A099378 -id:A099378 - cp's OEIS frontend

A296513 a(n) is the smallest subpart of the symmetric representation of sigma(n).

1, 3, 2, 7, 3, 1, 4, 15, 3, 9, 6, 5, 7, 12, 1, 31, 9, 2, 10, 3, 5, 18, 12, 13, 5, 21, 6, 1, 15, 3, 16, 63, 7, 27, 3, 10, 19, 30, 8, 11, 21, 4, 22, 42, 1, 36, 24, 29, 7, 15, 10, 49, 27, 3, 8, 9, 11, 45, 30, 6, 31, 48, 5, 127, 9, 1, 34, 63, 13, 13, 36, 7, 37, 57, 3

Offset: 1

Omar E. Pol, Feb 10 2018

nonn

If n is an odd prime (A065091) then a(n) = (n + 1)/2.
If n is a power of 2 (A000079) then a(n) = 2*n - 1.
If n is a perfect number (A000396) then a(n) = 1 assuming there are no odd perfect numbers.
a(n) is also the smallest number in the n-th row of the triangles A279391 and A280851.
a(n) is also the smallest nonzero term in the n-th row of triangle A296508.
The symmetric representation of sigma(n) has A001227(n) subparts.
For the definition of the "subpart" see A279387.
For a diagram with the subparts for the first 16 positive integers see A296508.
It appears that a(n) = 1 if and only if n is a hexagonal number (A000384). - Omar E. Pol, Sep 08 2021
The above conjecture is true. See A280851 for a proof. - Omar E. Pol, Mar 10 2022

			For n = 15 the subparts of the symmetric representation of sigma(15) are [8, 7, 1, 8], the smallest subpart is 1, so a(15) = 1.

Shares infinitely many terms with A241558, A241559, A241838, A296512 (and possibly more).

Cf. A000079, A000203 (sum of subparts), A000225, A000384, A000396, A001227 (number of subparts), A065091, A099378, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A279387, A279391, A280850, A280851, A296508.

(* a280851[] and support function are defined in A280851 *)
a296513[n_]:=Min[a280851[n]]
Map[a296513,Range[75]] (* Hartmut F. W. Hoft, Sep 05 2021 *)

More terms from Omar E. Pol, Aug 28 2021

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)).

A361783 Denominators of the harmonic means of the bi-unitary 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, 27, 9, 5, 10, 3, 8, 9, 12, 5, 13, 21, 10, 5, 15, 3, 16, 21, 4, 27, 12, 25, 19, 15, 14, 9, 21, 2, 22, 15, 1, 9, 24, 9, 25, 39, 6, 35, 27, 5, 18, 15, 20, 45, 30, 1, 31, 12, 20, 119, 21, 3, 34, 45, 8, 9, 36, 25, 37, 57

Offset: 1

Amiram Eldar, Mar 24 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011.

Cf. A188999, A222266, A286324, A286325 (positions of 1's), A361782 (numerators).

Similar sequences: A099378, A103340, A361317.

f[p_, e_] := p^e * If[OddQ[e], (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))]; a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); denominator(n * prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  if(e%2, (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))))); }

a(n) = denominator(n*A286324(n)/A188999(n)).

A335369 Harmonic numbers k such that k*p is not a harmonic number for all the primes p that do not divide k.

Original entry on oeis.org

1, 6, 140, 496, 672, 2970, 27846, 105664, 173600, 237510, 539400, 695520, 726180, 753480, 1421280, 1539720, 2229500, 2290260, 8872200, 11981970, 14303520, 15495480, 33550336, 50401728, 71253000, 80832960, 90409410, 144963000, 221557248, 233103780, 287425800, 318177800

Offset: 1

Amiram Eldar, Jun 03 2020

nonn

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, h(k) = k*tau(k)/sigma(k) = k*A000005(k)/A000203(k). The terms of this sequence are harmonic numbers k such that for all the divisors d of h(k), 2*d - 1 is either a nonprime or a prime divisor of k.

