A069088 -id:A069088 - cp's OEIS frontend

A069145 Numbers k such that A069088(k) divides k.

1, 4, 24, 81, 90, 112, 162, 324, 360, 384, 648, 810, 864, 960, 1024, 1944, 2592, 2688, 3072, 3240, 3456, 3969, 4320, 5040, 5488, 5760, 7128, 7168, 7776, 8640, 9000, 9072, 9504, 10368, 11025, 11200, 15360, 15876, 17280, 18144, 21168, 21504, 23296

Offset: 1

Benoit Cloitre, Apr 08 2002

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A069088.

f[p_, e_] := If[OddQ[e], (p + 1)*(e + 1)/2, (p + 1)*e/2 + 1]; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[25000], Divisible[#, s[#]] &] (* Amiram Eldar, Sep 03 2020 *)

for(n=1,35000,if(n%sumdiv(n,d,core(d))==0,print1(n,",")))

A074786 Numbers k such that phi(k) = Sum_{d|k} core(d) where core(x) is the squarefree part of x (A007913).

Original entry on oeis.org

1, 99, 1080, 1836, 4743, 5670, 7644, 8307, 14384, 14784, 15225, 15824, 16065, 20300, 21584, 25704, 29760, 34544, 46816, 71568, 94240, 128412, 169290, 264160, 266266, 312480, 331731, 364941, 404550, 445050, 454575, 458052, 479655, 497781

Offset: 1

Benoit Cloitre, Sep 07 2002

nonn

Numbers k such that A000010(k) = A069088(k). - Amiram Eldar, Apr 28 2025

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..500 from Donovan Johnson)

Cf. A000010, A007913, A069088.

f[p_, e_] := If[OddQ[e], (p+1)*(e+1)/2, (p+1)*e/2 + 1] / ((p-1)*p^(e-1)); q[1] = True; q[n_] := Times @@ f @@@ FactorInteger[n] == 1; Select[Range[500000], q] (* Amiram Eldar, Apr 28 2025 *)

isok(n) = eulerphi(n) == sumdiv(n, d, core(d)); \\ Michel Marcus, Aug 09 2013

isok(k) = {my(f = factor(k)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(e % 2, (p+1)*(e+1)/2, (p+1)*e/2 + 1) / ((p-1)*p^(e-1))) == 1;} \\ Amiram Eldar, Apr 28 2025

More terms from Michel Marcus, Aug 09 2013

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

cp's OEIS Frontend

A069145 Numbers k such that A069088(k) divides k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A074786 Numbers k such that phi(k) = Sum_{d|k} core(d) where core(x) is the squarefree part of x (A007913).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula