A061389 -id:A061389 - cp's OEIS frontend

A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 8, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 16, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2, 4, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 30 2018

The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (of possibly A092520 and A293443).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A061389, A318497 (numerators), A318499.

Cf. also A299150, A046644.

up_to = 65537;
A061389(n) = factorback(apply(e -> (1+eulerphi(e)),factor(n)[,2]));
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
A318497(n) = numerator(v318497_98[n]);
A318498(n) = denominator(v318497_98[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318499(n).

A069915 Sum of (1+phi)-divisors of n (cf. A061389).

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 7, 4, 18, 12, 12, 14, 24, 24, 11, 18, 12, 20, 18, 32, 36, 24, 28, 6, 42, 13, 24, 30, 72, 32, 31, 48, 54, 48, 12, 38, 60, 56, 42, 42, 96, 44, 36, 24, 72, 48, 44, 8, 18, 72, 42, 54, 39, 72, 56, 80, 90, 60, 72, 62, 96, 32, 35, 84, 144, 68, 54, 96, 144, 72

Offset: 1

Vladeta Jovovic, Apr 23 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A061389.

Cf. A027748, A124010, A038566, A051378.

a069915 n = product $ zipWith sum_1phi (a027748_row n) (a124010_row n)
   where sum_1phi p e = 1 + sum [p ^ k | k <- a038566_row e]
-- Reinhard Zumkeller, Mar 13 2012

a[1] = 1; a[p_?PrimeQ] = p + 1; a[n_] := Times @@ (1 + Sum[If[GCD[k, Last[#]] == 1, First[#]^k, 0], {k, 1, Last[#]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 04 2012 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sum(k = 1, f[i, 2], (gcd(k, f[i, 2]) == 1) * f[i, 1]^k));} \\ Amiram Eldar, Aug 15 2023

Multiplicative with a(p^e) = 1+Sum_{k=1..e, gcd(k, e)=1} p^k.

A318497 Numerators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 30 2018

No zeros among the first 2^20 terms. This is most probably multiplicative, like A318498.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A061389, A318314 (denominators).

up_to = 65537;
A061389(n) = factorback(apply(e -> (1+eulerphi(e)),factor(n)[,2]));
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
A318497(n) = numerator(v318497_98[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d 1.

A377711 Numbers that have a record number of (1+phi)-divisors (A061389).

Original entry on oeis.org

1, 2, 6, 24, 30, 96, 120, 210, 480, 840, 1920, 2310, 3360, 9240, 13440, 30030, 36960, 120120, 147840, 332640, 480480, 1330560, 1921920, 4324320, 8168160, 17297280, 30750720, 32672640, 73513440, 155195040, 294053760, 522762240, 620780160, 1396755360, 2646483840

Offset: 1

Amiram Eldar, Nov 04 2024

nonn

Indices of records in A061389.

The corresponding record values are 1, 2, 4, 6, 8, 10, 12, 16, 20, 24, 28, 32, 40, 48, 56, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..231
Amiram Eldar, Table of n, a(n), A363825(a(n)) for n = 1..231

Cf. A061389, A307004.

Subsequence of A025487.

f[p_, e_] := EulerPhi[e] + 1; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq

A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.

Original entry on oeis.org

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

Offset: 1

Yasutoshi Kohmoto

The exponential divisors of a number x = Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.
Wu gives a complicated Dirichlet g.f.
a(1) = 1 by convention. This is also required for a function to be multiplicative. - N. J. A. Sloane, Mar 03 2009
The inverse Moebius transform seems to be in A124315. The Dirichlet inverse appears to be related to A166234. - R. J. Mathar, Jul 14 2014

			a(8)=2 because 2 and 2^3 are e-divisors of 8.
The sets of e-divisors start as:
  1:{1}
  2:{2}
  3:{3}
  4:{2, 4}
  5:{5}
  6:{6}
  7:{7}
  8:{2, 8}
  9:{3, 9}
  10:{10}
  11:{11}
  12:{6, 12}
  13:{13}
  14:{14}
  15:{15}
  16:{2, 4, 16}
  17:{17}
  18:{6, 18}
  19:{19}
  20:{10, 20}
  21:{21}
  22:{22}
  23:{23}
  24:{6, 24}

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Andrew V. Lelechenko, Exponential and infinitary divisors, arXiv:1405.7597 [math.NT], 2014, sequence tau^(e).
David Moews, A database of aliquot cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
László Tóth and Nicuşor Minculete, Exponential unitary divisors, arXiv:0910.2798 [math.NT], 2009.
Tim Trudgian, The sum of the unitary divisor function, arXiv:1312.4615 [math.NT], 2013-2014, Section 3.
Eric Weisstein's World of Mathematics, e-Divisor.
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, J. Theor. Nombr. Bordeaux 7 (1) (1995) 133-141.

Row lengths of A322791.
Cf. A049599, A061389, A051377 (sum of e-divisors).
Partial sums are in A099593.
Cf. A124010, A000005, A049599, A072911, A124315, A166234, A327837, A361012.

A049419:=n->Product(List(Collected(Factors(n)), p -> Tau(p[2]))); List([1..10^4], n -> A049419(n)); # Muniru A Asiru, Oct 29 2017

a049419 = product . map (a000005 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

A049419 := proc(n)
    local a;
    a := 1 ;
    for pf in ifactors(n)[2] do
        a := a*numtheory[tau](op(2,pf)) ;
    end do:
    a ;
end proc:
seq(A049419(n),n=1..20) ; # R. J. Mathar, Jul 14 2014

a[1] = 1; a[p_?PrimeQ] = 1; a[p_?PrimeQ, e_] := DivisorSigma[0, e]; a[n_] := Times @@ (a[#[[1]], #[[2]]] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Jan 30 2012, after Vladeta Jovovic *)

a(n) = vecprod(apply(numdiv, factor(n)[,2])); \\ Amiram Eldar, Mar 27 2023

Multiplicative with a(p^e) = tau(e). - Vladeta Jovovic, Jul 23 2001

Sum_{k=1..n} a(k) ~ A327837 * n. - Vaclav Kotesovec, Feb 27 2023

More terms from Jud McCranie, May 29 2000

A072911 Number of "phi-divisors" of n.

