A068976 -id:A068976 - cp's OEIS frontend

A069144 Numbers k such that A068976(k) divides k.

1, 2, 12, 40, 120, 208, 280, 396, 440, 520, 624, 672, 680, 760, 920, 1040, 1160, 1240, 1456, 1480, 1640, 1720, 1880, 2120, 2288, 2360, 2440, 2680, 2840, 2920, 3120, 3160, 3320, 3360, 3392, 3536, 3560, 3880, 3952, 3960, 4040, 4120, 4280, 4360, 4368, 4520

Offset: 1

Benoit Cloitre, Apr 08 2002

For n > 4 A068976(a(n)) >= 40 and almost all terms k in the sequence are such that A068976(k) = 40.

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

Cf. A068976.

f[p_, e_] := If[OddQ[e], 2*(p^(e + 1) - 1)/(p^2 - 1), (p^(e + 2) + p^e - 2)/(p^2 - 1)]; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[5000], Divisible[#, s[#]] &] (* Amiram Eldar, Sep 03 2020 *)

for(n=1,10000,if(n%sumdiv(n,d,d/core(d))==0,print1(n,",")))

A069265 Numbers k such that Sum_{d|k} d/core(d) > k, where core(d) is the squarefree part of d.

Original entry on oeis.org

4, 8, 9, 16, 18, 25, 32, 36, 48, 49, 50, 64, 72, 81, 98, 100, 108, 121, 128, 144, 162, 169, 192, 196, 200, 225, 242, 256, 288, 289, 300, 324, 338, 361, 392, 400, 432, 441, 450, 484, 512, 529, 576, 578, 588, 625, 648, 676, 722, 729, 768, 784, 800, 841, 882, 900

Offset: 1

Benoit Cloitre, Apr 14 2002

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

Cf. A068976, A007913.

f[p_, e_] := If[OddQ[e], 2*(p^(e + 1) - 1)/(p^2 - 1), (p^(e + 2) + p^e - 2)/(p^2 - 1)]; Select[Range[2, 1000], Times @@ (f @@@ FactorInteger[#]) > # &] (* Amiram Eldar, Sep 03 2020 *)

k such that A068976(k) > k.

A069266 Numbers k such that Sum_{d|k} d/core(d) > 2*k, where core(d) is the squarefree part of d.

Original entry on oeis.org

576, 1296, 2304, 3600, 5184, 7056, 8100, 9216, 11664, 14400, 17424, 20736, 24336, 28224, 32400, 36864, 44100, 46656, 57600, 63504, 69696, 72900, 82944, 90000, 97344, 104976, 108900, 112896, 129600, 147456, 152100, 156816, 166464, 176400, 186624, 202500, 207936

Offset: 1

Benoit Cloitre, Apr 14 2002

All terms <= 10^17 are squares. Are there any nonsquare terms? - David A. Corneth, Sep 05 2020

All the terms are squares. Proof: Let f(n) = A068976(n)/n. f(n) is multiplicative with f(p^e) = (p^2 + 1 - 2/p^e)/(p^2-1) if e is even and 2*(p - 1/p^e)/(p^2-1) if e is odd. Both are strictly increasing with e, the limits as e -> oo are f_even(p) = (p^2+1)/(p^2-1) and f_odd(p) = 2*p/(p^2-1), respectively, and f_odd(p) < f_even(p) for all primes p. The upper bound on f(n) is being attained at even exponents: f(n) < lim_{e->oo} Product_{p prime} (p^2 + 1 - 2/p^e)/(p^2-1) = Product_{p prime} f_even(p) = 5/2. If k is not a square, then there is at least one prime q|k with an odd exponent. Replacing the factor f_even(q) with f_odd(q) in the infinite product, we get f(k) < (5/2) * f_odd(q)/f_even(q) = 5*q/(q^2+1) <= 2. Therefore, A068976(k) = f(k) * k < 2*k and k is not a term. - Amiram Eldar, Feb 11 2024

