A322857 -id:A322857 - cp's OEIS frontend

A383693 Exponential unitary abundant numbers: numbers k such that A322857(k) > 2*k.

900, 1764, 4356, 4500, 4900, 6084, 6300, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22500, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 36900, 38700, 40572, 42300, 42588, 44100, 47700, 47916, 49284, 49500

Offset: 1

Amiram Eldar, May 05 2025

First differs from its subsequence A383697 at n = 21.
All the terms are nonsquarefree numbers (A013929), since A322857(k) = k if k is a squarefree number (A005117).
If an exponential abundant number (A129575) is exponentially squarefree (A209061), then it is in this sequence. Terms of this sequence that are not exponentially squarefree are a(21) = 22500, a(77) = 86436, a(140) = 157500, etc..
The least odd term is a(202273) = 225450225, and the least term that is coprime to 6 is a(1.002..*10^18) = 1117347505588495206025.
The asymptotic density of this sequence is Sum_{n>=1} f(A383694(n)) = 0.00089722..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)).

			900 is a term since A322857(900) = 2160 > 2*900 = 1800.

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

Cf. A005117, A209061, A322857, A322858, A361255.
Subsequence of A013929 and A129575.
Subsequences: A383694, A383697, A383698.

f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Times @@ f @@@ FactorInteger[n] > 2 n; Select[Range[50000], q]

fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
isok(k) = {my(f = factor(k)); prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}

A383695 Exponential infinitary abundant numbers that are not exponential unitary abundant: numbers k such that A361175(k) > 2*k >= A322857(k).

Original entry on oeis.org

476985600, 815673600, 1018886400, 1177862400, 1493049600, 2014214400, 2373638400, 2712326400, 3756614400, 3863865600, 4744454400, 5218617600, 5246841600, 6234681600, 7928121600, 8108755200, 8245036800, 8972409600, 9062726400, 9824774400, 10502150400, 10603756800

Offset: 1

Amiram Eldar, May 05 2025

nonn

Exponential infinitary abundant numbers are numbers k such that A361175(k) > 2*k.
All the exponential unitary abundant numbers (A383693) are also exponential infinitary abundant numbers. There are numbers that are exponential infinitary abundant and not exponential unitary abundant. The least is: a(1) = 476985600, which is the 427970th exponential infinitary abundant number.
All the terms are nonsquarefree numbers (A013929), since A361175(k) = k if k is a squarefree number (A005117).
The asymptotic density of this sequence is Sum_{n>=1} f(A383696(n)) = 1.9875...*10^(-9), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). The relative density of this sequence within the exponential infinitary abundant numbers is 2.215... * 10^(-6).

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

Cf. A005117, A322857, A361175.
Subsequence of A013929 and A129575.
A383696 is a subsequence.

seq[max_] := Module[{prim = seqA383696[max], s = {}, sq}, Do[sq = Select[Range[Floor[max/p]], CoprimeQ[p, #] && SquareFreeQ[#] &]; s = Join[s, p*sq], {p, prim}]; Union[s]]; seq[10^10] (* using the function seqA383696 from A383696 *)

list(lim) = {my(p = listA383696(lim), s = []); for(i = 1, #p, s = concat(s, apply(x -> p[i]*x, select(x -> gcd(x, p[i]) == 1 && issquarefree(x), vector(lim\p[i], j, j))))); Set(s);} \\ using the function listA383696 from A383696

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 *)

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

A361175 The sum of the exponential infinitary divisors of n.

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, Mar 03 2023

First differs from A322857 at n = 256.
The exponential infinitary 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 an infinitary divisor of e(i).
The number of exponential infinitary divisors of n is A307848(n).

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).
Index entries for sequences related to divisors of numbers.

Cf. A077609, A307848.

