A099378 -id:A099378 - cp's OEIS frontend

A335368 Harmonic numbers k with a record number of primes p not dividing k such that k*p is also a harmonic number.

1, 28, 1638, 30240, 2178540, 2457000, 32997888, 142990848, 1307124000, 71271827200, 547929930240, 2198278051200, 2567400675840, 54409216942080

Offset: 1

Amiram Eldar, Jun 03 2020

The corresponding record values are 0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 17.

If k is a harmonic number (A001599) and p is a prime that does not divide k, then k*p is a harmonic number if and only if (p+1)/2 is a divisor of the harmonic mean of the divisors of k, k*tau(k)/sigma(k) = k*A000005(k)/A000203(k).

			1 is the first harmonic number, and it has 0 primes p such that 1*p = p is a harmonic number, since a prime number cannot be harmonic. The next harmonic number k with at least one prime p such that k*p is also a harmonic number is 28, since 28 * 5 = 140 is a harmonic number.

Mariano Garcia, On numbers with integral harmonic mean, The American Mathematical Monthly, Vol. 61, No. 2 (1954), pp. 89-96. See page 95.

Cf. A000005, A000203, A001599, A099377, A099378, A335369, A335370, A335371.

harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {, }][[;; , 2]]; harMean[n_] := n*DivisorSigma[0, n]/DivisorSigma[1, n]; primeCount[n_] := Module[{d = Divisors[harMean[n]]}, Length @ Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &]]; primeCountMax = -1; seq = {}; Do[If[(pc = primeCount[harmNums[[k]]]) > primeCountMax, primeCountMax = pc; AppendTo[seq, harmNums[[k]]]], {k, 1, Length[harmNums]}]; seq

A335370 Harmonic numbers m with a record number k of distinct prime numbers p_i (i = 1..k) that do not divide m such that mp_1, mp_1p_2, ... , mp_1...p_k are all harmonic numbers.

Original entry on oeis.org

1, 28, 1638, 6200, 2457000, 4713984, 1381161600, 10200236032

Offset: 1

Amiram Eldar, Jun 03 2020

If m is a harmonic number (A001599), then it is possible to generate a new harmonic number m*p if p is a prime number that does not divide m and (p+1)/2 is a divisor of the harmonic mean of the divisors of m, h(m) = m * tau(m)/sigma(m) = m * A000005(m)/A000203(m).
The terms of this sequence begin a chain of harmonic numbers of a record length. In each chain, each member, except the first, is generated from its predecessor by multiplying it by a prime that does not divide it.
The corresponding record values of k are 0, 1, 2, 3, 4, 6, 7, 8, ...
The list of primes or their order may not be unique.

			28 is the least harmonic number with one prime, p = 5, such that 28*p = 140 is also a harmonic number.
1638 is the least harmonic number with 2 primes, 5 and 29, such that 1638*5 = 8190 and 1638*5*29 = 237510 are also harmonic numbers.
.
n  a(n)         k   primes p_i, i = 1..k                 number of permutations
-------------------------------------------------------------------------------
1  1            0   -                                         -
2  28           1   5                                         1
3  1638         2   5, 29                                     1
4  6200         3   19, 37, 73                                1
5  2457000      4   11, 19, 37, 73                            4
6  4713984      6   5, 7, 13, 19, 37, 73                      15
                    5, 7, 19, 37, 73, 1021                    5
7  1381161600   7   11, 19, 37, 43, 73, 6277, 12553           10
                    11, 19, 37, 43, 3181, 6361, 12721         6
8  10200236032  8   3, 5, 79, 157, 313, 1877, 7507, 15013     5

Cf. A000005, A000203, A001599, A099377, A099378, A335368, A335369, A335371.

harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {, }][[;; , 2]]; harMean[n_] := n * DivisorSigma[0, n]/DivisorSigma[1, n]; harmGen[n_] := Module[{d = Divisors[harMean[n]]}, n * Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &]]; harmGens[s_] := Union @ Flatten[harmGen /@ s]; lenmax = -1; seq = {}; Do[len = -3 + Length @ FixedPointList[harmGens, {harmNums[[k]]}]; If[len > lenmax, lenmax = len; AppendTo[seq, harmNums[[k]]]], {k, 1, Length[harmNums]}]; seq

A335371 Harmonic numbers with a record number of harmonic numbers that can be generated from them using an iterative process of multiplying by primes (see Comments).

