A072911 -id:A072911 - cp's OEIS frontend

A331273 Sum of the iterated exponential totient function (A072911).

0, 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, 3, 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, Feb 25 2020

nonn

Analogous to A092693 with the exponential totient function ephi instead of the Euler totient function phi (A000010).

a(n) = 1 for n > 1 which is cubefree (A004709) and a(n) > 1 for n in A046099.

			a(8) = ephi(8) + ephi(ephi(8)) = 2 + 1 = 3 (where ephi is A072911).

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

Cf. A004709, A046099, A072911, A092693.

ephi[n_] := Times @@ EulerPhi[FactorInteger[n][[;; , 2]]]; s[n_] := Plus @@ FixedPointList[ephi, n] - n - 1; Array[s, 100]

A327838 Decimal expansion of the asymptotic mean of the exponential totient function (A072911).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 27 2019

			1.252707785375446126053750751934283060439237967108915...

László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-294.

Cf. A000010, A072911, A322887, A327837, A361013.

$MaxExtraPrecision = 500; m = 500; f[x_] := Log[1 + Sum[x^e * (EulerPhi[e] - EulerPhi[e - 1]), {e, 3, m}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[f[1/2] + NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals lim_{m->oo} (1/m) Sum_{k=1..m} A072911(k).

Equals Product_{p prime} (1 + Sum_{e >= 3} (phi(e) - phi(e-1))/p^e), where phi is the Euler totient function (A000010).

A349307 Numbers k such that A072911(k) = A072911(k+1) > 1.

Original entry on oeis.org

80, 135, 296, 343, 375, 567, 624, 728, 783, 944, 1160, 1431, 1592, 1624, 1647, 1808, 1863, 2024, 2240, 2295, 2456, 2511, 2624, 2727, 2888, 3087, 3320, 3429, 3536, 3591, 3624, 3752, 3992, 4023, 4184, 4239, 4375, 4400, 4455, 4624, 4671, 4887, 4912, 4913, 5048, 5103

Offset: 1

Amiram Eldar, Nov 14 2021

nonn

Without the restriction that A072911(k) > 1, all the terms of A340152 would be in this sequence.

In contrast to A001274, which has only one known pair of consecutive terms (5186 and 5187), this sequence seems to have many pairs of consecutive terms. The smaller members of these pairs are 4912, 5750, 6858, ...

			80 is a term since A072911(80) = A072911(81) = 2.

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

Cf. A072911, A340152.
Subsequence of A068140.
Similar sequences: A001274, A287055, A293184, A326403, A349308.

f[p_, e_] := EulerPhi[e]; ephi[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[5000], ephi[#] == ephi[# + 1] > 1 &]

A307004 Numbers k such that phi^e(k) > phi^e(m) for all m < k, where phi^e(k) = A072911(k) is the number of divisors d of k such that d and k are exponentially coprime.

Original entry on oeis.org

1, 8, 32, 128, 864, 2048, 3456, 7776, 31104, 279936, 497664, 1990656, 4478976, 17915904, 62208000, 97200000, 362797056, 559872000, 874800000, 1555200000, 6220800000, 13996800000, 55987200000, 349920000000, 895795200000, 1133740800000, 1399680000000, 4534963200000

Offset: 1

Amiram Eldar, Mar 19 2019

nonn

The corresponding record values of phi^e are 1, 2, 4, 6, 8, 10, 12, 16, 24, ... (see the link for more values).

József Sándor, On an exponential totient function, Studia Univ. Babees-Bolyai, Math., Vol. 41 (1996), pp. 91-94.

Amiram Eldar, Table of n, a(n) for n = 1..57
Amiram Eldar, Table of n, a(n), A072911(a(n)) for n = 1..57
László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2004), pp. 285-294.

Cf. A025487, A072911.

f[n_] := Times@@EulerPhi[FactorInteger[n][[All,2]]]; fm=0; s={}; Do[f1=f[n]; If[f1>fm, AppendTo[s,n]; fm=f1], {n,1,10^6}]; s

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

A325988 Number of covering (or complete) factorizations 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, 5, 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

Gus Wiseman, May 30 2019

nonn

First differs from A072911 at a(64) = 5, A072911(64) = 4.

A covering factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of some submultiset of the factors.

			The a(64) = 5 factorizations:
  (2*2*2*2*2*2)
  (2*2*2*2*4)
  (2*2*2*8)
  (2*2*4*4)
  (2*4*8)
The a(96) = 4 factorizations:
  (2*2*2*2*2*3)
  (2*2*2*3*4)
  (2*2*3*8)
  (2*3*4*4)

Cf. A002033, A103295, A126796, A188431.

Cf. A325684, A325781, A325786, A325788, A325986, A325989, A325990.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Union[Times@@@Subsets[#]]==Divisors[n]&]],{n,100}]

a(2^n) = A126796(n).

