A001599 -id:A001599 - cp's OEIS frontend

A335317 Harmonic numbers (A001599) with a record number of divisors.

1, 6, 28, 140, 270, 672, 2970, 8190, 30240, 332640, 2178540, 2457000, 11981970, 14303520, 17428320, 27027000, 163390500, 164989440, 191711520, 513513000, 1307124000, 2144862720, 2701389600, 3506025600, 5943057120, 13584130560, 14378364000, 29715285600, 45578332800

Offset: 1

Amiram Eldar, May 31 2020

nonn

The corresponding record values are 1, 4, 6, 12, 16, 24, 32, 48, 96, ... (see the link for more values).

			The first 7 harmonic numbers are 1, 6, 28, 140, 270, 496 and 672. Their numbers of divisors (A000005) are 1, 4, 6, 12, 16, 10 and 24. The record values, 1, 4, 6, 12, 16 and 24 occur at 1, 6, 28, 140, 270 and 672, the first 6 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..40 (terms below 10^14)
Amiram Eldar, Table of n, a(n), A000005(a(n)) for n = 1..40

Cf. A000005, A001599, A335316, A335318.

dm = 0; s = {}; Do[h = n * (d = DivisorSigma[0, n]) / DivisorSigma[1, n]; If[IntegerQ[h] && d > dm, dm = d; AppendTo[s, n]], {n, 1, 10^6}]; s

A325021 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820), tau(k) is the number of divisors of k (A000005), and sigma(k) is the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 6, 28, 496, 672, 8128, 30240, 32760, 332640, 695520, 2178540, 17428320, 23569920, 33550336, 45532800, 52141320, 142990848, 164989440, 318729600, 447828480, 481572000, 500860800, 540277920, 623397600, 644271264, 714954240, 995248800, 1047254400, 1307124000

Offset: 1

Jaroslav Krizek, Mar 27 2019

nonn

Numbers m such that m*tau(m)/sigma(m) is an integer g and simultaneously m*(m-tau(m))/sigma(m) is an integer h. Corresponding values of integers g: 1, 2, 3, 5, 8, 7, 24, 24, 44, 46, 54, 96, 80, 13, 96, ...
Corresponding values of integers h: 0, 1, 11, 243, 216, 4057, 7536, 8166, 76186, 166589, ...
Even perfect numbers from A000396 are terms.
Complement of A325022 with respect to A001599.
Intersection of A325020 and A001599.

			Harmonic number 28 is a term because 28*tau(28)/sigma(28) = 28*6/56 = 3 (integer) and simultaneously 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..255 (terms below 10^14)

Cf. A000005, A000203, A001599, A007691, A049820, A325020, A325022, A325023, A325024.

[n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n) * n) / SumOfDivisors(n)) and IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n))]

Select[Range[10^6], And[IntegerQ@ HarmonicMean@ #2, IntegerQ[#1 (#1 - #3)/#4]] & @@ Join[{#}, {Divisors@ #}, DivisorSigma[{0, 1}, #]] &] (* Michael De Vlieger, Mar 27 2019 *)

isok(m) = my(d=numdiv(m), s=sigma(m)); !frac(m*d/s) && !frac(m*(m-d)/s); \\ Michel Marcus, Mar 27 2019

from itertools import count, islice
from math import prod
from functools import reduce
from sympy import factorint
def A325021_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
        if not (n*n%s or reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s):
            yield n
A325021_list = list(islice(A325021_gen(),10)) # Chai Wah Wu, Feb 14 2023

A325022 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is not an integer, where k-tau(k) = the number of nondivisors of k (A049820), tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

140, 270, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 360360, 539400, 726180, 753480, 950976, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000, 2845800, 4358600, 4713984, 4754880, 5772200, 6051500

Offset: 1

Jaroslav Krizek, Mar 28 2019

nonn

Numbers m such that sigma(m) divides m*tau(m) but sigma(m) does not divide m*(m-tau(m)).

Complement of A325021 with respect to A001599.

