 130856, 218276, 229160, 242210, 242306, 438914, 454552, 484420, 484640, 485512, 496904, 507032, 518688, 522848, 523400, 811556, 877636

Offset: 1

Antti Karttunen, Feb 17 2019

nonn

Sequence A243071(A001599(k)), k >= 1, sorted into ascending order. See also comments in A324185.

Also positions of zeros in A324189.

Antti Karttunen, Table of n, a(n) for n = 1..164 (all terms up to size 2^26)

Cf. A001599, A243071, A324185, A324187, A324189.

Cf. A324200 (subsequence).

for(n=0,2^21,if(0==A324187(n),print1(n,", "))); \\ Uses code from A324187.

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A046987 Multiply perfect numbers that are neither harmonic numbers nor arithmetic numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A090240 Numbers which occur as the harmonic mean of the divisors of m for some m.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions

A046986 Multiply perfect numbers that are also harmonic numbers but are not arithmetic numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A324051 a(n) = A106315(A156552(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A349474 a(n) is the length of the continued fraction of the harmonic mean of the divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A047728 Intersection of A046985 and A033950: multiply perfect, refactorable numbers with integer average divisor dividing the number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A074266 Numbers k such that the harmonic mean of the divisors of k is the square of a rational number.

Views

Author

Keywords