A166374 -id:A166374 - 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 *)

A342413 a(n) = gcd(phi(n), A003415(n)), where A003415(n) is the arithmetic derivative of n, and phi is Euler totient function.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 6, 1, 1, 4, 1, 3, 8, 8, 1, 3, 1, 8, 2, 1, 1, 4, 10, 3, 9, 4, 1, 1, 1, 16, 2, 1, 12, 12, 1, 3, 8, 4, 1, 1, 1, 4, 3, 1, 1, 16, 14, 5, 4, 8, 1, 9, 8, 4, 2, 1, 1, 4, 1, 3, 3, 32, 6, 1, 1, 8, 2, 1, 1, 12, 1, 3, 5, 4, 6, 1, 1, 16, 54, 1, 1, 4, 2, 3, 8, 20, 1, 3, 4, 4, 2, 1, 24, 16, 1, 7, 15, 20

Offset: 1

Antti Karttunen, Mar 11 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12500
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000010, A003415, A003557, A166374, A342009, A342414, A342415, A342416.

Cf. also A009195, A085731.

Array[GCD[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]], EulerPhi[#]] &@ Abs[#] &, 100] (* Michael De Vlieger, Mar 11 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342413(n) = gcd(eulerphi(n), A003415(n));

a(n) = gcd(A000010(n), A003415(n)).
a(n) = A003415(n) / A342414(n) = A000010(n) / A342415(n).
a(n) = A003557(n) * A342416(n).

A345051 Numbers k such that A345048(k) is equal to A345049(k).

Original entry on oeis.org

2, 6, 9, 15, 28, 496, 625, 1225, 3993, 8128, 117649, 218491, 857375, 3788435, 4259571, 33550336, 69302975, 136410197, 200533921, 313742585, 603439225, 1516358753, 2563893625, 3326174929, 5655792025, 8589869056, 10214476341

Offset: 1

Antti Karttunen, Jun 06 2021

Numbers k for which A342001(n) * A051709(n) = A173557(n) * A345001(n).

Conjecture: Sequence is a disjoint union of A000396 and A166374, i.e., there are no terms of any other kind.

Index entries for sequences where odd perfect numbers must occur, if they exist at all.

Cf. A051709, A173557, A342001, A345001, A345048, A345049.
Positions of zeros in A345050.
Cf. A000396, A166374 (subsequences).
Cf. also A345003.

PARI
```
isA345051(n) = (0==A345050(n));
```

a(21)-a(27) from Amiram Eldar, Dec 08 2023

A342009 Numbers k such that the arithmetic derivative of k is a multiple of phi(k).

Original entry on oeis.org

1, 2, 4, 8, 9, 12, 15, 16, 20, 32, 36, 48, 64, 81, 108, 112, 128, 144, 180, 189, 192, 196, 225, 256, 320, 324, 400, 432, 500, 512, 528, 576, 625, 729, 768, 972, 1024, 1225, 1296, 1300, 1360, 1452, 1728, 2048, 2160, 2304, 2700, 2816, 2916, 3024, 3072, 3375, 3564, 3840, 3888, 3993, 4096, 4800, 5120, 5184, 5292, 5616, 6000

Offset: 1

Antti Karttunen, Mar 11 2021

nonn

Numbers k for which A000010(k) is a divisor of A003415(k), or equally, k for which A173557(k) is a divisor of A342001(k).

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A000010, A003415, A166374 (subsequence), A173557, A342001, A342008.

Positions of ones in A342415.

Select[Range[6000], Mod[If[Abs[#] < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]], EulerPhi[#]] == 0 &] (* Michael De Vlieger, Mar 11 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA342009(n) = !(A003415(n)%eulerphi(n));

A301939 Integers whose arithmetic derivative is equal to their Dedekind function.

Original entry on oeis.org

8, 81, 108, 2500, 2700, 3375, 5292, 13068, 15625, 18252, 31212, 38988, 57132, 67228, 90828, 94500, 103788, 147852, 181548, 199692, 231525, 238572, 303372, 375948, 401868, 484812, 544428, 575532, 674028, 713097, 744012, 855468, 1016172, 1058841, 1101708, 1145772

Offset: 1

Paolo P. Lava, Mar 29 2018

If n = Product_{k=1..j} p_k ^ i_k with each p_k prime, then psi(n) = n * Product_{k=1..j} (p_k + 1)/p_k and n' = n*Sum_{k=1..j} i_k/p_k.
Thus every number of the form p^(p+1), where p is prime, is in the sequence.
The sequence also contains every number of the form 108*p^2 where p is a prime > 3, or 108*p^3*(p+2) where p > 3 is in A001359. - Robert Israel, Mar 29 2018

			5292 = 2^2 * 3^3 * 7^2.
n' = 5292*(2/2 + 3/3 + 2/7) = 12096,
psi(n) = 5292*(1 + 1/2)*(1 + 1/3)*(1 + 1/7) = 12096.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A001359, A001615, A003415, A166374, A342458. A345005 (gives the odd terms).

Subsequence of A345003.

with(numtheory): P:=proc(n) local a,p; a:=ifactors(n)[2];
if add(op(2,p)/op(1,p),p=a)=mul(1+1/op(1,p),p=a) then n; fi; end:
seq(P(i),i=1..10^6);

selQ[n_] := Module[{f = FactorInteger[n], p, e}, Product[{p, e} = pe; p^e + p^(e-1), {pe, f}] == Sum[{p, e} = pe; (n/p)e, {pe, f}]];
Select[Range[10^6], selQ] (* Jean-François Alcover, Oct 16 2020 *)

dpsi(f) = prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
ader(n, f) = sum(i=1, #f~, n/f[i, 1]*f[i, 2]);
isok(n) = my(f=factor(n)); dpsi(f) == ader(n, f); \\ Michel Marcus, Mar 29 2018

Solutions of the equation n' = psi(n).

