A049419 -id:A049419 - cp's OEIS frontend

A359411 a(n) is the number of divisors of n that are both infinitary and exponential.

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, Dec 30 2022

First differs from A318672 and A325989 at n = 32.
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 an infinitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e and the bitwise AND of e and k is equal to k.
The least term that is higher than 2 is a(216) = 4.
The position of the first appearance of a prime p in this sequence is 2^A359081(p), if A359081(p) > -1. E.g., 2^39 = 549755813888 for p = 3, 2^175 = 4.789...*10^52 for p = 5, and 2^1275 = 6.504...*10^383 for p = 7.
This sequence is unbounded since A246600 is unbounded (see A359082).

			a(8) = 2 since 8 has 2 divisors that are both infinitary and exponential: 2 and 8.

Cf. A037445, A049419, A077609, A080948, A138302, A246600, A322791, A359081, A359082, A359412.

Cf. A318672, A325989.

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = sumdiv(n, d, bitand(d, n)==d);
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i,2]));}

from math import prod
from sympy import divisors, factorint
def A359411(n): return prod(sum(1 for d in divisors(e,generator=True) if e|d == e) for e in factorint(n).values()) # Chai Wah Wu, Sep 01 2023

Multiplicative with a(p^e) = A246600(e).
a(n) = 1 if and only if n is in A138302.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} A246600(k)/p^k) = 1.135514937... .

A348964 Exponential harmonic (or e-harmonic) numbers of type 2: numbers k such that the harmonic mean of the exponential divisors of k is an integer.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94

Offset: 1

Amiram Eldar, Nov 05 2021

nonn

Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 2.
Equivalently, numbers k such that A348963(k) | k * A049419(k).
Apparently, most exponential harmonic numbers of type 1 (A348961) are also terms of this sequence. Those that are not exponential harmonic numbers of type 2 are 1936, 5808, 9680, 13552, 17424, 29040, ...

			The squarefree numbers are trivial terms. If k is squarefree, then it has a single exponential divisor, k itself, and thus the harmonic mean of its exponential divisors is also k, which is an integer.
12 is a term since its exponential divisors are 6 and 12, and their harmonic mean, 8, is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicuşor Minculete, Contribuţii la studiul proprietăţilor analitice ale funcţiilor aritmetice - Utilizarea e-divizorilor, Ph.D. thesis, Academia Română, 2012. See section 4.3, pp. 90-94.
József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

A005117 and A348965 are subsequences.
Cf. A049419, A322791, A348961, A348963.
Similar sequences: A001599, A006086, A063947, A286325, A319745.

f[p_, e_] := p^e * DivisorSigma[0, e] / DivisorSum[e, p^(e-#) &]; ehQ[1] = True; ehQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], ehQ]

A353898 a(n) is the number of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, May 10 2022

First differs from A049599 and A282446 at n=32.

			The divisors of 8 are 1, 2 = 2^1, 4 = 2^2 and 8 = 2^3. 3 of these divisors, 1, 2 and 4, are in A138302. Therefore, a(8) = 3.

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

Cf. A000005, A004709, A049599, A138302, A282446, A353897.

Similar sequences: A034444, A048105, A286324, A049419.

f[p_, e_] := Floor[Log2[e]] + 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Multiplicative with a(p^e) = floor(log_2(e)) + 2.
a(n) > 1 for n > 1 and a(n) = 2 if and only if n is a prime.
a(n) = A000005(n) if and only if n is cubefree (A004709).

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).

A130897 Numbers that are not exponentially squarefree.