			140 is a term because 140*(140-tau(140))/sigma(140) = 140*(140-12)/336 = 160/3.

Amiram Eldar, Table of n, a(n) for n = 1..682 (terms below 10^14)

Cf. A000005, A000203, A001599, A049820, A325020, A325021, A325023, A325024.

[n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n) * n) / SumOfDivisors(n)) and not IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n))]

Select[Range[10^5], And[IntegerQ@ HarmonicMean@ #4, ! IntegerQ[#1 (#1 - #2)/#3]] & @@ Append[{#}~Join~DivisorSigma[{0, 1}, #], Divisors@ #] &] (* Michael De Vlieger, Mar 30 2019 *)

isok(m) = my(d=numdiv(m), s=sigma(m)); !frac(m*d/s) && frac(m*(m-d)/s); \\ Michel Marcus, Mar 28 2019

from itertools import count, islice
from math import prod
from functools import reduce
from sympy import factorint
def A325022_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
        if n*n%s and not reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s:
            yield n
A325022_list = list(islice(A325022_gen(),10)) # Chai Wah Wu, Feb 14 2023

A325025 Numbers that are multi-perfect (A007691) and simultaneously harmonic (A001599).

Original entry on oeis.org

1, 6, 28, 496, 672, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 8589869056, 14182439040, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 13661860101120

Offset: 1

Jaroslav Krizek, Mar 24 2019

nonn

Multi-perfect numbers from A007691 that are harmonic numbers (A001599). Complement of A325026 with respect to A001599.
Harmonic numbers from A001599 that are multi-perfect numbers (A007691). Complement of A140798 with respect to A007691.
Numbers m such that sigma(m)/m is an integer g and simultaneously m*tau(m)/sigma(m) is an integer h, where tau(k) is the number of the divisors of k (A000005) and sigma(k) is the sum of the divisors of k (A000203). Corresponding values of integers g: 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 2, 4, 4, 3, 4, 2, 5, ... Corresponding values of integers h: 1, 2, 3, 5, 8, 7, 24, 24, 54, 80, 13, 96, 120, ...
Even perfect numbers from A000396 are terms.

			28 is a term because 28*tau(28)/sigma(28) = 28*6/56 = 3 (integer) and simultaneously 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..528

Cf. A000005, A000203, A000396, A001599, A007691, A140798, A325026.

A325021 and A325023 are closely related sequences. - N. J. A. Sloane, May 03 2019

[n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n)) * n / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]

Select[Range[10^6], And[Mod[DivisorSigma[1, #], #] == 0, IntegerQ@ HarmonicMean@ Divisors@ #] &] (* Michael De Vlieger, Mar 24 2019 *)

isok(n) = my(s=sigma(n)); !frac(s/n) && !frac(n*numdiv(n)/s); \\ Michel Marcus, Mar 24 2019

A090945 Harmonic numbers (A001599) which are not perfect (A000396).

Original entry on oeis.org

1, 140, 270, 672, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720

Offset: 1

N. J. A. Sloane, Feb 28 2004

nonn

			A001599(4) = 140, but 336 = sigma(140) <> 2*140 = 280. Thus, 140 is a harmonic number which is not perfect. - _Muniru A Asiru_, Nov 26 2018

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.

Amiram Eldar, Table of n, a(n) for n = 1..930 (terms below 10^14; terms 1..77 from Muniru A Asiru)
T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.

Cf. A001599, A003601. Different from A007340.

For the associated harmonic means, see A102408.

Concatenation([1],Filtered([2,4..2000000],n->Sigma(n)<>2*n and IsInt(n*Tau(n)/Sigma(n)))); # Muniru A Asiru, Nov 26 2018

Select[Range[2 10^7], IntegerQ[HarmonicMean[Divisors[#]]] && !DivisorSigma[1, #]==2 # &] (* Vincenzo Librandi, Nov 27 2018 *)

isok(n) = my(sn = sigma(n)); (frac(n*numdiv(n)/sn) == 0) && (sn != 2*n); \\ Michel Marcus, Nov 28 2018

A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).

