A278908 -id:A278908 - cp's OEIS frontend

A307845 Exponential unitary highly composite numbers: where the number of exponential unitary divisors (A278908) increases to a record.

1, 4, 36, 576, 14400, 705600, 57153600, 6915585600, 1168733966400, 337764116289600, 121932845980545600, 64502475523708622400, 40314047202317889000000, 33904113697149344649000000, 32581853262960520207689000000, 44604557116992952164326241000000, 74980260513665152588232411121000000

Offset: 1

Amiram Eldar, May 01 2019

nonn

Subsequence of A025487.
All the terms have prime factors with multiplicities which are primorials > 1 (the primorials, A002110, are the unitary highly composite numbers), similarly to exponential highly composite numbers (A318278) whose prime factors have multiplicities which are highly composite numbers (A002182). Thus all the terms are squares. Their square roots are 1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 18378360, 349188840, 8031343320, 200783583000, 5822723907000, 180504441117000, ...
First differs from A306736 at n = 107. - Georg Fischer, Aug 13 2025

Amiram Eldar, Table of n, a(n) for n = 1..201

Cf. A002110, A002182, A025487, A278908, A306736, A318278.

f[p_, e_] := 2^PrimeNu[e]; a[n_] := Times @@ (f @@@ FactorInteger[n]); s = {}; am = 0; Do[a1 = a[n]; If[a1 > am, am = a1; AppendTo[s, n]], {n, 1, 10^6}]; s

A278908(a(n)) = 2^(n-1).

A322857 a(1) = 1; a(n) = sum of exponential unitary divisors of n for n > 1.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 18, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 54, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67

Offset: 1

Amiram Eldar, Dec 29 2018

The exponential unitary (or e-unitary) divisors of n = Product p(i)^a(i) are all the numbers of the form Product p(i)^b(i) where b(i) is a unitary divisor of a(i).

Ivan Neretin, 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. A361255, A051377, A077610, A278908 (number of exponential unitary divisors).

f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#]==1 &]; eusigma[n_] := Times @@ f @@@ FactorInteger[n]; Array[eusigma, 100]

ff(p, e) = sumdiv(e, d, if (gcd(d, e/d)==1, p^d));
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = ff(f[k,1], f[k,2]); f[k,2] = 1); factorback(f); \\ Michel Marcus, Dec 29 2018

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

A307848 The number of exponential infinitary divisors of 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, 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, May 01 2019

The exponential infinitary divisors of Product p(i)^r(i) are all the numbers of the form Product p(i)^s(i) where s(i) if an infinitary divisor of r(i) for all i.
Differs from A278908 at n = 256, 768, 1280, 1792, 2304, 2816, ...
Differs from A323308 at n = 64, 192, 256, 320, 448, 576, 704, ...

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

Cf. A037445, A278908, A323308.

di[1] = 1; di[n_] := Times @@ Flatten[ 2^DigitCount[#, 2, 1]&  /@ FactorInteger[n][[All, 2]] ]; fun[p_,e_] := di[e]; a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); Array[a, 100] (* after Jean-François Alcover at A037445 *)

Multiplicative with a(p^e) = A037445(e).

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Product_{p prime} (1 + Sum_{k>=2} (d(k) - d(k-1))/p^k) = 1.5482125828..., where d(k) = A037445(k). - Amiram Eldar, Nov 08 2020

A361255 Triangle read by rows: row n lists the exponential unitary divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 2, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 16, 17, 6, 18, 19, 10, 20, 21, 22, 23, 6, 24, 5, 25, 26, 3, 27, 14, 28, 29, 30, 31, 2, 32, 33, 34, 35, 6, 12, 18, 36, 37, 38, 39, 10, 40, 41, 42, 43, 22, 44, 15, 45, 46, 47, 6, 48, 7, 49, 10, 50, 51, 26, 52, 53, 6, 54, 55, 14, 56, 57, 58, 59

Offset: 1

R. J. Mathar, Mar 06 2023

Starts to differ from A322791 in row n=16, where 4 is an exponential divisor but not an exponential unitary divisor.

Amiram Eldar, Table of n, a(n) for n = 1..15331 (first 10000 rows, flattened)
Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.

Cf. A322857 (row sums), A278908 (row lengths), A322791 (includes non-unitary exp divs).

A361255 := proc(n)
    local expundivs ,d,isue,p,ai,bi;
    expudvs := {} ;
    for d in numtheory[divisors](n) do
        isue := true ;
        for p in numtheory[factorset](n) do
            ai := padic[ordp](n,p) ;
            bi := padic[ordp](d,p) ;
            if bi > 0 then
                if modp(ai,bi) <>0 or igcd(bi,ai/bi) <> 1 then
                    isue := false;
                end if;
            else
                isue := false ;
            end if;
        end do;
        if isue then
            expudvs := expudvs union {d} ;
        end if;
    end do:
    sort(expudvs) ;
end proc:
seq(op(A361255(n)),n=1..60) ;

udivQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m] && CoprimeQ[m, n/m]);
expuDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[udivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expuDivs[1] = {1};
expuDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expuDivQ[n, #] &]];
Table[expuDivs[n], {n, 1, 70}] // Flatten (* Amiram Eldar, Mar 11 2023 *)

A358260 a(n) is the number of infinitary square divisors of 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, 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, Nov 06 2022

First differs from A007424 at n = 36, from A323308 at n = 64, and from A278908 and A307848 at n = 128.