Similar sequences: A051377, A322857, A323309, A361174.

idivs[1] = {1}; idivs[n_] := Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
f[p_, e_] := Total[p^idivs[e]]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); } \\ Michel Marcus at A077609
ff(p, e) = sumdiv(e, d, if(isidiv(d, factor(e)), 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 infinitary divisor of e} p^d.

A322858 List of e-perfect numbers that are not e-unitary perfect.

Original entry on oeis.org

17424, 87120, 121968, 226512, 296208, 331056, 400752, 505296, 540144, 609840, 644688, 714384, 749232, 818928, 923472, 1028016, 1062864, 1132560, 1167408, 1237104, 1271952, 1306800, 1376496, 1446192, 1481040, 1550736, 1585584, 1655280, 1690128, 1759824

Offset: 1

Amiram Eldar, Dec 29 2018

nonn

The e-unitary perfect numbers are numbers k such that the sum of their exponential unitary divisors (A322857) equals 2k. Apparently most of the e-perfect numbers (A054979) are also e-unitary perfect numbers: the first 150 e-perfect numbers are also the first 150 e-unitary perfect numbers. But A054979(151) = 17424 is not e-unitary perfect.
Minculete and Tóth ask if there is any e-unitary perfect number which is not e-perfect.
The asymptotic density of this sequence is Sum_{n>=1} f(b(n)) = 0.000016169..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) and b = {17424, 1306800, 54531590400, ...} is the sequence of primitive e-perfect numbers (A054980) that are not e-unitary perfect. - Amiram Eldar, May 06 2025

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. A051377, A054979, A054980, A322857.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[n_] := Times @@ f @@@ FactorInteger[n]; ePerfectQ[n_] := esigma[n] == 2n; fu[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#]==1 &]; eusigma[n_] := Times @@ fu @@@ FactorInteger[n]; euPerfectQ[n_] := eusigma[n] == 2n; aQ[n_] := ePerfectQ[n] && !euPerfectQ[n]; Select[Range[125000], aQ]

A323310 List of e-unitary perfect numbers that are not e-semiproper perfect numbers.

Original entry on oeis.org

4769856, 23849280, 52468416, 81087552, 90627264, 109706688, 138325824, 147865536, 176484672, 195564096, 205103808, 224183232, 252802368, 262342080, 281421504, 290961216, 319580352, 338659776, 348199488, 357739200, 376818624, 395898048, 405437760, 424517184

Offset: 1

Amiram Eldar, Jan 10 2019

nonn

The e-unitary perfect numbers are numbers k such that the sum of their exponential unitary divisors (A322857) equals 2k. The e-semiproper perfect numbers are numbers k such that the sum of their exponential semiproper divisors (A323309) equals 2k. Apparently most of the e-unitary perfect numbers are also e-semiproper perfect numbers: The first 41393 e-unitary perfect numbers are also the first 41393 e-semiproper perfect numbers, but the 41394th e-unitary perfect number is 4769856 which is not e-semiproper perfect. This number, which is the first term of this sequence, was found by Minculete.

The powerful (A001694) terms of this sequence are the primitive terms, i.e., if k is a powerful term, then m*k is a term for any squarefree (A005117) number m that is coprime to k. The only primitive terms below 10^18 are 4769856 and 357739200. If S is the sequence of primitive terms, then the asymptotic density of this sequence is Sum_{n>=1} f(S(n)) = 5.235...*10^(-8), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, May 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicusor Minculete, A new class of divisors: the exponential semiproper divisors, Bulletin of the Transilvania University of Brasov, Mathematics, Informatics, Physics, Series III, Vol. 7, No. 1 (2014), pp. 37-46.

Cf. A322857, A322858, A323308, A323309.

fs[p_, e_] := If[e==1, p, p^e + p]; a[1]=1; essigma[n_] := Times @@ fs @@@ FactorInteger[n]; esPerfectQ[n_] := essigma[n]==2n; fu[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#]==1 &]; eusigma[n_] := Times @@ fu @@@ FactorInteger[n]; euPerfectQ[n_] := eusigma[n] == 2n; aQ[n_] := euPerfectQ[n] && !esPerfectQ[n]; Select[Range[1, 10^8], aQ]

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]

A383694 Primitive exponential unitary abundant numbers: the powerful terms of A383693.

