A222266 -id:A222266 - cp's OEIS frontend

A361782 Numerators of the harmonic means of the bi-unitary divisors of the positive integers.

1, 4, 3, 8, 5, 2, 7, 32, 9, 20, 11, 12, 13, 7, 5, 64, 17, 12, 19, 8, 21, 22, 23, 16, 25, 52, 27, 14, 29, 10, 31, 64, 11, 68, 35, 72, 37, 38, 39, 32, 41, 7, 43, 44, 3, 23, 47, 32, 49, 100, 17, 104, 53, 18, 55, 56, 57, 116, 59, 4, 61, 31, 63, 384, 65, 11, 67, 136

Offset: 1

Amiram Eldar, Mar 24 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
Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011.

Cf. A188999, A222266, A286324, A361783 (denominators).

Similar sequences: A099377, A103339, A361316.

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_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); numerator(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) = numerator(n*A286324(n)/A188999(n)).

f(n) = a(n)/A361783(n) is multiplicative with f(p^e) = (e+1)*(p-1)/(p^(e+1)-1) if e is odd, and e/((p^(e+1)-1)/(p-1) - p^(e/2)) if e is even.

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)).

A379027 Irregular table read by rows in which the n-th row lists the modified exponential divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 3, 4, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 4, 5, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 6, 8, 24, 1, 25, 1, 2, 13, 26, 1, 3, 27, 1, 4, 7, 28

Offset: 1

Amiram Eldar, Dec 14 2024

If the prime factorization of n is Product_{i} p_i^e_i, then the modified exponential divisors of n are all the divisors of n that are of the form Product_{i} p_i^b_i such that 1 + b_i | 1 + e_i for all i.

			The table starts:
  1;
  1, 2;
  1, 3;
  1, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 8;
  1, 9;
  1, 2, 5, 10;
  1, 11;
  1, 3, 4, 12;

Amiram Eldar, Table of n, a(n) for n = 1..11627 (first 2000 rows)
Andrew V. Lelechenko, The Quest for the Generalized Perfect Numbers, Theoretical and Applied Aspects of Cybernetics, Proceedings, The 4th International Scientific Conference of Students and Young Scientists, Kyiv, 2014.
David Moews, Perfect, amicable and sociable numbers.

Cf. A379028 (row lengths), A241405 (row sums).

Similar tables: A027750 (all divisors), A077609 (infinitary), A077610 (unitary), A222266 (bi-unitary), A322791 (exponential), A361255 (exponential unitary).

modexpDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[Divisible, {f[[;; , 2]] + 1, IntegerExponent[d, f[[;; , 1]]] + 1}]]; row[1] = {1}; row[n_] := Select[Divisors[n], modexpDivQ[n, #] &]; Table[row[n], {n, 1, 28}] // Flatten

ismodexpdiv(f, d) = {my(e); for(i=1, #f~, e = valuation(d, f[i, 1]); if((f[i, 2]+1) % (e+1), return(0))); 1; }
row(n) = {my(f = factor(n), d = divisors(f), mediv = [1]); if(n == 1, return(mediv)); for(i=2, #d, if(ismodexpdiv(f, d[i]), mediv = concat(mediv, d[i]))); mediv; }

A331970 The sum of the proper bi-unitary divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 6, 1, 7, 1, 8, 1, 8, 1, 10, 9, 11, 1, 12, 1, 10, 11, 14, 1, 36, 1, 16, 13, 12, 1, 42, 1, 31, 15, 20, 13, 14, 1, 22, 17, 50, 1, 54, 1, 16, 15, 26, 1, 60, 1, 28, 21, 18, 1, 66, 17, 64, 23, 32, 1, 60, 1, 34, 17, 55, 19, 78, 1, 22, 27, 74, 1, 78, 1

Offset: 1

Amiram Eldar, Feb 03 2020

nonn

First differs from A126168 at n = 16.

			a(6) = 6 since A188999(6) - 6 = 12 - 6 = 6.