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

Cf. A000120, A048881, A037445, A077609, A358261.
Similar sequences: A046951, A056624, A056626.
Sequences with the same initial terms: A007424, A278908, A307848, A323308.

f[p_, e_] := 2^DigitCount[If[OddQ[e], e - 1, e], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = 2^A000120(e) if e is even, and 2^A000120(e-1) if e is odd.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((1-1/p) * Sum_{k>=1} a(p^k)/p^k) = 1.55454884667440993654... .

A361174 The sum of the exponential squarefree exponential divisors (or e-squarefree e-divisors) of n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 6, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 18, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67, 102

Offset: 1

Amiram Eldar, Mar 03 2023

The exponential squarefree exponential divisors (or e-squarefree e-divisors) of n = Product_i p(i)^e(i) are all the numbers of the form Product_i p(i)^d(i) where d(i) is a squarefree divisor of e(i).

The number of exponential squarefree exponential divisors of n is A278908(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.
Xiangzhen Zhao, Min Liu, and Yu Huang, Mean value for the function t^(e)(n) over square-full numbers, Scientia Magna, Vol. 8, No. 3 (2012), pp. 110-114.
Index entries for sequences related to divisors of numbers.

Cf. A278908.

Similar sequences: A051377, A322857, A323309, A361175.

f[p_, e_] := DivisorSum[e, p^# &, SquareFreeQ[#] &]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

ff(p, e) = sumdiv(e, d, if(issquarefree(d), p^d, 0));
a(n) = {my(f=factor(n)); prod(i=1, #f~, ff(f[i, 1], f[i, 2]));}

Multiplicative with a(p^e) = Sum_{d|e, d squarefree} p^d.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, c = Product_{p prime} (1 + Sum_{k>=2} (a(p^k) - p*a(p^(k-1)))/p^(2*k)) = 1.08989220899432387559... . - Amiram Eldar, Feb 13 2024

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... .

A369163 a(n) = A000005(A000688(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, 3, 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, 2, 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 15 2024

First differs from A007424, A278908, A307848, A323308, A358260 and A365549 at n = 36.

The sums of the first 10^k terms, for k = 1, 2, ..., are 13, 143, 1486, 15054, 151067, 1511982, 15123465, 151245456, 1512484372, 15124927227, ... . From these values the asymptotic mean of this sequence, whose existence was proven by Ivić (1983) (see the Formula section), can be empirically evaluated by 1.512... .

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 73.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137. See p. 131, eq. 4.3.

Cf. A000005, A000688.
Cf. A048109, A059956, A271971.
Cf. A369162, A369164, A369165.
Cf. A007424, A278908, A307848, A323308, A358260, A365549.

Table[DivisorSigma[0, FiniteAbelianGroupCount[n]], {n, 1, 100}]

a(n) = numdiv(vecprod(apply(numbpart, factor(n)[, 2])));

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^4), where c = Sum_{k>=1} d(k) * A000005(k) is a constant, d(k) is the asymptotic density of the set {m | A000688(m) = k} (e.g., d(1) = A059956, d(2) = A271971, d(3) appears in A048109) (Ivić, 1983).

A383863 The number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a unitary divisor of the p-adic valuation of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 3, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 6, 3, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, May 12 2025

First differs from A073184 at n = 64.
First differs from A383865 at n = 256.
The number of divisors d of n such that each is a unitary divisor of an exponential unitary divisor of n (see A361255).	
Analogous to the number of (1+e)-divisors (A049599) as exponential unitary divisors (A361255, A278908) are analogous to exponential divisors (A322791, A049419).
The sum of these divisors is A383864(n).
Also, the number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a squarefree divisor of the p-adic valuation of n. The sum of these divisors is A383867(n).

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

Cf. A001221, A034444, A049419, A049599, A073184, A209061, A278908, A322791, A361255, A383864, A383865, A383867.

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

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

Multiplicative with a(p^e) = 1 + 2^A001221(e) = 1 + A034444(e).

a(n) <= A049599(n), with equality if and only if n is an exponentially squarefree number (A209061).

A349026 Exponential unitary harmonic numbers: numbers k such that the harmonic mean of the exponential unitary 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 06 2021

nonn

First differs from A348964 at n = 102. a(102) = 144 is not an exponential harmonic number of type 2.
The exponential unitary divisors of n = Product p(i)^e(i) are all the numbers of the form Product p(i)^b(i) where b(i) is a unitary divisor of e(i) (see A278908).
Equivalently, numbers k such that A349025(k) | k * A278908(k).

			The squarefree numbers are trivial terms. If k is squarefree, then it has a single exponential unitary divisor, k itself, and thus the harmonic mean of its exponential unitary divisors is also k, which is an integer.
144 is a term since its exponential unitary divisors are 6, 18, 48 and 144, and their harmonic mean, 16, 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.

Cf. A278908 (number of exponential unitary divisors), A322857, A322858, A323310, A349025, A349027.

Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.

f[p_, e_] := p^e * 2^PrimeNu[e] / DivisorSum[e, p^(e - #) &, CoprimeQ[#, e/#] &]; euhQ[1] = True; euhQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], euhQ]

cp's OEIS Frontend

A307845 Exponential unitary highly composite numbers: where the number of exponential unitary divisors (A278908) increases to a record.

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A322857 a(1) = 1; a(n) = sum of exponential unitary divisors of n for n > 1.

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A307848 The number of exponential infinitary divisors of n.

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A361255 Triangle read by rows: row n lists the exponential unitary divisors of n.

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A358260 a(n) is the number of infinitary square divisors of n.

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A361174 The sum of the exponential squarefree exponential divisors (or e-squarefree e-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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A369163 a(n) = A000005(A000688(n)).

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