The even perfect numbers, 2^(p-1)*(2^p - 1) where p is a Mersenne exponent (A000043), have harmonic mean of divisors p. Therefore, they are in this sequence if p = 2 or if 2*p - 1 is composite (i.e., not in A172461). Of the first 47 Mersenne exponents there are 37 such primes (p = 2, 5, 13, 17, ...), with the corresponding even perfect numbers 6, 496, 33550336, 8589869056, ...

			1 is a term since it is a harmonic number, and there is no prime p such that 1*p = p is a harmonic number (if p is a prime, h(p) = 2*p/(p+1) cannot be an integer).

Amiram Eldar, Table of n, a(n) for n = 1..246 (terms below 10^14)
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, A000043, A000203, A000396, A001599, A099377, A099378, A172461, A335368, A335370, A335371.

harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {, }][[;; , 2]]; harMean[n_] := n * DivisorSigma[0, n]/DivisorSigma[1, n]; primeCountQ[n_] := Module[{d = Divisors[harMean[n]]}, Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &] == {}]; Select[harmNums, primeCountQ]

A348658 Numbers whose numerator and denominator of the harmonic mean of their divisors are both Fibonacci numbers.

Original entry on oeis.org

1, 3, 5, 6, 15, 21, 28, 140, 182, 496, 546, 672, 918, 1890, 2016, 4005, 4590, 24384, 52780, 55860, 68200, 84812, 90090, 105664, 145782, 186992, 204600, 381654, 728910, 907680, 1655400, 2302344, 2862405, 3828009, 3926832, 5959440, 21059220, 33550336, 33839988, 42325920

Offset: 1

Amiram Eldar, Oct 28 2021

nonn

Terms that also Fibonacci numbers are 1, 3, 5, 21, and no more below Fibonacci(300).

			3 is a term since the harmonic mean of its divisors is 3/2 = Fibonacci(4)/Fibonacci(3).
15 is a term since the harmonic mean of its divisors is 5/2 = Fibonacci(5)/Fibonacci(3).

Cf. A000045, A099377, A099378.

Similar sequences: A074266, A123193, A272412, A272440, A348659.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[{5 n^2 - 4, 5 n^2 + 4}]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := fibQ[Numerator[(hn = h[n])]] && fibQ[Denominator[hn]]; Select[Range[1000], q]

from itertools import islice
from sympy import integer_nthroot, gcd, divisor_sigma
def A348658(): # generator of terms
    k = 1
    while True:
        a, b = divisor_sigma(k), divisor_sigma(k,0)*k
        c = gcd(a,b)
        n1, n2 = 5*(a//c)**2-4, 5*(b//c)**2-4
        if (integer_nthroot(n1,2)[1] or integer_nthroot(n1+8,2)[1]) and (integer_nthroot(n2,2)[1] or integer_nthroot(n2+8,2)[1]):
            yield k
        k += 1
A348658_list = list(islice(A348658(),10)) # Chai Wah Wu, Oct 28 2021

A348659 Numbers whose numerator and denominator of the harmonic mean of their divisors are both prime numbers.

Original entry on oeis.org

3, 5, 13, 14, 15, 37, 42, 61, 66, 73, 92, 114, 157, 182, 193, 258, 277, 308, 313, 397, 402, 421, 457, 476, 477, 541, 546, 570, 613, 661, 673, 733, 744, 757, 812, 877, 978, 997, 1093, 1148, 1153, 1201, 1213, 1237, 1266, 1278, 1321, 1381, 1428, 1453, 1621, 1657

Offset: 1

Amiram Eldar, Oct 28 2021

nonn

The prime terms of this sequence are the primes p such that (p+1)/2 is also a prime (A005383).

If p is in A109835, then p*(2*p-1) is a semiprime term.

			3 is a term since the harmonic mean of its divisors is 3/2 and both 2 and 3 are primes.

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

Cf. A005383, A099377, A099378, A109835.

Similar sequences: A023194, A048968, A074266, A348659.

q[n_] := Module[{h = DivisorSigma[0, n]/DivisorSigma[-1, n]}, And @@ PrimeQ[{Numerator[h], Denominator[h]}]]; Select[Range[2000], q]

A349476 Numbers k such that the continued fraction of the harmonic mean of the divisors of k contains a single distinct element.