Original entry on oeis.org

900, 1764, 4356, 4500, 4900, 6084, 10404, 12348, 12996, 19044, 22500, 30276, 34596, 44100, 47916, 49284, 60516, 66564, 79092, 79524, 86436, 88200, 101124, 108900, 112500, 125316, 132300, 133956, 152100, 161604, 176400, 176868, 181476, 191844, 213444, 217800, 220500

Offset: 1

Amiram Eldar, May 05 2025

nonn

First differs from its subsequence A383698 at n = 11.
For squarefree numbers k, eusigma(k) = k, where eusigma is the sum of exponential unitary divisors function (A322857). Thus, if m is a term (eusigma(m) > 2*m) and k is a squarefree number coprime to m, then eusigma(k*m) = eusigma(k) * eusigma(m) = k * eusigma(m) > 2*k*m, so k*m is an exponential unitary abundant number. Therefore, the sequence of exponential unitary abundant numbers (A383693) can be generated from this sequence by multiplying with coprime squarefree numbers.
The least odd term is a(1455) = 225450225, and the least term that is coprime to 6 is 1117347505588495206025.

			900 is a term since eusigma(900) = 2160 > 2 * 900, and 900 = 2^2 * 3^2 * 5^2 is a powerful number.
6300 is exponential unitary abundant, since eusigma(6300) = 15120 > 2 * 6300, but it is not a powerful number: 6300 = 2^2 * 3^2 * 5^2 * 7. Thus it is not in this sequence. It can be generated as a term of A383693 from a(1) = 900 by 7 * 900 = 6300, since 7 is squarefree and gcd(7, 900) = 1.

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

Intersection of A001694 and A383693.
A383698 is a subsequence.
Cf. A005117, A322857, A328136.

fun[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Min[(f = FactorInteger[n])[[;; , 2]]] > 1 && Times @@ fun @@@ f > 2*n; Select[Range[250000], q]

fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
isok(k) = {my(f = factor(k)); ispowerful(f) && prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}

A383864 The sum 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, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 19, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 76, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144

Offset: 1

Amiram Eldar, May 12 2025

First differs from A383866 at n = 256.
The sum of divisors d of n such that each is a unitary divisor of an exponential unitary divisor of n (see A361255).
Analogous to the sum of (1+e)-divisors (A051378) as exponential unitary divisors (A361255, A322857) are analogous to exponential divisors (A322791, A051377).
The number of these divisors is A383863(n).

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

Cf. A051377, A209061, A322791, A322857, A361255, A383863, A383866.

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

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sumdiv(f[i, 2], d, if(gcd(d, f[i, 2]/d) == 1, f[i, 1]^d)));}

Multiplicative with a(p^e) = 1 + Sum_{d|e, gcd(d, e/d) = 1} p^d.
a(n) <= A051378(n), with equality if and only if n is an exponentially squarefree number (A209061).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.52168352620962354041..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d|k, gcd(d, k/d)=1} x^(2*k-d))).

cp's OEIS Frontend

A383693 Exponential unitary abundant numbers: numbers k such that A322857(k) > 2*k.

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A383695 Exponential infinitary abundant numbers that are not exponential unitary abundant: numbers k such that A361175(k) > 2*k >= A322857(k).

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A361255 Triangle read by rows: row n lists the exponential unitary 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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A361175 The sum of the exponential infinitary divisors of n.

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A322858 List of e-perfect numbers that are not e-unitary perfect.

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A323310 List of e-unitary perfect numbers that are not e-semiproper perfect numbers.

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A349026 Exponential unitary harmonic numbers: numbers k such that the harmonic mean of the exponential unitary divisors of k is an integer.

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