A342008 -id:A342008 - cp's OEIS frontend

A166374 Numbers whose arithmetic derivative is equal to Euler totient function: n' = phi(n).

2, 9, 15, 625, 1225, 3993, 117649, 218491, 857375, 3788435, 4259571, 69302975, 136410197, 200533921, 313742585, 603439225, 1516358753, 2563893625, 3326174929, 5655792025, 10214476341, 25937424601, 29677977573, 59797108943, 283867750439, 715167055525

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Oct 13 2009

nonn

The sequence is infinite. If n=prod(pi^ei) with each pi prime, then phi(n) = n*prod((pi-1)/pi) and n' = n*sum(ei/pi). Thus every number of the form p^(p-1), where p is prime, is in this sequence. - Nathaniel Johnston, Nov 27 2010

If p > q are primes and q does not divide p-1, there is a solution in positive integers of (p-1)*(q-1) = a*p + b*q, and then p^b*q^a is in the sequence. - Robert Israel, Aug 21 2014

Giovanni Resta, Table of n, a(n) for n = 1..31 (terms < 10^13)

Cf. A000010, A003415, A036878 (p^(p-1)).

Intersection of A342008 and A342009.

A003415:= n -> n*add(f[2]/f[1],f=ifactors(n)[2]):
select(numtheory:-phi = A003415, [$0..10^5]); # Robert Israel, Aug 21 2014

(*Run the Mathematica program given in A003415 first, to define the function a as the arithmetic derivative.*) Select[Range[0, 10000], EulerPhi[ # ] == a[ # ] &]

from sympy import factorint, totient
A166374 = [n for n in range(1,10**6) if sum([int(n*e/p) for p,e in factorint(n).items()]) == totient(n)] # Chai Wah Wu, Aug 22 2014, edited by Antti Karttunen, Mar 13 2021

A166374_list = lambda n: filter(lambda k: euler_phi(k) == A003415(k), range(n))
A166374_list(10^6) # Peter Luschny, Aug 23 2014

Two terms added by Alonso del Arte, Oct 20 2009
Offset corrected and a(12)-a(16) from Donovan Johnson, Nov 03 2010
a(17)-a(18) from Donovan Johnson, May 09 2011
a(19)-a(24) from Donovan Johnson, Oct 01 2012
a(25)-a(28) from Giovanni Resta, Mar 13 2014
Term a(1)=0 removed and the indices in the above comments decremented by one. - Antti Karttunen, Mar 13 2021

A342414 a(n) = A003415(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

0, 1, 1, 2, 1, 5, 1, 3, 1, 7, 1, 4, 1, 3, 1, 4, 1, 7, 1, 3, 5, 13, 1, 11, 1, 5, 3, 8, 1, 31, 1, 5, 7, 19, 1, 5, 1, 7, 2, 17, 1, 41, 1, 12, 13, 25, 1, 7, 1, 9, 5, 7, 1, 9, 2, 23, 11, 31, 1, 23, 1, 11, 17, 6, 3, 61, 1, 9, 13, 59, 1, 13, 1, 13, 11, 20, 3, 71, 1, 11, 2, 43, 1, 31, 11, 15, 4, 7, 1, 41, 5, 24, 17, 49, 1, 17

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, A342001, A342008 (positions of ones), A342413, A342415, A342416.

Array[#1/GCD[#1, #2] & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], EulerPhi[#]} &, 96] (* 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]));
A342414(n) = { my(u=A003415(n)); (u/gcd(eulerphi(n),u)); };

a(n) = A003415(n) / A342413(n) = A003415(n) / gcd(A000010(n),A003415(n)).

a(n) = A342001(n) / A342416(n).

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));

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)));

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A166374 Numbers whose arithmetic derivative is equal to Euler totient function: n' = phi(n).

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A342414 a(n) = A003415(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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A342009 Numbers k such that the arithmetic derivative of k is a multiple of phi(k).

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

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