A322483 -id:A322483 - cp's OEIS frontend

A322484 Semi-unitary highly composite numbers: where the number of semi-unitary divisors of n (A322483) increases to a record.

1, 2, 6, 24, 30, 120, 210, 840, 2310, 7560, 9240, 30030, 83160, 120120, 480480, 1081080, 1921920, 2042040, 8168160, 18378360, 32672640, 38798760, 155195040, 349188840, 620780160, 892371480, 3569485920, 8031343320, 14277943680, 25878772920, 103515091680

Offset: 1

Amiram Eldar, Dec 11 2018

nonn

The record numbers of semi-unitary divisors are 1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 128, 144, 160, 192, 256, 288, 320, 384, 512, 576, 640, 768, 1024, 1152, 1280, 1536, 2048, ... (see the link for more values).

Amiram Eldar, Table of n, a(n), A322483(a(n)) for n = 1..63

Analogous sequences: A002182 (regular divisors), A002110 (unitary divisors), A293185 (bi-unitary).

Cf. A322483.

f[p_, e_] := Floor[(e+3)/2]; sud[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; seq={}; sm=0; Do[s = sud[k]; If[s > sm, AppendTo[seq, k]; sm = s], {k, 1, 100000}]; seq

nbu(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = (f[k,2]+3)\2; f[k,2] = 1;); factorback(f);} \\ A322483
lista(nn) = {my(m = 0, nb); for (n=1, nn, nb = nbu(n); if (nb > m, m = nb; print1(n, ", ")););} \\ Michel Marcus, Dec 14 2018

A380601 Decimal expansion of the asymptotic mean of the ratio A322483(k)/A000005(k).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jan 27 2025

			0.85980677933034363311244767594941832466515809551385...

Cf. A000005, A322483, A380602 (mean of the inverse ratio).

Similar constants: A308043, A361060, A361062.

$MaxExtraPrecision = 1000; m = 1000; f[x_] := (2*x + 3*(x-1)*Log[1 - x] + (x-1)*Log[1+x])/(4*x); c = Rest[CoefficientList[Series[Log[f[x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k], {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

default(realprecision, 120); default(parisize, 30000000);
my(m = 1024, x = 'x + O('x^m), v); v = Vec((2*x + 3*(x-1)*log(1-x) + (x-1)*log(1+x))/(4*x)); prodeulerrat(sum(i=1, #v, v[i]/p^(i-1)))

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A322483(k)/A000005(k).

Equals Product_{p prime} (1/2 - ((p-1)/4) * (3*log(1-1/p) + log(1+1/p))).

A380602 Decimal expansion of the asymptotic mean of the ratio A000005(k)/A322483(k).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 27 2025

			1.23587977752617354871093805318945110447752750370305...

Cf. A000005, A322483, A380601 (mean of the inverse ratio).

Similar constants: A307869, A361059, A361061.

$MaxExtraPrecision = 1000; m = 1000; f[x_] := 2/x - (1 + 1/x - 2/x^2)*Log[1-x^2]/x; c = Rest[CoefficientList[Series[Log[f[x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k], {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

default(realprecision, 120); default(parisize, 30000000);
my(m = 1024, x = 'x + O('x^m), v); v = Vec(2/x - (1 + 1/x - 2/x^2)*log(1-x^2)/x); prodeulerrat(sum(i=1, #v, v[i]/p^(i-1)))

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A000005(k)/A322483(k).

Equals Product_{p prime} (2*p - (1+p-2*p^2)*log(1-1/p^2)*p).

A325837 The number of coreful divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 07 2019

First differs from A050361 at n = 64.
From Amiram Eldar, Sep 08 2023: (Start)
The number of exponentially odd divisors of n is A322483(n), and their sum is A033634(n).
A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n. (End)
Also, the number of divisors of n that are cubefull exponentially odd numbers (A335988). - Amiram Eldar, Feb 11 2024

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

Cf. A033634, A050361, A065487, A268335, A295316, A335988, A350390.

Cf. A003557, A005361 (number of coreful divisors), A046951, A268335.

