A190402 -id:A190402 - cp's OEIS frontend

A190403 Number n for which sigma(n)=sigma(n'), where sigma is the sum of divisors and n' the arithmetic derivative of n.

4, 27, 60, 84, 132, 140, 204, 220, 228, 260, 270, 340, 372, 378, 444, 492, 564, 572, 580, 620, 644, 702, 708, 740, 804, 812, 820, 836, 860, 884, 918, 945, 1026, 1068, 1180, 1242, 1276, 1284, 1292, 1308, 1316, 1364, 1420, 1460, 1484, 1485, 1508, 1564, 1566

Offset: 1

Paolo P. Lava, May 10 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000203, A003415, A189057, A190402.

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 sigma(n)=sigma(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 < 2000, n++, If[DivisorSigma[1, n] == DivisorSigma[1, d[n]], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Apr 22 2015 *)

from sympy import factorint, totient
A190402 = [n for n in range(2,10**3) if totient(int(sum([n*e/p for p,e in factorint(n).items()]))) == totient(n)] # Chai Wah Wu, Aug 21 2014

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);

A260961 Numbers n such that phi(n') = phi(n)', where phi(n) is the Euler totient function of n and n' is the arithmetic derivative of n.

Original entry on oeis.org

1, 3, 8, 28, 32, 36, 40, 50, 56, 80, 128, 176, 184, 228, 324, 368, 448, 472, 504, 560, 576, 664, 704, 1328, 1336, 1512, 1620, 1686, 1816, 2048, 2544, 2580, 2864, 3008, 3064, 3540, 3776, 3832, 3888, 4024, 5184, 5744, 6048, 6904, 7096, 7128, 8192, 9216, 10264, 10456

Offset: 1

Paolo P. Lava, Aug 06 2015

			Arithmetic derivative of 56 is 92 and phi(92) = 44, while phi(56) = 24 and the arithmetic derivative of 24 is 44.

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from Paolo P. Lava)

Cf. A000010, A003415, A190402.

with(numtheory): P:=proc(q) local a,b,c,n,p; for n from 1 to q do
a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); b:=phi(n);
c:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]);
if phi(a)=c then print(n); fi; od; end: P(10^9);

f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]; Select[Range@ 12000, EulerPhi@ f@ # == f@ EulerPhi@ # &] (* Michael De Vlieger, Aug 07 2015, after Michael Somos at A003415 *)

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)

A260962 Numbers k such that phi(k) = phi'(k'), where phi(k) is the Euler totient function of k and k' is the arithmetic derivative of k.

Original entry on oeis.org

8, 26, 122, 351, 31195, 47201, 51243, 118265, 300985, 472491, 672147, 673863, 850969, 931383, 1440625, 3000927, 3669213, 3740755, 4688645, 4822143, 4864175, 11224565, 13897079, 13949343, 16362857, 16744355, 18844265, 19536205, 35580099, 38656975, 42056215, 46294105

Offset: 1

Paolo P. Lava, Aug 06 2015

			Arithmetic derivative of 26 is 15, phi(15) = 8 and 8' = 12 that is equal to phi(26).

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000010, A003415, A190402, A260961.

with(numtheory):P:=proc(q) local a,b,n,p;
for n from 1 to q do a:=phi(n*add(op(2,p)/op(1,p),p=ifactors(n)[2]));
b:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]);
if phi(n)=b then print(n); fi; od; end: P(10^9);

f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]; Select[Range@ 100000, f@ EulerPhi@ f@ # == EulerPhi@ # &] (* Michael De Vlieger, Aug 07 2015, after Michael Somos at A003415 *)

A352331 Numbers k for which phi(k) = phi(k''), where phi is the Euler totient function (A000010) and k'' the second arithmetic derivative of k (A068346).

Original entry on oeis.org

4, 27, 104, 260, 296, 405, 525, 740, 910, 945, 1460, 1806, 1818, 2504, 3125, 3140, 3176, 3656, 3860, 4563, 5540, 6056, 6930, 7016, 8420, 8636, 9224, 10820, 12573, 13256, 14024, 15140, 15464, 15944, 16136, 19940, 20456, 21690, 21860, 22856, 23336, 24020, 24260

Offset: 1

Marius A. Burtea, Apr 09 2022

nonn

If m is a term in A051674, then m'' = m, phi(m'') = phi(m) so the sequence is infinite.
If p > 3 is at the intersection of A023208 and A005383 then m = 8*p is a term. Indeed, m'' = (8*p)'' = (12*p + 8)' = (4*(3*p + 2))' = 12*(p + 1) and phi(m'') = phi(12*(p + 1)) = phi(24*(p + 1)/2) = 8*(p - 1)/2 = 4*(p - 1) and phi(m) = phi(8*p) = 4*(p - 1).
If p > 5 is at the intersection of A023221 and A005383 then m = 20*p is a term. Indeed, m'' = (20*p)'' = (24*p + 20)' = (4*(6*p + 5))' = 4*(6*p + 6)  = 24*(p + 1) and phi(m'') = phi(24*(p + 1)) = phi(48*(p + 1)/2) = 16*(p - 1)/2 = 8*(p - 1) and phi(m) = phi(20*p) = 8*(p - 1).