Original entry on oeis.org

16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 256, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 512, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 768, 784, 810, 816, 848, 880, 891, 912, 944, 976

Offset: 1

Laszlo Toth, Mar 18 2011

nonn

A positive integer is called exponentially squarefree (e-squarefree) if in its prime power factorization all the exponents are squarefree.
a(n) is the sequence of positive integers in which prime power factorization there is at least one nonsquarefree exponent.
n is non-e-squarefree iff f(n)=0, where f(n) is the exponential Moebius function A166234.
Product_{k = 1..A001221(n)} A008966(A124010(n,k)) = 0. - Reinhard Zumkeller, Mar 13 2012
The density of {a(n)} is 0.04407699... (see comment in A209061). - Peter J. C. Moses and Vladimir Shevelev, Sep 08 2015

			16=2^4, 48=2^4*3, 256=2^8 are non-e-squarefree, since 4 and 8 are nonsquarefree.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. V. Subbarao, On some arithmetic convolutions, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics No. 251, 247-271, Springer, 1972, doi:10.1007/BFb0058796.
Laszlo Toth, On certain arithmetic functions involving exponential divisors, II., Annales Univ. Sci. Budapest., Sect. Comp., 27 (2007), 155-166.

Complement of A209061; subsequence of A013929, A046099, and A046101.

Cf. A005117, A166234, A049419, A051377, A008966, A124010.

a130897 n = a130897_list !! (n-1)
a130897_list = filter
   (any (== 0) . map (a008966 . fromIntegral) . a124010_row) [1..]
-- Reinhard Zumkeller, Mar 13 2012

filter:=n ->  not andmap(t -> numtheory:-issqrfree(t[2]), ifactors(n)[2]);
select(filter, [$1..1000]); # Robert Israel, Sep 03 2015

Select[Range@ 1000, ! AllTrue[Last /@ FactorInteger@ #, SquareFreeQ] &] (* Michael De Vlieger, Sep 07 2015, Version 10 *)

is(n)=my(f=factor(n)[, 2]); for(i=1, #f, if(!issquarefree(f[i]), return(1))); 0 \\ Charles R Greathouse IV, Sep 03 2015

A166234 The inverse of the constant 1 function under the exponential convolution (also called the exponential Möbius function).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 0, 1, -1, 1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, 1, 1, 0, -1, -1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 0, 0, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1

Offset: 1

Laszlo Toth, Oct 09 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Xiaodong Cao and Wenguang Zahi, Some arithmetic functions involving exponential divisors, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.7.
Andrew V. Lelechenko, Exponential and infinitary divisors, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; arXiv preprint, arXiv:1405.7597 [math.NT], 2014, function mu^(E)(n).
M. V. Subbarao, On some arithmetic convolutions, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions, Lecture Notes in Mathematics No. 251, Springer, 1972, pp. 247-271; alternative link.
László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-296; arXiv preprint, arXiv:math/0610274 [math.NT], 2006-2009.
László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Cf. A008683, A049419, A051377, A124010, A209802 (partial sums).

a166234 = product . map (a008683 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

A166234 := proc(n)
    local a,p;
    a := 1;
    if n =1 then
        ;
    else
        for p in ifactors(n)[2] do
                    a := a*numtheory[mobius](op(2,p)) ;
        end do:
    end if;
    a ;
end proc:# R. J. Mathar, Nov 30 2016

a[n_] := Times @@ MoebiusMu /@ FactorInteger[n][[All, 2]];
Array[a, 100] (* Jean-François Alcover, Nov 16 2017 *)

a(n)=factorback(apply(moebius, factor(n)[,2])) \\ Charles R Greathouse IV, Sep 02 2015

Multiplicative, a(p^e) = mu(e) for any prime power p^e (e>=1), where mu is the Möbius function A008683.
a(A130897(n)) = 0; a(A209061(n)) <> 0. - Reinhard Zumkeller, Mar 13 2012
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Product_{p prime} (1 + Sum_{k>=2} (mu(k) - mu(k-1))/p^k) = 0.3609447238... (Tóth, 2007). - Amiram Eldar, Nov 08 2020

A362852 The number of divisors of n that are both bi-unitary and exponential.