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

a(n) = vecprod(apply(x -> (x+1)\2, factor(n)[, 2])); \\ Amiram Eldar, Sep 01 2023

Multiplicative with a(p^e) = floor((e+1)/2).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - Amiram Eldar, Sep 10 2022
a(n) = A046951(A350390(n)) (the number of squares dividing the largest exponentially odd divisor of n). - Amiram Eldar, Sep 01 2023
From Amiram Eldar, Sep 08 2023: (Start)
a(n) = A046951(A003557(n)).
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)). (End)

Name corrected by Amiram Eldar, Sep 08 2023

A368979 The number of exponential divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 11 2024

First differs from A367516 at n = 128, and from A359411 at n = 512.

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

Cf. A001227, A008578, A049419, A138302, A268335, A322791, A368980.
Cf. A359411, A367516.
Similar sequences: A055076, A322483, A363825, A368977.

f[p_, e_] := DivisorSigma[0, e/2^IntegerExponent[e, 2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> numdiv(x >> valuation(x, 2)), factor(n)[, 2]));

Multiplicative with a(p^e) = A001227(e).
a(n) >= 1, with equality if and only if n is in A138302.
a(n) <= A049419(n), with equality if and only if n is noncomposite (A008578).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=2} (d(k) - d(k-1))/p^k) = 1.13657098749361390865..., where d(k) is the number of odd divisors of k (A001227).

A322485 The sum of the semi-unitary divisors of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 11, 10, 18, 12, 20, 14, 24, 24, 19, 18, 30, 20, 30, 32, 36, 24, 44, 26, 42, 31, 40, 30, 72, 32, 39, 48, 54, 48, 50, 38, 60, 56, 66, 42, 96, 44, 60, 60, 72, 48, 76, 50, 78, 72, 70, 54, 93, 72, 88, 80, 90, 60, 120, 62, 96, 80, 71, 84, 144

Offset: 1

Amiram Eldar, Dec 11 2018

A semi-unitary divisor of n is defined as the largest divisor d of n such that the largest divisor of d that is a unitary divisor of n/d is 1 (see A322483).

			The semi-unitary divisors of 8 are 1, 2, 8 (4 is not semi-unitary divisor since the largest divisor of 4 that is a unitary divisor of 8/4 = 2 is 2 > 1), and their sum is 11, thus a(8) = 11.

J. Chidambaraswamy, Sum functions of unitary and semi-unitary divisors, J. Indian Math. Soc., Vol. 31 (1967), pp. 117-126.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Pentti Haukkanen, Basic properties of the bi-unitary convolution and the semi-unitary convolution, Indian J. Math, Vol. 40 (1998), pp. 305-315.
D. Suryanarayana and V. Siva Rama Prasad, Sum functions of k-ary and semi-k-ary divisors, Journal of the Australian Mathematical Society, Vol. 15, No. 2 (1973), pp. 148-162.
Laszlo Tóth, Sum functions of certain generalized divisors, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., Vol. 41 (1998), pp. 165-180.

Cf. A000203, A005117, A034448, A183699, A188999, A322483.

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

a(n) = {my(f = factor(n)); for (k=1, #f~, my(p=f[k,1], e=f[k,2]); f[k,1] = (p^((e+1)\2) - 1)/(p-1) + p^e; f[k,2] = 1;); factorback(f);} \\ Michel Marcus, Dec 14 2018

Multiplicative with a(p^e) = sigma(p^floor((e-1)/2)) + p^e = (p^floor((e+1)/2) - 1)/(p-1) + p^e.
In particular a(p) = p + 1, a(p^2) = p^2 + 1, a(p^3) = p^3 + p + 1.
a(n) <= A000203(n) with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)*zeta(3)/2) * Product_{p prime} (1 - 2/p^3 + 1/p^5) = 0.7004703314... . - Amiram Eldar, Nov 24 2022

A365549 The number of exponentially odd divisors of the square root of the largest square dividing n.

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, 2, 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, 3, 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, Sep 08 2023

First differs from A278908, A307848, A323308 and A358260 at n = 64.

