A306633 -id:A306633 - cp's OEIS frontend

A362984 Decimal expansion of the asymptotic mean of the abundancy index of the powerful numbers (A001694).

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

Offset: 1

Amiram Eldar, May 12 2023

The abundancy index of a positive integer k is A000203(k)/k = A017665(k)/A017666(k).

The asymptotic mean of the abundancy index over all the positive integers is lim_{m->oo} (1/m) * Sum_{k=1..m} A000203(k)/k = Pi^2/6 = zeta(2) = 1.644934... (A013661).

			2.14968690301526765128219042105109416145987653275100999873...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (6).

Cf. A000203, A001694, A017665, A017666, A180114.

Similar constants (the asymptotic mean of the abundancy index of other sequences): A013661 (all positive integers), A082020 (cubefree), A111003 (odd), A157292 (5-free), A157294 (7-free), A157296 (9-free), A240976 (squares), A245058 (even), A306633 (squarefree), A362985 (cubefull).

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{2, -3, 4, -6, 7, -7, 7, -6, 5, -3, 2, -1}, {0, 0, 0, 4, 5, 6, 0, -12, -9, -5, 0, 22}, m]; RealDigits[(2^4 + 2^2 + 2^(3/2) - 1)/(2^4 - 2)*(3^4 + 3^2 + 3^(3/2) - 1)/(3^4 - 3) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - 1/2^(n/2) - 1/3^(n/2))/n, {n, 4, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

prodeulerrat((p^8 + p^4 + p^3 - 1)/(p^8 - p^2), 1/2)

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A180114(k)/A001694(k).

Equals Product_{p prime} (p^4 + p^2 + p^(3/2) - 1)/(p^4 - p) = Product_{p prime} (1 + (p^2 + p^(3/2) + p - 1)/(p^4 - p)) (Jakimczuk and Lalín, 2022).

A362985 Decimal expansion of the asymptotic mean of the abundancy index of the cubefull numbers (A036966).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 12 2023

			2.48217919642235952546167643674687698536368940971930468354...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (6).

Cf. A000203, A036966, A017665, A017666, A362986.

Similar constants (the asymptotic mean of the abundancy index of other sequences): A013661 (all positive integers), A082020 (cubefree), A111003 (odd), A157292 (5-free), A157294 (7-free), A157296 (9-free), A245058 (even), A240976 (squares), A306633 (squarefree), A362984 (powerful).

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{2, -1, -2, 3, -2, -1, 3, -2, -2, 3, -1, -2, 3, -1, -1, 1}, {0, 0, 0, -4, 0, 6, 7, 4, 9, 0, -11, -22, -26, -21, -15, 20}, m]; RealDigits[((2^5 + 2^(10/3) + 2^3 + 2^(8/3) - 1)/(2^(10/3)*(2^(5/3) + 2^(1/3) + 1)))*((3^5 + 3^(10/3) + 3^3 + 3^(8/3) - 1)/(3^(10/3)*(3^(5/3) + 3^(1/3) + 1))) * Zeta[4/3] * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/3] - 1/2^(n/3) - 1/3^(n/3))/n, {n, 4, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

zeta(4/3) * prodeulerrat((p^15 + p^10 + p^9 + p^8 - 1)/(p^10 * (p^5 + p + 1)), 1/3)

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A362986(k)/A036966(k).

Equals zeta(4/3) * Product_{p prime} ((p^5 + p^(10/3) + p^3 + p^(8/3) - 1)/(p^(10/3) * (p^(5/3) + p^(1/3) + 1))).

A365209 The sum of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 26, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 50, 78, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A000005 at n = 25.

The number of these divisors is A365208(n).

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

Cf. A000203, A003586, A034448, A065330, A065331, A365208.

Cf. A002117, A013661, A306633.

f[p_, e_] := If[p <= 3, (p^(e+1)-1)/(p-1), 1 + p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] <= 3, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1), 1 + f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) for p = 2 or 3, and a(p^e) = 1 + p^e for a prime p >= 5.
a(n) <= A000203(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) >= A034448(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) = A000203(A065331(n)) * A034448(A065330(n)).
Dirichlet g.f.: (4^s/(4^s-2)) * (9^s/(9^s-3)) * zeta(s)*zeta(s-1)/zeta(2*s-1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (54/91) * zeta(2)/zeta(3) = (54/91) * A306633 = 0.812037... .