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, 2, 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, May 05 2023

First differs from A061704 at n = 128, and from A304327 and abs(A307428) at n = 64.
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 bi-unitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e but k != e/2.
The least term that is higher than 2 is a(64) = 3.
This sequence is unbounded. E.g., a(2^(2^prime(n))) = prime(n).

			a(8) = 2 since 8 has 2 divisors that are both bi-unitary and exponential: 2 and 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Biunitary Divisor.
Eric Weisstein's World of Mathematics, e-Divisor.

Cf. A000005, A004709, A049419, A188999, A222266, A286324, A322791, A359411, A362853 (indices of records), A362854.

Cf. A061704, A304327, A307428.

f[p_, e_] := DivisorSigma[0, e] - If[OddQ[e], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, numdiv(f[i, 2]) - !(f[i, 2] % 2));}

Multiplicative with a(p^e) = d(e) if e is odd, and d(e)-1 if e is even, where d(k) is the number of divisors of k (A000005).
a(n) = 1 if and only if n is cubefree (A004709).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} (d(k)+(k mod 2)-1)/p^k) = 1.1951330849... .

A145353 Sum of the number of e-divisors of all numbers from 1 up to n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 12, 13, 14, 16, 17, 18, 19, 22, 23, 25, 26, 28, 29, 30, 31, 33, 35, 36, 38, 40, 41, 42, 43, 45, 46, 47, 48, 52, 53, 54, 55, 57, 58, 59, 60, 62, 64, 65, 66, 69, 71, 73, 74, 76, 77, 79, 80, 82, 83, 84, 85, 87, 88, 89, 91, 95, 96, 97, 98, 100, 101, 102, 103, 107

Offset: 1

Jaroslav Krizek and N. J. A. Sloane, Mar 03 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, J. Theor. Nombr. Bordeaux 7 (1) (1995) 133-141.

Equals partial sums of A049419.
Cf. A099353, A327837.
Different from A013936 (which does not contain 52).

f[p_, e_]  := DivisorSigma[0, e]; ediv[n_] := Times @@ (f @@@ FactorInteger[n]); Accumulate[Array[ediv, 100]] (* Amiram Eldar, Jun 23 2019 *)

d(n) = {my(f = factor(n)); prod(i = 1, #f~, numdiv(f[i,2]));}
lista(nmax) = {my(s = 0); for(n = 1, nmax, s += d(n); print1(s, ", ")); } \\ Amiram Eldar, Dec 08 2022

a(n) ~ c * n, where c = A327837. - Amiram Eldar, Dec 08 2022

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

A340233 a(n) is the least number with exactly n exponential divisors.

Original entry on oeis.org

1, 4, 16, 36, 65536, 144, 18446744073709551616, 576, 1296, 589824

Offset: 1

Amiram Eldar, Jan 01 2021

nonn

a(11) = 2^(2^10) has 309 digits and is too large to be included in the data section.

See the link for more values of this sequence.

			a(2) = 4 since 4 is the least number with 2 exponential divisors, 2 and 4.

Amiram Eldar, Table of n, a(n) for n = 1..12
Amiram Eldar, Table of n, a(n) for n = 1..100 (given by prime factorizations)

Subsequence of A025487.
Cf. A049419, A318278, A322791.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A309181 (nonunitary), A340232 (bi-unitary).

f[p_, e_] := DivisorSigma[0, e]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]);  max = 6; s = Table[0, {max}]; c = 0; n = 1;  While[c < max, i = d[n]; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* ineffective for n > 6 *)

A049419(a(n)) = n and A049419(k) != n for all k < a(n).

cp's OEIS Frontend

A359411 a(n) is the number of divisors of n that are both infinitary and exponential.

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A348964 Exponential harmonic (or e-harmonic) numbers of type 2: numbers k such that the harmonic mean of the exponential divisors of k is an integer.

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A353898 a(n) is the number of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

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

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A130897 Numbers that are not exponentially squarefree.

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A166234 The inverse of the constant 1 function under the exponential convolution (also called the exponential Möbius function).

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A362852 The number of divisors of n that are both bi-unitary and exponential.

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