A361012 Multiplicative with a(p^e) = sigma(e), where sigma = A000203.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 3, 1, 1, 1, 4, 3, 1, 4, 3, 1, 1, 1, 6, 1, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 3, 3, 1, 1, 7, 3, 3, 1, 3, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 3, 12, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 3, 3, 1, 1, 1, 7, 7, 1, 1, 3, 1, 1

Offset: 1

Vaclav Kotesovec, Feb 28 2023

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

Cf. A000203, A049419, A145353, A072911, A327837, A327838.

g[p_, e_] := DivisorSigma[1, e]; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(sigma, factor(n)[, 2])); \\ Amiram Eldar, Jan 07 2025

from math import prod
from sympy import divisor_sigma, factorint
def A361012(n): return prod(divisor_sigma(e) for e in factorint(n).values()) # Chai Wah Wu, Feb 28 2023

Dirichlet g.f.: Product_{p prime} (1 + Sum_{e>=1} sigma(e) / p^(e*s)).

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 + Sum_{e>=2} (sigma(e) - sigma(e-1)) / p^e) = 2.96008030202494141048182047811089469392843909592516341... = A361013

A321167 The e-unitary Euler function: a(1) = 1, a(n) = Product uphi(e(i)) for n = Product p(i)^e(i), where uphi is the unitary totient function (A047994).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 10 2019

The unitary version of A072911.

For n = Product p(i)^e(i) > 1, a(n) is the number of divisors d of n such that d and n are exponentially unitary coprime, i.e., d = Product p(i)^f(i) where 1 <= f(i) <= e(i) and uGCD(f(i), e(i)) = 1 for any i, where uGCD(m, n) is the largest divisor of m that is a unitary divisor of n.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp. Vol. 35 (2011), pp. 205-216.

Cf. A047994, A072911, A358658.

f[p_, e_] := p^e-1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; fe[p_, e_] := uphi[e]; euphi[n_] := Times @@ fe @@@ FactorInteger[n]; Array[euphi, 100]

uphi(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^f[i,2]-1)};
a(n) = {my(f=factor(n)); prod(i=1, #f~, uphi(f[i,2]))}; \\ Amiram Eldar, Nov 29 2022

Sum_{k=1..n} a(k) ~ c_1 * n + c_2 * n^(1/3) + O(n^(1/4 + eps)), where c_1 = A358658 and c_2 is a constant (see Minculete and Tóth, 2011). - Amiram Eldar, Nov 29 2022

A308056 a(1) = 1, a(n) is the sum of the divisors d of n such that d and n are exponentially coprime.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 6, 3, 10, 11, 6, 13, 14, 15, 10, 17, 6, 19, 10, 21, 22, 23, 18, 5, 26, 12, 14, 29, 30, 31, 30, 33, 34, 35, 6, 37, 38, 39, 30, 41, 42, 43, 22, 15, 46, 47, 30, 7, 10, 51, 26, 53, 24, 55, 42, 57, 58, 59, 30, 61, 62, 21, 34, 65, 66, 67, 34, 69

Offset: 1

Amiram Eldar, May 10 2019

The sequence of the number of those divisors is A072911.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-294; arXiv preprint, arXiv:math/0610274v2 [math.NT], 2006-2009.

Cf. A072911.

fun[p_, e_] := Sum[If[GCD[i,e]==1, p^i, 0], {i,1,e}]; a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, sum(k = 1, f[i,2], (gcd(k, f[i, 2]) == 1) * f[i,1]^k));} \\ Amiram Eldar, Feb 13 2024

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

Sum_{k=1..n} a(k) = c * n^2 / 2 + O(x^(3/2) * exp(A * log(n)^(3/5) * log(log(n))^(-1/5))), where A is a constant and c = Product_{p prime} (1 + Sum_{k>=2} (a(p^k) - p*a(p^(k-1)))/p^(2*k)) = 0.77693509739103041486... (Tóth, 2004). - Amiram Eldar, Feb 13 2024

A331407 Numbers at which the sum of the iterated exponential totient function (A331273) attains a record.

Original entry on oeis.org

1, 2, 8, 32, 128, 864, 3456, 7776, 31104, 279936, 497664, 1990656, 4478976, 17915904, 62208000, 97200000, 559872000, 874800000, 1555200000, 6220800000, 13996800000, 55987200000

Offset: 1

Amiram Eldar, Feb 25 2020

Analogous to A181659 with the exponential totient function (A072911) instead of the Euler totient function phi (A000010).

The corresponding record values are 0, 1, 3, 5, 7, 11, 13, 19, 27, 37, 43, 51, 61, 75, 83, 101, 123, 147, 165, 195, 243, 293, ...

Cf. A072911, A181659, A331273.

ephi[n_] := Times @@ EulerPhi[FactorInteger[n][[;; , 2]]]; s[n_] := Plus @@ FixedPointList[ephi, n] - n - 1; seq = {}; smax = -1; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 5000}]; seq

cp's OEIS Frontend

A331273 Sum of the iterated exponential totient function (A072911).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A327838 Decimal expansion of the asymptotic mean of the exponential totient function (A072911).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A349307 Numbers k such that A072911(k) = A072911(k+1) > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A307004 Numbers k such that phi^e(k) > phi^e(m) for all m < k, where phi^e(k) = A072911(k) is the number of divisors d of k such that d and k are exponentially coprime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A325988 Number of covering (or complete) factorizations of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A361012 Multiplicative with a(p^e) = sigma(e), where sigma = A000203.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A321167 The e-unitary Euler function: a(1) = 1, a(n) = Product uphi(e(i)) for n = Product p(i)^e(i), where uphi is the unitary totient function (A047994).

Views