A342008 Numbers k such that Euler totient phi(k) is a multiple of the arithmetic derivative of k.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 209, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281

Offset: 1

Antti Karttunen, Mar 12 2021

nonn

Numbers k for which A000010(k) is a multiple of A003415(k), or equally, k for which A173557(k) is a multiple of A342001(k).

Cf. A000010, A003415, A173557, A342001, A342009.
Subsequences: A000040, A166374, A342418 (composite terms).
Positions of ones in A342414.

Select[Range[2, 281], Mod[EulerPhi[#], If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]] ] &@ Abs[#]] == 0 &] (* Michael De Vlieger, Mar 12 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA342008(n) = ((n>1)&&!(eulerphi(n)%A003415(n)));
for(n=2,2^8,if(isA342008(n),print1(n,", ")));

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)

A217715 Numbers equal to the Euler totient function of their arithmetic derivative: k = phi(k').

Original entry on oeis.org

16, 30, 32, 48, 54, 64, 80, 120, 176, 288, 368, 432, 464, 656, 848, 858, 864, 1328, 1424, 1722, 1808, 1944, 2096, 2768, 2864, 2916, 3056, 3728, 3824, 4016, 4496, 4688, 5744, 5832, 6704, 6896, 7088, 7856, 8144, 9488, 10256, 10448, 10544, 10928, 11504, 11888

Offset: 1

Paolo P. Lava, Mar 21 2013

nonn

If p is a Sophie Germain prime (A005384) then m = 16*p is a term. Indeed: m' = (16*p)' = 32*p + 16 = 16*(2*p + 1) and phi(m') = phi(32*p + 16) = phi(16*(2*p + 1)) = 8*phi(2*p + 1) = 8*2*p = m for odd p. If p = 2 then m = 16*2 = 32 is a term. - Marius A. Burtea, Apr 10 2022

			For k=368, k'=752 and phi(752)=368.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from Paolo P. Lava)

Cf. A000010, A003415, A005384, A166374.

f:=func;  [k:k in [2..12000]| k eq EulerPhi(Floor(f(k)))]; // Marius A. Burtea, Apr 09 2022

with(numtheory);
A217715:= proc(q) local n,p;
for n from 1 to q do
if phi(n*add(op(2,p)/op(1,p),p=ifactors(n)[2]))=n then print(n); fi; od; end:
A217715(10^6);

aQ[1]=1; aQ[n_] := EulerPhi[n * Total[#2/#1 & @@@ FactorInteger[n]]] == n; Select[Range[10000], aQ] (* Amiram Eldar, Jul 11 2019 *)

A248815 Numbers equal to the arithmetic derivative of their Euler totient function.

Original entry on oeis.org

31, 32, 80, 81, 112, 176, 192, 244, 368, 752, 859, 912, 944, 1296, 1328, 1712, 1723, 2672, 2864, 3024, 3632, 4208, 5552, 5744, 6128, 6156, 7472, 7664, 8048, 8748, 9008, 9392, 11504, 13424, 13808, 14192, 15728, 16304, 18992, 20412, 20528, 20912, 21104, 21872

Offset: 1

Paolo P. Lava, Oct 15 2014

nonn

Solutions of the equation n = (phi(n))’.

			Euler totient function of 32 is 16 and the arithmetic derivative of 16 is 32.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Paolo P. Lava)

Cf. A000005, A003415, A166374.

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

deriv[n_] := n*Total[#2/#1 & @@@ FactorInteger[n]]; aQ[1] = 1; aQ[n_] := deriv[EulerPhi[n]] == n; Select[Range[25000], aQ] (* Amiram Eldar, Jul 11 2019 *)

A342418 Composite numbers k such that Euler totient phi(k) is a multiple of the arithmetic derivative of k.

Original entry on oeis.org

9, 15, 25, 35, 49, 95, 119, 121, 143, 169, 209, 287, 289, 319, 323, 343, 361, 377, 527, 529, 559, 625, 779, 841, 899, 923, 961, 989, 1007, 1189, 1199, 1225, 1343, 1349, 1369, 1681, 1763, 1849, 1919, 2159, 2197, 2209, 2507, 2759, 2809, 2911, 3239, 3481, 3599, 3721, 3827, 3993, 4031, 4489, 4607, 5041, 5183, 5207, 5249

Offset: 1

Antti Karttunen, Mar 12 2021

nonn

The term 343 is the first one that does not occur in A070160, and 625 is the second.

Cf. A000010, A003415, A166374 (a subsequence after its initial terms).
Subsequence of A342008.
Cf. also A070160.

Select[Range[5300], And[CompositeQ@ #, Mod[EulerPhi[#], Times @@ If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]] ]] == 0] &] (* Michael De Vlieger, Mar 12 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA342418(n) = ((n>1)&&!isprime(n)&&!(eulerphi(n)%A003415(n)));

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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A342413 a(n) = gcd(phi(n), A003415(n)), where A003415(n) is the arithmetic derivative of n, and phi is Euler totient function.

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A345051 Numbers k such that A345048(k) is equal to A345049(k).

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A342009 Numbers k such that the arithmetic derivative of k is a multiple of phi(k).

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A301939 Integers whose arithmetic derivative is equal to their Dedekind function.

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A342008 Numbers k such that Euler totient phi(k) is a multiple of the arithmetic derivative of k.

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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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A217715 Numbers equal to the Euler totient function of their arithmetic derivative: k = phi(k').

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