Original entry on oeis.org

1, 28, 1638, 6200, 950976, 2178540, 2457000, 4713984, 45532800, 142990848, 459818240, 1381161600, 10200236032, 57575890944, 109585986048, 513480135168, 1553357978368, 10881843388416, 43947421401888

Offset: 1

Amiram Eldar, Jun 03 2020

If m is a harmonic number (A001599), then it is possible to generate a new harmonic number m*p if p is a prime number that does not divide m and (p+1)/2 is a divisor of the harmonic mean of the divisors of m, h(m) = m * tau(m)/sigma(m) = m * A000005(m)/A000203(m).
Given a harmonic number m, in the first iteration a finite set of new harmonic numbers, {m*p_1, m*p_2, ...} is being generated. In the second iteration, a new set of harmonic number is being generated from each of the harmonic numbers from the previous iteration, a union of these sets is be calculated (removing duplicates). The process is terminated when no more harmonic numbers can be generated. The total number of harmonic numbers from all the iterations is being counted. The terms of this sequence have a record count of new harmonic numbers.
The corresponding record values of k are 0, 1, 3, 5, 8, 12, 17, 36, 38, 40, 44, 62, 70, 82, 156, 226, 281, 335, 358, ...

			1638 is a term since a record number of 3 new harmonic numbers can be generated from it. In the first iteration 2 new harmonic numbers can be generated: 1638 * 5 = 8190, and 1638 * 17 = 27846. In the second iteration, a new harmonic number can be generated from 8190: 8190 * 29 = 237510.

Cf. A000005, A000203, A001599, A099377, A099378, A335368, A335369, A335370.

harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {, }][[;; , 2]]; harMean[n_] := n * DivisorSigma[0, n]/DivisorSigma[1, n]; harmGen[n_] := Module[{d = Divisors[harMean[n]]}, n * Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &]]; harmGens[s_] := Union@Flatten[harmGen /@ s]; lenmax = -1; seq = {}; Do[len = Length @ Union @ Flatten @ FixedPointList[harmGens, {harmNums[[k]]}]; If[len > lenmax, lenmax = len; AppendTo[seq, harmNums[[k]]]], {k, 1, Length[harmNums]}]; seq

A348414 Numbers with record values of the denominator of the harmonic mean of their divisors.

Original entry on oeis.org

1, 2, 4, 8, 16, 36, 64, 100, 128, 144, 256, 324, 400, 512, 576, 900, 1296, 1600, 1936, 2304, 3600, 5184, 6400, 8100, 9216, 10404, 11664, 14400, 17424, 20736, 22500, 30276, 32400, 41616, 46656, 57600, 76176, 90000, 108900, 129600, 166464, 202500, 230400, 260100

Offset: 1

Amiram Eldar, Oct 17 2021

nonn

The corresponding record values are 1, 3, 7, 15, 31, 91, 127, 217, 255, 403, ... (see the link for more values).

			The first 4 terms of A099378 are 1, 3, 2 and 7. The record values, 1, 3 and 7, occur at 1, 2 and 4, the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..200
Amiram Eldar, Table of n, a(n), A099378(a(n)) for n = 1..200

Cf. A099377, A099378.

d[n_] := Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]]; dm = 0; s = {}; Do[dn = d[n]; If[dn > dm, dm = dn; AppendTo[s, n]], {n, 1, 10^6}]; s

A348825 Numbers k such that the denominator of the harmonic mean of the divisors of k is larger than k.

Original entry on oeis.org

2, 4, 8, 9, 16, 25, 36, 64, 80, 81, 100, 104, 121, 128, 144, 208, 225, 256, 272, 289, 320, 324, 400, 484, 512, 529, 576, 625, 729, 841, 900, 1024, 1088, 1089, 1156, 1250, 1296, 1300, 1332, 1575, 1600, 1664, 1681, 1856, 1936, 2025, 2116, 2196, 2209, 2304, 2368

Offset: 1

Amiram Eldar, Nov 01 2021

nonn

Numbers k such that A099378(k) > k.
This sequence is infinite. For example, if p is a prime of the form 6*k-1, then p^2 is a term.
2 is the only prime term, since the denominator of the harmonic mean of the divisors of an odd prime p is (p+1)/2 < p.

			2 is a term since the harmonic mean of the divisors of 2 is 4/3 and 3 > 2.
4 is a term since the harmonic mean of the divisors of 4 is 12/7 and 7 > 4.