Original entry on oeis.org

1, 6, 40, 126, 154, 204, 1716, 1914, 2772, 8580, 11264, 12090, 12540, 50960, 62790, 64350, 77748, 83200, 104720, 152320, 186116, 193440, 331890, 382720, 432768, 518364, 648788, 684684, 753480, 817344, 895356, 1083852, 1113840, 1619352, 1675044, 1743588, 1759680, 1991340, 2060322, 2360484, 2492028, 2621080, 2932800

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Numbers k for which A336314(k) = A323173(k).
Sequence A122111(A001599(n)), n >= 1, sorted into ascending order. Positions of zeros in A323174 (corresponding to perfect numbers similarly mapped) is a subsequence.
Note that all terms after 1 seem to be present in A102750. This observation is equal to Ore's conjecture that there are no odd Harmonic numbers larger than one.
Also, all terms after 1 seem to be even, which would imply that apart from its initial 1, A001599 were a subsequence of A102750. However, this is false, as there are terms of A001599 not in A102750, for example 8011798098793361832960 found by David A. Corneth. Note that A122111(8011798098793361832960) = 96922193555635754403846044921625, which is thus an odd term of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..61
Index entries for sequences computed from indices in prime factorization

Cf. A001599, A088902, A102750, A122111, A323173, A323174, A336314, A336315, A336353, A336397 (the intersection with A001599).

isA001599(n) = !((sigma(n,0)*n)%sigma(n,1));
isA336317(n) = isA001599(A122111(n)); \\ Program for A122111 given under that entry.

\\ Standalone program:
isA336317(n) = if(1==n,1,my(f=factor(n),es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,d=1,s=1,x=1,p,e); for(i=1, #es, pri += es[i]; p = prime(pri); e = 1+is[i]-is[1+i]; d *= e; s *= ((p^e)-1)/(p-1); x *= (p^(e-1))); !((x*d)%s));

A325026 Multi-perfect numbers from A007691 that are not harmonic (A001599).

Original entry on oeis.org

120, 523776, 1476304896, 31998395520, 518666803200, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 69357059049509038080, 87934476737668055040, 170206605192656148480, 1161492388333469337600, 1802582780370364661760, 1940351499647188992000

Offset: 1

Jaroslav Krizek, Mar 24 2019

nonn

Multi-perfect numbers m such that m*tau(m)/sigma(m) is not an integer, where tau(k) is the number of the divisors of k (A000005) and sigma(k) is the sum of the divisors of k (A000203).
Supersequence of A046987.
Complement of A325025 with respect to A007691.

			120 is a term because 120*tau(120)/sigma(120) = 120*16/360 = 16/3.

Amiram Eldar, Table of n, a(n) for n = 1..202

Cf. A000005, A000203, A001599, A007691, A325025.

Cf. A140798 (harmonic numbers that are not multi-perfect).

[n: n in [1..1000000] | not IsIntegral((NumberOfDivisors(n)) * n / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]

isok(n) = my(s=sigma(n)); !frac(s/n) && frac(n*numdiv(n)/s); \\ Michel Marcus, Mar 24 2019

A335316 Harmonic numbers (A001599) with a record harmonic mean of divisors.

Original entry on oeis.org

1, 6, 28, 140, 270, 672, 1638, 2970, 8190, 27846, 30240, 167400, 237510, 332640, 695520, 1421280, 2178540, 2457000, 11981970, 14303520, 17428320, 23963940, 27027000, 46683000, 56511000, 71253000, 142990848, 163390500, 164989440, 191711520, 400851360, 407386980

Offset: 1

Amiram Eldar, May 31 2020

nonn

The corresponding record values are 1, 2, 3, 5, 6, 8, 9, 11, 15, ... (see the link for more values).

The terms 1, 6, 30240 and 332640 are also terms of A179971.

