A190403 -id:A190403 - cp's OEIS frontend

A190402 Number n for which phi(n) = phi(n'), where phi is the Euler totient function and n' the arithmetic derivative of n.

2, 4, 8, 14, 20, 27, 45, 52, 75, 148, 195, 244, 292, 364, 628, 729, 772, 1108, 1196, 1215, 1252, 1406, 1552, 1588, 1684, 1701, 1828, 2164, 2452, 2644, 2692, 2924, 2932, 3028, 3125, 3508, 3825, 3982, 3988, 4372, 4462, 4612, 4804, 4852, 4948, 5284, 5524

Offset: 1

Paolo P. Lava, May 10 2011

nonn

Nathaniel Johnston, Table of n, a(n) for n = 1..2500

Cf. A000010, A003415, A166374, A190403.

with(numtheory);
P:=proc(i)
local f,n,p,pfs;
for n from 1 to i do
    pfs:=ifactors(n)[2];
    f:=n*add(op(2,p)/op(1,p),p=pfs);
    if phi(n)=phi(f) then print(n); fi;
od;
end:
P(1000);

d[0] = d[1] = 0; d[n_] := n*Total[f = FactorInteger[n]; f[[All, 2]]/f[[All, 1]] ]; Reap[For[n = 1, n < 6000, n++, If[EulerPhi[n] == EulerPhi[d[n]], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Apr 22 2015 *)

A209870 Numbers n for which tau(n) = tau(n'), where tau is the number of divisors of n and n' the arithmetic derivative of n.

Original entry on oeis.org

4, 15, 21, 26, 27, 28, 33, 38, 48, 50, 57, 62, 69, 72, 74, 80, 85, 93, 99, 106, 129, 133, 134, 145, 156, 166, 176, 177, 178, 200, 205, 207, 213, 217, 218, 226, 237, 249, 253, 254, 262, 265, 276, 278, 308, 309, 314, 348, 362, 364, 368, 380, 393, 398, 410, 417

Offset: 1

Paolo P. Lava, Mar 15 2012

			Divisors of 26 are four: 1, 2, 13 and 26. Arithmetic derivative of 26 is 15 and its divisors are four: 1, 3, 5 and 15.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000005, A003415, A190402, A190403

with(numtheory);
A209870:=proc(n)
local a,i,p,pfs;
for i from 1 to n do
  pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs);
  if tau(a)=tau(i) then print(i); fi;
od; end:
A209870(1000);

A189057 Numbers n for which phi(n)=sigma(n'), where phi is the Euler totient function, sigma is the sum of divisors and n' the arithmetic derivative of n.

Original entry on oeis.org

2, 57, 175, 357, 381, 543, 777, 903, 2379, 3027, 6807, 25823, 47047, 74333, 82621, 136213, 153425, 163471, 194873, 230547, 257799, 259555, 265111, 269545, 285439, 289009, 302403, 305305, 311395, 354365, 416005, 484169, 569245, 718333, 755885, 781501, 1012505

Offset: 1

Paolo P. Lava, May 17 2011

nonn

			phi(57)=36. 57'=22 and sigma(22)=36
phi(1012505)=725760. 1012505'=310156 and sigma(310156)=725760

Donovan Johnson, Table of n, a(n) for n = 1..300

Cf. A000010, A000203, A003415, A166374, A190402, A190403.

with(numtheory);
P:=proc(i)
local f, n, p, pfs;
for n from 1 by 1 to i do
    pfs:=ifactors(n)[2];
    f:=n*add(op(2, p)/op(1, p), p=pfs);
    if phi(n)=sigma(f) then print(n); fi;
od;
end:
P(1000000)

cp's OEIS Frontend

A190402 Number n for which phi(n) = phi(n'), where phi is the Euler totient function and n' the arithmetic derivative of n.

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A209870 Numbers n for which tau(n) = tau(n'), where tau is the number of divisors of n and n' the arithmetic derivative of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A189057 Numbers n for which phi(n)=sigma(n'), where phi is the Euler totient function, sigma is the sum of divisors and n' the arithmetic derivative of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple