A323309 -id:A323309 - cp's OEIS frontend

A323308 The number of exponential semiproper divisors of n.

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, 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 10 2019

An exponential semiproper divisor of n is a divisor d such that rad(d) = rad(n) and gcd(d/rad(n), n/d) = 1, where rad(n) is the largest squarefree divisor of n (A007947).
a(n) is also the number of divisors of n that are squares of squarefree numbers (A062503). - Amiram Eldar, Oct 08 2022
a(n) is also the number of unitary divisors of n that are powerful (A001694). - Amiram Eldar, Feb 18 2023
The smallest integer that has exactly 2^n exponential semiproper divisors is A061742(n). - Bernard Schott, Feb 20 2023

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.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to divisors of numbers.

Cf. A000188, A001694, A007947, A008833, A034444, A057521, A061742, A062503, A082020, A323309, A323310.

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

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = min(f[k,2], 2); f[k,2] = 1); factorback(f); \\ Michel Marcus, Jan 11 2019

a(n) = A034444(n/A007947(n)).
Multiplicative with a(p^e) = 1 for e = 1 and 2 otherwise.
Asymptotic mean: Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k) = 15/Pi^2 = 1.5198177546... (A082020). - Amiram Eldar, Nov 08 2020
a(n) = Sum_{d^2|n} mu(d)^2. - Wesley Ivan Hurt, Feb 13 2022
Dirichlet g.f.: zeta(s) * zeta(2*s) / zeta(4*s). - Werner Schulte, Dec 29 2022
a(n) = A034444(A000188(n)) = A034444(A008833(n)) (the number of unitary divisors of the largest square dividing n). - Amiram Eldar, Sep 03 2023
a(n) = A034444(A057521(n)) (the number of unitary divisors of the powerful part of n). - Amiram Eldar, Oct 03 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.

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]

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A323308 The number of exponential semiproper 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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A323310 List of e-unitary perfect numbers that are not e-semiproper perfect numbers.

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