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

Cf. A007528, A099377, A099378.

q[n_] := Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]] > n; Select[Range[2500], q]

isok(k) = my(d=divisors(k)); (denominator(#d/sum(i=1, #d, 1/d[i])) > k); \\ Michel Marcus, Nov 01 2021

A349477 Numbers k such that the sequence of elements of the continued fraction of the harmonic mean of the divisors of k is palindromic.

Original entry on oeis.org

1, 6, 8, 10, 15, 16, 21, 28, 30, 39, 48, 56, 57, 64, 93, 111, 129, 140, 183, 184, 192, 195, 200, 201, 210, 219, 220, 237, 270, 291, 309, 327, 345, 381, 417, 453, 471, 489, 496, 543, 545, 574, 579, 597, 600, 633, 645, 669, 672, 687, 723, 765, 792, 795, 798, 813

Offset: 1

Amiram Eldar, Nov 19 2021

nonn

All the harmonic numbers (A001599) are terms of this sequence.

			8 is a term since the sequence of elements of the continued fraction of the harmonic mean of the divisors of 8, 32/15 = 2 + 1/(7 + 1/2), is {2, 7, 2}, which is palindromic.

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

A001599 and A349476 are subsequences.

Cf. A099377, A099378, A349473.

q[n_] := PalindromeQ[ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]]; Select[Range[1000], q]

A330606 Numbers k such that k*d(k) and sigma(k) are relatively prime, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 25, 36, 64, 81, 100, 121, 128, 144, 225, 256, 289, 324, 400, 484, 512, 529, 576, 625, 729, 841, 900, 1024, 1089, 1156, 1250, 1296, 1600, 1681, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2809, 3025, 3364, 3481, 3600, 4096, 4356, 4624, 4761

Offset: 1

Amiram Eldar, Dec 20 2019

nonn

If p is prime and p == 2 (mod 3) then p^2 is in the sequence.
Let E(x) = #{n | a(n) <= x} be the number of terms of this sequence up to x. Kanold proved that there are two constants 0 < c1 < c2 and a positive number x_0 such that c1 < E(x)/sqrt(x/log(x)) < c2 for x > x_0. De Koninck and Kátai proved that there is a positive constant c such that E(x) = c * (1 + o(1)) * sqrt(x/log(x)).
Apparently most of the terms are squares or powers of 2. Terms that are not included 1250, 4802, 31250, 57122, ...
Numbers k such that A099377(k) = A038040(k) and A099378(k) = A000203(k). - Amiram Eldar, Nov 02 2021

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 75.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Imre Kátai, On an estimate of Kanold, Int. J. Math. Anal., Vol. 5, No. 8 (2007), pp. 1-12.
Hans-Joachim Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl II, Mathematische Annalen, Vol. 134, No. 3 (1958), pp. 225-231.

Cf. A000005, A000203, A038040, A099377, A099378, A324121.

Subsequence of A014567 and A046678.

[k:k in [1..5000]| Gcd(k*NumberOfDivisors(k),DivisorSigma(1,k)) eq 1]; // Marius A. Burtea, Dec 20 2019

Select[Range[10^4], CoprimeQ[# * DivisorSigma[0, #], DivisorSigma[1, #]] &]

A348411 Numbers whose divisors have a harmonic mean with a denominator 2.

Original entry on oeis.org

3, 15, 42, 84, 135, 308, 420, 546, 1428, 1488, 1890, 2295, 2660, 3780, 6210, 7440, 9424, 12180, 13392, 18018, 20832, 24384, 24570, 43152, 43400, 64260, 66960, 77490, 90090, 98420, 121920, 127710, 155610, 200340, 204600, 227664, 316992, 348688, 353400, 461776, 483210

Offset: 1

Amiram Eldar, Oct 17 2021

nonn

Numbers k such that A099378(k) = 2.
The odd terms seem to be relatively rare: 3, 15, 135, 2295, 544635, 9258795, 22330035, 39118408875, ...
If k is in this sequence, then 2*k is in A348412.

			3 is a term since the harmonic mean of its divisors, {1, 3}, is 3/2.
15 is a term since the harmonic mean of its divisors, {1, 3, 5, 15}, is 5/2.

Amiram Eldar, Table of n, a(n) for n = 1..310

Cf. A001599, A099377, A099378, A348412.