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

Offset: 1

Yasutoshi Kohmoto, Aug 21 2002

If n = Product p(i)^r(i), d = Product p(i)^s(i), = s(i)<=r(i) and gcd(s(i), r(i)) = 1, then d is a phi-divisor of n.

The integers n = Product_{i=1..r} p_i^{a_i} and m = Product_{i=1..r} p_i^{b_i}, a_i, b_i >= 1 (1 <= i <= r) having the same prime factors are called exponentially coprime, if gcd(a_i, b_i) = 1 for every 1 <= i <= r, i.e., the only common exponential divisor of n and m is Product_{i=1..r} p_i = the common squarefree kernel of n and m, cf. A049419, A007947. The terms of this sequence count the divisors d of n such that d and n are exponentially coprime. - Laszlo Toth, Oct 06 2008

József Sándor, On an exponential totient function, Studia Univ. Babes-Bolyai, Math., 41 (1996), 91-94. [Laszlo Toth, Oct 06 2008]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., 27 (2004), 285-294; arXiv:math/0610274 [math.NT], 2006-2009. [From _Laszlo Toth_, Oct 06 2008]

Cf. A061389, A327838.

Cf. A124010, A000010, A049599, A049419, A361012.

a072911 = product . map (a000010 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

A072911 := proc(n)
    local a, p;
    a := 1 ;
    for p in ifactors(n)[2] do
        a := a*numtheory[phi](op(2, p)) ;
    od:
    a ;
end:
seq(A072911(n),n=1..100) ; # R. J. Mathar, Sep 25 2008

a[n_] := Times @@ EulerPhi[FactorInteger[n][[All, 2]]];
Array[a, 105] (* Jean-François Alcover, Nov 16 2017 *)

If n = Product p(i)^r(i) then a(n) = Product (phi(r(i))), where phi(k) is the Euler totient function of k, cf. A000010.

Sum_{k=1..n} a(k) ~ c_1 * n + c_2 * n^(1/3) + O(n^(1/5+eps)), where c_1 = A327838 (Tóth, 2004). - Amiram Eldar, Oct 30 2022

More terms from R. J. Mathar, Sep 25 2008

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).

A380922 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s + 1/p^(3*s)).

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, 6, 2, 8, 4, 4, 4, 4, 4, 6, 2, 4, 4, 4

Offset: 1

Vaclav Kotesovec, Apr 22 2025

First differs from A061389 at n = 32.
First differs from A322483 at n = 32.
First differs from A372380 at n = 128 (next differences are at n=128*k, n=2187*k, ...).
The number of divisors of n that are both biquadratefree (A046100) and exponentially odd (A268335), i.e., in A336591. - Amiram Eldar, Apr 22 2025

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

Cf. A073184, A365498, A383292.
Cf. A322483, A372380, A061389.
Cf. A046100, A268335, A336591.

f[p_, e_] := If[e < 3, 2, 3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 22 2025 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X + X^3))[n], ", "))

Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 1/p^(4*s)).
Dirichlet g.f.: zeta(s)^2 * f(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 + 1/p^3 - 1/p^4) = 0.684286924186862318141968725791218083472312736723163777284618226290055...,
f'(1) = f(1) * Sum_{p prime} (2*p^2 - 3*p + 4) * log(p) / ((p-1) * (p^3 + p^2 + 1)) = f(1) * 0.85825768698295295413525347933038488513032293516964600096226328323449...
and gamma is the Euler-Mascheroni constant A001620.
Multiplicative with a(p^e) = 2 if e <= 2 and 3 otherwise. - Amiram Eldar, Apr 22 2025

A095724 Fixed points of 1+Phi power sigma function 1PhiPsigma: integers m such that 1PhiPsigma(m) = m, where for j = Product p_i^r_i, 1PhiPsigma(j) = Product_i Sum_{0 <= s_i <= r_i, s_i is 0 or coprime to r_i} p_i^s_i.

Original entry on oeis.org

12, 56, 528, 992, 6720, 16256, 666624, 67100672

Offset: 1

Yasutoshi Kohmoto, Jul 08 2004

Factorizations: 2^2*3, 2^3*7, 2^4*3*11, 2^5*31, 2^6*3*5*7, 2^7*127, 2^10*3*7*31.
If m is a perfect number then 2*m is a term of the sequence. Examples: 2^2*3, 2^3*7, 2^5*31, 2^7*127, ....
If a(n)=2^r*k, GCD(2^r,k)=1, then k is squarefree.

			1PhiPsigma(2^5*3^4) = (1 + 2 + 2^2 + 2^3 + 2^4)*(1 + 3 + 3^3) = 961.

Cf. A061389, A095723.

a(8) from Jud McCranie, Jul 16 2004

Edited by Max Alekseyev, Dec 04 2024

cp's OEIS Frontend

A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

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A069915 Sum of (1+phi)-divisors of n (cf. A061389).

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A318497 Numerators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

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A377711 Numbers that have a record number of (1+phi)-divisors (A061389).

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A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.

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A072911 Number of "phi-divisors" of 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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