			3600 is in the sequence as its divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, ..., 400, 450, 600, 720, 900, 1200, 1800, 3600 and the respective values d/core(d) are 1, 1, 1, 4, 1, 1, 4, 9, ... , 900, 400, 900, 3600. The sum of this latter list is 7722 which is greater than 2*3600. - _David A. Corneth_, Sep 05 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 5000 terms from Amiram Eldar)

Cf. A068976, A007913.

f[p_, e_] := If[OddQ[e], 2*(p^(e + 1) - 1)/(p^2 - 1), (p^(e + 2) + p^e - 2)/(p^2 - 1)]; Select[Range[2, 150000], Times @@ (f @@@ FactorInteger[#]) > 2*# &] (* Amiram Eldar, Sep 05 2020 *)

is(n) = sumdiv(n, d, d/core(d)) > n << 1 \\ David A. Corneth, Sep 05 2020

Numbers k such that A068976(k) > 2*k.

More terms from Amiram Eldar, Sep 05 2020

A198286 a(n) = Sum_{d|n} (A053143(d) or smallest square divisible by d).

Original entry on oeis.org

1, 5, 10, 9, 26, 50, 50, 25, 19, 130, 122, 90, 170, 250, 260, 41, 290, 95, 362, 234, 500, 610, 530, 250, 51, 850, 100, 450, 842, 1300, 962, 105, 1220, 1450, 1300, 171, 1370, 1810, 1700, 650, 1682, 2500, 1850, 1098, 494, 2650, 2210, 410, 99, 255, 2900, 1530, 2810

Offset: 1

Antonio Roldán, Oct 23 2011

Multiplicative function with a(p^e) = 1+2*(p^(e+2)-p^2)/(p^2-1) if e is even else a(p^e)=(1+p^2)((p^(e+1)-1)/(p^2-1)). Examples: a(9)=a(3^2)=1+2*((81-9)/(9-1))=1+2*9=19; a(8)=a(2^3)=(1+4)((16-1)/(4-1))=5*5=25.

Another definition of a(n): Sum_{d|n} (d*core(d)), where core(d) is the squarefree part of d (A007913), i.e., inverse Mobius transform of A053143.

			a(18) = 95 because 18=2*3^2, so a(18) = (1+4)(1+9+9) = 5*19 = 95.
a(20) = 234 because 20=2^2*5, so a(20) = (1+4+4)(1+25) = 9*26 = 234.

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

Similar to A068976 (sum of square part of d) and A069088 (sum of squarefree part of d).

Cf. A007913, A053143.

ssq[n_] := For[k=1, True, k++, If[ Divisible[s = k^2, n], Return[s]]]; a[n_] := Sum[ ssq[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 53}] (* Jean-François Alcover, Sep 03 2012 *)
f[p_, e_] := If[OddQ[e], (1+p^2)((p^(e+1)-1)/(p^2-1)), 1+2*(p^(e+2)-p^2)/(p^2-1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 53] (* Amiram Eldar, Sep 05 2020 *)

a(n)=sumdiv(n,d,d*core(d)) \\ Charles R Greathouse IV, Oct 30 2011

Dirichlet g.f.: zeta(s)*zeta(s-2)*zeta(2s-2)/zeta(2s-4). - R. J. Mathar, Mar 12 2012

Sum_{k=1..n} a(k) ~ Pi^2 * Zeta(3) * n^3 / 45. - Vaclav Kotesovec, Feb 02 2019

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A069144 Numbers k such that A068976(k) divides k.

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A069265 Numbers k such that Sum_{d|k} d/core(d) > k, where core(d) is the squarefree part of d.

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A069266 Numbers k such that Sum_{d|k} d/core(d) > 2*k, where core(d) is the squarefree part of d.

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A198286 a(n) = Sum_{d|n} (A053143(d) or smallest square divisible by d).

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