Similar sequences: A159907, A330598.

filter:= proc(n) local L,h;
  L:= map(t->1/t,numtheory:-divisors(n));
  denom(nops(L)/convert(L,`+`))=2;
end proc:
select(filter, [$1..10^6]); # Robert Israel, Oct 17 2021

Select[Range[10^5], Denominator[DivisorSigma[0, #]/DivisorSigma[-1, #]] == 2 &]
Select[Range[500000],Denominator[HarmonicMean[Divisors[#]]]==2&] (* Harvey P. Dale, Apr 06 2023 *)

isok(m) = my(d=divisors(m)); denominator(#d/sum(k=1, #d, 1/d[k])) == 2; \\ Michel Marcus, Oct 18 2021

from sympy import gcd, divisor_sigma
A348411_list = [n for n in range(1,10**3) if (lambda x, y: 2*gcd(x,y*n)==x)(divisor_sigma(n),divisor_sigma(n,0))] # Chai Wah Wu, Oct 20 2021

A348415 Numbers k such that k and k+1 have the same denominator of the harmonic means of their divisors.

Original entry on oeis.org

12, 88, 180, 266, 321, 604, 4277, 4364, 8632, 15861, 18720, 28461, 47613, 63546, 97412, 98907, 135078, 137333, 154132, 179621, 185776, 192699, 203709, 265489, 284883, 344217, 383466, 517610, 604197, 876469, 1089604, 1277518, 1713865, 1839123, 1893268, 2349390

Offset: 1

Amiram Eldar, Oct 17 2021

nonn

Numbers k such that A099378(k) = A099378(k+1).
The common denominators of k and k+1 are 7, 45, 91, 30, 36, 133, 96, 637, ...
Can 3 consecutive numbers have the same denominator of harmonic mean of divisors? There are no such numbers below 10^10.

			12 is a term since the harmonic means of the divisors of 12 and 13 are 18/7 and 13/7, respectively, and both have the denominator 7.

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A099377, A099378.

Similar sequences: A002961, A238380.

dh[n_] := Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]]; Select[Range[10^6], dh[#] == dh[# + 1] &]

f(n) = my(d=divisors(n)); denominator(#d/sum(k=1, #d, 1/d[k])); \\ A099378
isok(k) = f(k) == f(k+1); \\ Michel Marcus, Oct 20 2021

A348694 a(n) is the least number k such that the numerator of the harmonic mean of the divisors of k is equal to n, or -1 if no such k exists.

Original entry on oeis.org

1, 6, 3, 2, 5, 270, 7, 672, 84, 30, 11, 4, 13, 18620, 420, 24, 17, 12, 19, 10, 21, 22, 23, 30240, 1550, 78, 9, 168, 29, 60, 31, 8, 132, 102, 35, 18, 37, 38, 39, 3360, 41, 3724, 43, 7392, 45, 15456, 47, 1080, 49, 6051500, 153, 26, 53, 540, 55, 56, 57, 174, 59, 90

Offset: 1

Amiram Eldar, Oct 30 2021

nonn

			a(2) = 6 since the harmonic mean of the divisors of 6 is 2.
a(3) = 3 since the harmonic mean of the divisors of 3 is 3/2.

Amiram Eldar, Table of n, a(n) for n = 1..179
Amiram Eldar, Table of n, a(n) for n = 1..1000 with 9 holes marked with 0

Cf. A099377, A099378, A348695.

num[n_] := Numerator[DivisorSigma[0, n]/DivisorSigma[-1, n]]; seq[m_] := Module[{s = Table[0, {m}], c = 0, n = 1, i}, While[c < m, i = num[n]; If[i <= m && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[100]

cp's OEIS Frontend

A335368 Harmonic numbers k with a record number of primes p not dividing k such that k*p is also a harmonic number.

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A335370 Harmonic numbers m with a record number k of distinct prime numbers p_i (i = 1..k) that do not divide m such that m*p_1, m*p_1*p_2, ... , m*p_1*...*p_k are all harmonic numbers.

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A335371 Harmonic numbers with a record number of harmonic numbers that can be generated from them using an iterative process of multiplying by primes (see Comments).

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A348414 Numbers with record values of the denominator of the harmonic mean of their divisors.

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A348825 Numbers k such that the denominator of the harmonic mean of the divisors of k is larger than k.

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A349477 Numbers k such that the sequence of elements of the continued fraction of the harmonic mean of the divisors of k is palindromic.

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A330606 Numbers k such that k*d(k) and sigma(k) are relatively prime, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

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A348411 Numbers whose divisors have a harmonic mean with a denominator 2.

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A335370 Harmonic numbers m with a record number k of distinct prime numbers p_i (i = 1..k) that do not divide m such that mp_1, mp_1p_2, ... , mp_1...p_k are all harmonic numbers.