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

Cf. A001065, A034460, A126164, A126168, A188999, A222266.

fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := bsigma[n] = Times @@ (fun @@@ FactorInteger[n]); bs[n_] := bsigma[n] - n; Array[bs, 100]

a(n) = A188999(n) - n.

A363332 a(n) is the number of divisors of n that are both coreful and bi-unitary.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 28 2023

For the definition of a coreful divisor see A307958, and for the definition of a bi-unitary divisor see A222266.
If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a coreful and an bi-unitary divisor, the exponent of the highest power of p dividing d is a number k >= 1 that is not equal to e/2.
All the terms are odd.

			a(8) = 3 since 8 has 4 divisors, 1, 2, 4 and 8, all are bi-unitary and 3 of them (2, 4 and 8) are also coreful.

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

Cf. A004709, A005361 (number of coreful divisors), A222266, A286324, A362852.

f[p_, e_] := If[OddQ[e], e, e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 120]

a(n)={my(e = factor(n)[,2]); prod(i=1, #e, e[i] - 1 + e[i] % 2);}

Multiplicative with a(p^e) = e - 1 + (e mod 2).
a(n) = 1 if and only if n is cubefree (A004709).
a(n) >= A362852(n), with equality if and only if n is cubefree.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 2/(p^3-p)) = 1.48264570900305853294... .
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^(2*s) + 2/p^(3*s)). - Amiram Eldar, Sep 24 2023

A368978 The number of bi-unitary divisors of n that are squares (A000290).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 11 2024

First differs from A007424, A278908, A307848, A323308, A358260 and A365549 at n = 32.

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

Cf. A000005, A000290, A005117, A013662, A062503, A222266, A286324, A368977.
Cf. A007424, A278908, A307848, A323308, A365549.
Similar sequences: A046951, A056624, A358260, A368980.

f[p_, e_] := If[OddQ[e], (e + 1)/2, 2*Floor[(e+2)/4]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2, (x+1)/2, 2*((x+2)\4)), factor(n)[, 2]));

Multiplicative with a(p^e) = (e + 1)/2 if e is odd, and 2*floor((e+2)/4) if e is even.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A286324(n), with equality if and only if n is in A062503.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4 + 1/p^5) = 1.58922450321701775833... .

A331971 a(n) is the number of values of m such that the sum of proper bi-unitary divisors of m (A331970) is n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 4, 1, 6, 1, 4, 2, 4, 2, 5, 0, 3, 1, 4, 2, 5, 1, 4, 2, 4, 1, 6, 2, 5, 2, 5, 2, 8, 1, 6, 1, 4, 2, 7, 1, 5, 3, 5, 2, 8, 0, 5, 1, 6, 1, 8, 2, 5, 3, 6, 3, 9, 0, 6, 2, 5, 1, 9, 1, 7, 1

Offset: 2

Amiram Eldar, Feb 03 2020

nonn

The bi-unitary version of A048138.

The offset is 2 as in A048138 since there are infinitely many numbers k (the primes and squares of primes) for which A331970(k) = 1.

			a(8) = 2 since 8 is the sum of the proper bi-unitary divisors of 2 numbers: 10 (1 + 2 + 5) and 12 (1 + 3 + 4).

Amiram Eldar, Table of n, a(n) for n = 2..30000

Cf. A048138, A188999, A222266, A324938, A331970, A331973.

fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); bs[n_] := bsigma[n] - n; m = 300; v = Table[0, {m}]; Do[b = bs[k]; If[2 <= b <= m, v[[b]]++], {k, 1, m^2}]; Rest @ v

A335385 The number of tri-unitary divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Jun 04 2020

A divisor d of k is tri-unitary if the greatest common bi-unitary divisor of d and k/d is 1.

Differs from A037445 at n = 32, 96, 128, 160, 224, ...

			a(4) = 2 since 4 has 2 tri-unitary divisors, 1 and 4. 2 is not a tri-unitary divisor of 4 since the greatest common bi-unitary divisor of 2 and 4/2 = 2 is 2 and not 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.
Pentti Haukkanen, On the k-ary convolution of arithmetical functions, The Fibonacci Quarterly, Vol. 38, No. 5 (2000), pp. 440-445.