A316148 Number of non-congruent solutions of x^2+y^2 == z^2+w^2 (mod n).

Original entry on oeis.org

1, 8, 33, 96, 145, 264, 385, 896, 945, 1160, 1441, 3168, 2353, 3080, 4785, 7680, 5185, 7560, 7201, 13920, 12705, 11528, 12673, 29568, 18625, 18824, 26001, 36960, 25201, 38280, 30721, 63488, 47553, 41480, 55825, 90720, 51985, 57608, 77649, 129920, 70561, 101640, 81313

Offset: 1

R. J. Mathar, Jun 25 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index to sequences related to sums of squares.

Cf. A096018, A240547, A306633.

A316148 := proc(n)
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := a*p^(2*e+1)*(p^e-1) ;
        else
            a := a*p^(2*e-1)*(p^(e+1)+p^e-1) ;
        end if;
    end do:
    a ;
end proc:
seq(A316148(n),n=1..100) ;

f[2, e_] :=  2^(2*e+1)*(2^e-1); f[p_, e_] := p^(3*e)+p^(3*e-1)-p^(2*e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p == 2, 2^(2*e+1)*(2^e-1), p^(3*e)+p^(3*e-1)-p^(2*e-1)));} \\ Amiram Eldar, Dec 18 2023

Multiplicative with a(2^e) = 2^(2e+1)*(2^e-1), a(p^e) = p^(3e)+p^(3e-1)-p^(2e-1) for odd primes p.

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(2)/zeta(3) = 1.368432... (A306633). - Amiram Eldar, Dec 18 2023

A351435 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j + 1)).

Original entry on oeis.org

1, 9, 16, 27, 36, 144, 64, 81, 64, 324, 144, 432, 196, 576, 576, 243, 324, 576, 400, 972, 1024, 1296, 576, 1296, 216, 1764, 256, 1728, 900, 5184, 1024, 729, 2304, 2916, 2304, 1728, 1444, 3600, 3136, 2916, 1764, 9216, 1936, 3888, 2304, 5184, 2304, 3888, 512, 1944, 5184, 5292, 2916

Offset: 1

Ilya Gutkovskiy, Feb 11 2022

Cf. A003557, A003958, A003959, A048250, A064549, A065478, A306633, A326297, A327564, A351434.

a:= n-> mul((i[1]+1)^(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..53);  # Alois P. Heinz, Feb 11 2022

f[p_, e_] := (p + 1)^(e + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Table[a[n], {n, 1, 53}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]++; f[k,2]++); factorback(f); \\ Michel Marcus, Feb 11 2022

a(n) = A003959(n) * A048250(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = 1/(3 * Product_{p prime} (1 - p/(p^3-1))) = 1 /(3 * A065478) = 0.5787439255... . - Amiram Eldar, Nov 19 2022
Sum_{n>=1} 1/a(n) = zeta(2)/zeta(3) (A306633). - Amiram Eldar, Dec 15 2023

A308045 Numbers k such that usigma(k) = round(zeta(2)/zeta(3)*k), where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

1, 2, 3, 4, 35, 44, 111, 123, 1105, 1900, 2920, 12452, 17889, 34200, 65067, 716148, 14134055, 179040201, 221709100, 221743300, 221766100, 221788900, 1120968741, 1272582040, 1441454511, 7339101375

Offset: 1

Amiram Eldar, May 10 2019

The unitary version of A072355.
zeta(2)/zeta(3) is the asymptotic mean of the unitary abundancy index usigma(k)/k (A306633).
a(27) > 10^10.

			35 is in the sequence since usigma(35) = 48, and (zeta(2)/zeta(3)) * 35 = 47.895... has a round value of 48.

Cf. A002117, A013661, A034448, A072355, A074920, A306633.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); meanAb = Zeta[2]/Zeta[3]; Select[Range[10^6], usigma[#] == Round[meanAb*#] &]

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A362984 Decimal expansion of the asymptotic mean of the abundancy index of the powerful numbers (A001694).

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A362985 Decimal expansion of the asymptotic mean of the abundancy index of the cubefull numbers (A036966).

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A365209 The sum of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

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A316148 Number of non-congruent solutions of x^2+y^2 == z^2+w^2 (mod n).

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A351435 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j + 1)).

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A308045 Numbers k such that usigma(k) = round(zeta(2)/zeta(3)*k), where usigma(k) is the sum of unitary divisors of k (A034448).

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