The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n).

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

Cf. A000188, A005117, A013662, A046951, A322483.

Cf. A278908, A307848, A323308, A358260.

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

a(n) = vecprod(apply(x -> 2 + (x-2)\4, factor(n)[, 2]));

a(n) = A322483(A000188(n)).
a(n) >= 1 with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = 2 + floor((e-2)/4).
Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 1/p^(2*s) - 1/p^(4*s)).
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) = 1.54211628314015874165... .

A372380 The number of divisors of n that are numbers whose number of divisors is a power of 2.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 29 2024

First differs from A061389 and A322483 at n = 32.

First differs from A380922 at n = 128. - Vaclav Kotesovec, Apr 22 2025

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

Cf. A000005, A005117, A036537, A372379.

Cf. A061389, A322483, A380922.

f[p_, e_] := Floor[Log2[e + 1]] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> exponent(x+1)+1, factor(n)[, 2]));

from math import prod
from sympy import factorint
def A372380(n): return prod((e+1).bit_length() for e in factorint(n).values()) # Chai Wah Wu, Apr 30 2024

Multiplicative with a(p^e) = floor(log_2(e+1)) + 1.

a(n) = A000005(n) if and only if n is squarefree (A005117).

A365491 The number of divisors of the smallest number whose 4th power is divisible by n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 05 2023

First differs from A365210 at n = 25 and from A034444 at n = 32.

The number of divisors of the smallest 4th divisible by n, A053167(n), is A365492(n).

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

Cf. A000005, A001620, A005117, A053166, A053167, A365492, A365499.

Cf. A019554, A034444, A322483, A365210.

f[p_, e_] := Ceiling[e/4] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
With[{c=Range[100]^4},Table[DivisorSigma[0,Surd[SelectFirst[c,Mod[#,n]==0&],4]],{n,90}]] (* Harvey P. Dale, Jul 09 2024 *)

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

a(n) = A000005(A053166(n)).
Multiplicative with a(p^e) = ceiling(e/4) + 1.
a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 1/p^s - 1/p^(4*s)).
From Vaclav Kotesovec, Sep 06 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * zeta(4*s) * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(4*s) + 1/p^(5*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(4*s) + 1/p^(5*s)).
Sum_{k=1..n} a(k) ~ zeta(4) * f(1) * n * (log(n) + 2*gamma - 1 + 4*zeta'(4)/zeta(4) + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.57615273538566705952061107826411727540624711680289618854325028459572487...,
f'(1) = f(1) * Sum_{p prime} (-5 + 4*p + 2*p^3) * log(p) / (1 - p - p^3 + p^5) = f(1) * 1.3011434396559802378314782600747661399223385669839998680418996210...
and gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = A322483(A019554(n)) (the number of exponentially odd divisors of the smallest number whose square is divisible by n). - Amiram Eldar, Sep 08 2023

A365552 The number of exponentially odd divisors of the powerful part of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 08 2023

First differs from A095691 at n = 512.

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

Cf. A013661, A095691, A057521, A322483.

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

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

a(n) = A322483(A057521(n)).
Multiplicative with a(p) = 1 and a(p^e) = floor((e+3)/2) for e >= 2.
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^(3*s) - 1/p^(4*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 1/p^3 - 1/p^4) = 1.80989829762278336163... .

cp's OEIS Frontend

A322484 Semi-unitary highly composite numbers: where the number of semi-unitary divisors of n (A322483) increases to a record.

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A380601 Decimal expansion of the asymptotic mean of the ratio A322483(k)/A000005(k).

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A380602 Decimal expansion of the asymptotic mean of the ratio A000005(k)/A322483(k).

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A325837 The number of coreful divisors of n that are exponentially odd numbers (A268335).

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A368979 The number of exponential divisors of n that are exponentially odd numbers (A268335).

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A322485 The sum of the semi-unitary divisors of n.

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A365549 The number of exponentially odd divisors of the square root of the largest square dividing n.

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A372380 The number of divisors of n that are numbers whose number of divisors is a power of 2.

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