			The first 7 harmonic numbers are 1, 6, 28, 140, 270, 496 and 672. Their harmonic means of divisors (A001600) are 1, 2, 3, 5, 6, 5 and 8. The record values, 1, 2, 3, 5, 6 and 8 occur at 1, 6, 28, 140, 270 and 672, the first 6 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..70 (terms below 10^14)
Amiram Eldar, Table of n, a(n), A099377(a(n)) for n = 1..70

Cf. A001599, A001600, A099377, A099378, A179971, A335317, A335318.

h[n_] := n * DivisorSigma[0, n] / DivisorSigma[1, n]; hm = 0; s = {}; Do[h1 = h[n];  If[IntegerQ[h1] && h1 > hm, hm = h1; AppendTo[s, n]], {n, 1, 10^6}]; s

A157849 Numbers k such that are not harmonic means of divisors of harmonic (Ore) numbers (harmonic (Ore) numbers is A001599).

Original entry on oeis.org

4, 12, 16, 18, 20, 22, 23, 28, 30, 32, 33, 34, 36, 38, 40, 43, 52, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 74, 76, 79, 90, 93, 95, 98, 100, 103, 104, 109, 111, 112, 113, 119, 122, 123, 124, 126, 129, 131, 133, 134, 136, 137, 138, 141, 142, 146, 148, 151, 154, 157, 162, 170

Offset: 1

Jaroslav Krizek, Mar 07 2009

nonn

a(n) non-occurring in A001600(m) = A001599(m)*tau(A001599(m))/sigma(A001599(m)) = A001599(m)*A000005(A001599(m))/A000203(A001599(m)).

Robert G. Wilson v, Table of n, a(n) for n = 1..412
Takeshi Goto, On Ore's harmonic numbers, slides, Tokyo University of Science, Japan.

Cf. A001600, A001599, A000005, A000203.

More terms from Robert G. Wilson v, Aug 18 2013

A328944 Arithmetic numbers (A003601) that are not harmonic (A001599).

Original entry on oeis.org

3, 5, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101

Offset: 1

Jaroslav Krizek, Oct 31 2019

nonn

Numbers m such that the arithmetic mean of the divisors of m is an integer but the harmonic mean of the divisors of m is not an integer.
Numbers m such that A(m) = A000203(m)/A000005(m) is an integer but H(m) = m * A000005(m)/A000203(m) is not an integer.
Corresponding values of A(m): 2, 3, 4, 6, 7, 6, 6, 9, 10, 7, 8, 9, 12, 10, 15, 9, 16, 12, 12, 19, 15, 14, 21, 12, 22, ...
Corresponding values of H(m): 3/2, 5/3, 7/4, 11/6, 13/7, 7/3, 5/2, 17/9, 19/10, 20/7, 21/8, 22/9, ...
Complement of A007340 with respect to A003601.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A003601, A007340, A046999, A328945.

[m: m in [1..10^5] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and not IsIntegral(m * NumberOfDivisors(m) / SumOfDivisors(m))];

harm:= proc(S) local s; nops(S)/add(1/s,s=S) end proc:
filter:= proc(n) local S;
  S:= numtheory:-divisors(n);
  (convert(S,`+`)/nops(S))::integer and not harm(S)::integer
end proc:
select(filter, [$1..200]); # Robert Israel, May 04 2025

Select[Range[100], Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] && !Divisible[# * DivisorSigma[0, #], DivisorSigma[1, #]] &] (* Amiram Eldar, Nov 01 2019 *)

cp's OEIS Frontend

A335317 Harmonic numbers (A001599) with a record number of divisors.

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A325021 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820), tau(k) is the number of divisors of k (A000005), and sigma(k) is the sum of the divisors of k (A000203).

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A325022 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is not an integer, where k-tau(k) = the number of nondivisors of k (A049820), tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

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A325025 Numbers that are multi-perfect (A007691) and simultaneously harmonic (A001599).

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A090945 Harmonic numbers (A001599) which are not perfect (A000396).

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A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).

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A325026 Multi-perfect numbers from A007691 that are not harmonic (A001599).

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