Cf. A222266, A324706, A324707, A324708, A324709.

Cf. A000005, A034444, A037445, A048105, A049419, A286324, A322483.

f[p_, e_] := If[e == 3 || e == 6, 4, 2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100]

a(n) = vecprod(apply(x -> if(x == 3 || x == 6, 4, 2), factor(n)[, 2])); \\ Amiram Eldar, Dec 18 2023

Multiplicative with a(p^e) = 4 if e = 3 or 6, and a(p^e) = 2 otherwise.

A368977 The number of bi-unitary divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 3, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 4, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 6, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 3, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 6, 3, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 11 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A005117, A062503, A222266, A268335, A286324, A368978.

Similar sequences: A055076, A322483, A363825, A368979.

f[p_, e_] := If[OddQ[e], (e+3)/2, 2*Floor[e/4]+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2, (x+3)/2, 2*(x\4)+1), factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2 + 2*X^3 - X^4)/(1 - X - X^4 + X^5))[n], ", ")) \\ Vaclav Kotesovec, Jan 11 2024

Multiplicative with a(p^e) = (e+3)/2 if e is odd, and 2*floor(e/4)+1 if e is even.
a(n) >= 1, with equality if and only if n is in A062503.
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
From Vaclav Kotesovec, Jan 11 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - (1 - p^s + 2*p^(2*s)) / (p^s*(1 + p^s)*(1 + p^(2*s)))).
Let f(s) = Product_{p prime} (1 - (1 - p^s + 2*p^(2*s)) / (p^s*(1 + p^s)*(1 + p^(2*s)))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (1 - p + 2*p^2) / (p*(1 + p)*(1 + p^2))) = 0.5715031234451924252215041182933420817059774181158824297150124265420835...,
f'(1) = f(1) * Sum_{p prime} (4*p^5 - p^4 + 2*p^3 + 2*p + 1) * log(p) / (p^7 + 2*p^6 + p^5 + 3*p^4 + p^3 + p - 1) = f(1) * 1.1422556395248477875508983912036578244050011522937179465478688905880430...
and gamma is the Euler-Mascheroni constant A001620. (End)

A293618 Numbers n that equal the sum of their first k consecutive aliquot bi-unitary divisors, but not all of them (i.e k < A286324(n)-1).

Original entry on oeis.org

24, 360, 432, 1344, 2016, 19440, 45360, 68544, 714240, 864000, 1468800, 1571328, 1900800, 2391120, 2888704, 3057600, 4586400, 5241600, 103194000

Offset: 1

Amiram Eldar, Oct 13 2017

The bi-unitary version of Erdős-Nicolas numbers (A194472).

If all the aliquot bi-unitary divisors are permitted (i.e. k <= A286324(n)-1), then the 3 bi-unitary perfect numbers, 6, 60 and 90, are included.

			24 is in the sequence since its aliquot bi-unitary divisors are 1, 2, 3, 4, 6, 8, 12 and 24 and 1 + 2 + 3 + 4 + 6 + 8 = 24.

Cf. A188999, A194472, A222266, A286324.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n, Drop[bdiv[n], -2]]; a = {}; Do[ If[selDivs[n] == 0, AppendTo[a, n]; Print[n]], {n, 2, 10^6}]; a (* after Alonso del Arte at A194472 *)

cp's OEIS Frontend

A361782 Numerators of the harmonic means of the bi-unitary 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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A379027 Irregular table read by rows in which the n-th row lists the modified exponential divisors of n.

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A331970 The sum of the proper bi-unitary divisors of n.

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A363332 a(n) is the number of divisors of n that are both coreful and bi-unitary.

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A368978 The number of bi-unitary divisors of n that are squares (A000290).

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A331971 a(n) is the number of values of m such that the sum of proper bi-unitary divisors of m (A331970) is n.

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A335385 The number of tri-unitary divisors of n.

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