Original entry on oeis.org

1, 6, 15, 28, 30, 140, 270, 496, 545, 672, 792, 1365, 1638, 2970, 3515, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 44950, 46359, 55860, 59670, 105664, 117800, 167400, 173600, 237510, 242060, 253539, 332640, 360360, 539400, 681156, 691782, 695520, 726180, 753480, 950976

Offset: 1

Amiram Eldar, Nov 19 2021

nonn

All the harmonic numbers (A001599) are terms of this sequence.
The least term with m elements in the continued fraction of the harmonic mean of its divisors for m = 1, 2, 3, and 4 is 1, 15, 792 and 545, respectively.
Are there terms with more than 4 elements? There are no such terms below 2*10^9.

			15 is a term since the harmonic mean of its divisors is 5/2 = 2 + 1/2.
545 is a term since the harmonic mean of its divisors is 109/33 = 3 + 1/(3 + 1/(3 + 1/3)).
792 is a term since the harmonic mean of its divisors is 528/65 = 8 + 1/(8 + 1/8).

Amiram Eldar, Table of n, a(n) for n = 1..150

Cf. A099377, A099378, A349473.

c[n_] := ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]; q[n_] := Length[Union[c[n]]] == 1; Select[Range[10^6], q]

A379946 Irregular triangle read by rows: T(n, k) is the denominator of the harmonic mean of all positive divisors of n except the k-th of them.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 3, 1, 1, 1, 1, 5, 11, 1, 1, 7, 11, 13, 7, 2, 5, 2, 4, 13, 8, 17, 1, 1, 4, 11, 2, 5, 13, 9, 1, 1, 5, 17, 11, 23, 1, 19, 7, 23, 15, 23, 27, 29, 15, 1, 1, 7, 1, 11, 2, 37, 19, 1, 1, 11, 8, 37, 19, 2, 41, 11, 25, 29, 31, 7, 25, 17, 35, 1, 1, 3, 2, 13, 9, 1, 19, 29, 59

Offset: 2

Stefano Spezia, Jan 07 2025

			The irregular triangle begins as:
  1,  1;
  1,  1;
  3,  5,  3;
  1,  1;
  1,  1,  5, 11;
  1,  1;
  7, 11, 13,  7;
  2,  5,  2;
  4, 13,  8, 17;
  ...
The irregular triangle of the related fractions begins as:
     2,     1;
     3,     1;
   8/3,   8/5,   4/3;
     5,     1;
     3,     2,   9/5,  18/11;
   7,1;
  24/7, 24/11, 24/13,   12/7;
   9/2,   9/5,   3/2;
  15/4, 30/13,  15/8,  30/17;
  ...

Stefano Spezia, Table of n, a(n) for n = 2..10371 (first 1400 rows of the triangle)
Jaba Kalita and Helen K. Saikia, A note on near harmonic divisor number and associated concepts, Palestine Journal of Mathematics, Vol. 13(4), 2024.

Cf. A000005, A000203, A001599, A027750, A099378, A379945 (numerator).

T[n_,k_]:=Denominator[n(DivisorSigma[0,n]-1)/(DivisorSigma[1,n]-n/Part[Divisors[n],k])]; Table[T[n,k],{n,2,24},{k,DivisorSigma[0,n]}]//Flatten

T(n, k) = denominator(n*(tau(n) - 1)/(sigma(n) - n/A027750(n, k))).

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

cp's OEIS Frontend

A296513 a(n) is the smallest subpart of the symmetric representation of sigma(n).

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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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A361783 Denominators of the harmonic means of the bi-unitary divisors of the positive integers.

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A335369 Harmonic numbers k such that k*p is not a harmonic number for all the primes p that do not divide k.

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A348658 Numbers whose numerator and denominator of the harmonic mean of their divisors are both Fibonacci numbers.

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A348659 Numbers whose numerator and denominator of the harmonic mean of their divisors are both prime numbers.

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A349476 Numbers k such that the continued fraction of the harmonic mean of the divisors of k contains a single distinct element.

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