			phi(4'') = phi(4) because 4'' = 4, so 4 is a term.
phi (27'') = phi(27) because 27'' = 27, so 27 is a term.
phi(104'') = phi(164') = phi(168) = phi (8*3*7) = 4*2*6 = 48 and phi(104) = phi(8*13) = 4*12 = 48, so 104 is a term.

Cf. A000010, A005383, A051674, A068346, A023208, A023221, A190402,

f:=func; [n:n in [2..24300]| not IsPrime(n) and EulerPhi(n) eq EulerPhi(Floor(f(Floor(f(n))))) ];

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[25000], EulerPhi[#] == EulerPhi[d[d[#]]] &] (* Amiram Eldar, Apr 10 2022 *)

A359330 Composite k for which phi(k) + phi(k') = k, where k' is the arithmetic derivative of k (A003415).

Original entry on oeis.org

4, 6, 8, 10, 12, 18, 22, 28, 34, 58, 60, 72, 82, 84, 88, 108, 112, 118, 124, 132, 140, 142, 202, 204, 214, 216, 220, 228, 260, 274, 298, 324, 340, 358, 372, 382, 394, 444, 454, 478, 492, 508, 538, 562, 564, 572, 580, 620, 622, 644, 694, 708, 740, 804, 812, 820

Offset: 1

Marius A. Burtea, Jan 28 2023

nonn

Composite numbers k for which phi(k') = cototient(k) (A051953).
The sequence refers only to composite numbers because for any prime number p we obtain phi(p) + phi(p') = p - 1 + phi(1) = p.
If p = 2^k - 1 is a Mersenne prime (A000668), then m = 4*p is a term. Indeed, m' = 4*(p + 1) = 4*2^k = 2^(k + 2) and phi(m) + phi(m') = phi(4*p) + phi(2^(k + 2)) = 2*(p-1) + 2^(k+1) = 2*(p - 1) + 2*(p + 1) = 4*p = m, so m is a term.
If p, q and p*q + p + q are prime numbers then m = 4*p*q is a term. Indeed, m'= 4*(p*q + p + q) and phi(m) + phi(m') = phi(4*p*q) + phi(4*(p*q + p + q)) = 2*(p - 1)*(q - 1) + 2*(p*q + p + q - 1) = 4*p*q.
If p is in A023221 then m = 20*p is a term. Indeed, m' = 24*p + 20 = 4*(6*p + 5) and phi(m) + phi(m') = phi(20*p) + phi(4*(6*p + 5)) = 8*(p-1) + 2*(6*p + 4) = 20*p = m, so m is a term.

			If m = 4 then m' = 4 and phi(m) + phi(m') = phi(4) + phi(4) = 2 + 2 = 4, so 4 is a term.
If m = 8 then m' = 12 and phi(m) + phi(m') = phi(8) + phi(12) = 4 + 4 = 8, so 8 is a term.
14 is not a term because phi(14) + phi(14') = 6 + phi(9) = 6 + 6 = 12 <> 14.

Cf. A000010, A001359, A002808, A003415, A051953, A066938, A023221, A190402.

f:=func;  [n:n in [2..850]|not IsPrime(n) and n eq EulerPhi(Floor(f(n))) + EulerPhi(n)];

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
q:= n-> not isprime(n) and (p-> p(n)+p(d(n))=n)(numtheory[phi]):
select(q, [$4..1000])[];  # Alois P. Heinz, Jan 29 2023

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[1000], CompositeQ[#] && EulerPhi[#] + EulerPhi[d[#]] == # &] (* Amiram Eldar, Jan 29 2023 *)

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A190403 Number n for which sigma(n)=sigma(n'), where sigma is the sum of divisors 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.

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A260961 Numbers n such that phi(n') = phi(n)', where phi(n) is the Euler totient function of n and n' is the arithmetic derivative of n.

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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.

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A260962 Numbers k such that phi(k) = phi'(k'), where phi(k) is the Euler totient function of k and k' is the arithmetic derivative of k.

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A352331 Numbers k for which phi(k) = phi(k''), where phi is the Euler totient function (A000010) and k'' the second arithmetic derivative of k (A068346).

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A359330 Composite k for which phi(k) + phi(k') = k, where k' is the arithmetic derivative of k (A003415).

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