A036878 -id:A036878 - 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

A380888 Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = -1.

Original entry on oeis.org

2, 9, 75, 625, 1029, 1365, 8575, 11375, 24843, 32955, 73815, 117649, 156065, 207025, 274625, 483153, 599781, 615125, 866481, 1008273, 1252815, 1337505, 1343433, 1553937, 1782105, 1955085, 2061345, 2840383, 3051015, 3432165, 3737085, 3767855, 4026275, 4998175

Offset: 1

Paolo P. Lava, Feb 07 2025

nonn

2 is the only even term. - Chai Wah Wu, Apr 24 2025

			73815 = 3*5*7*19*37 = 73815/(3-1) + 73815/(5-1) + 73815/(7-1) + 73815/(19-1) + 73815/(37-1);
599781 = 3*7*13^4 = 599781/(3-1) + 599781/(7-1) + 599781*4/(13-1).

Robert Israel, Table of n, a(n) for n = 1..213

Cf. A380889, A380900, A380901, A380923-A380928.

Contains A036878.

with(numtheory): P:=proc(q,h) local k,n,v; v:=[];
for n from 1 to q do if n=add(n*k[2]/(k[1]+h),k=ifactors(n)[2]) then v:=[op(v),n]; fi;
od; op(v); end: P(4998175,-1);

A276092 a(n) = Product_{i=1..n} prime(i)^(prime(i)-1), a(0)=1.

Original entry on oeis.org

1, 2, 18, 11250, 1323551250, 34329510752434301250, 799811863723341113907011901401250, 38919798565076223182552300534870824616780123359001250, 4052615498709835178737678586220586796222761283625319842830388618784835051250, 3679152532021669595137666762315244807517735994898621013565758767014111825486079213219685771368099483111250

Offset: 0

Antti Karttunen, Aug 22 2016

nonn

Cumulative product of A036878 (after a(0)). - Rick L. Shepherd, Aug 23 2016

			For n=0 we have an empty product, thus a(0) = 1.
For n=1, a(1) = 2^1.
For n=2, a(2) = 2^1 * 3^2 = 18.
For n=3, a(3) = 2^1 * 3^2 * 5^4 = 11250.

Cf. A000040, A002110, A057588, A036878, A276086.

Subsequence of A048103.

Table[Product[Prime[i]^(Prime[i] - 1), {i, n}], {n, 0, 9}] (* Michael De Vlieger, Aug 31 2016 *)

A276092(n) = prod(i=1, n, prime(i)^(prime(i) - 1)) \\ Rick L. Shepherd, Aug 23 2016

(define (A276092 n) (let outloop ((i n) (t 1)) (if (zero? i) t (let ((p (A000040 i))) (let inloop ((j (- p 1)) (t t)) (if (zero? j) (outloop (- i 1) t) (inloop (- j 1) (* t p))))))))
;; Or as a recurrence:
(definec (A276092 n) (if (zero? n) 1 (* (A276092 (- n 1)) (expt (A000040 n) (- (A000040 n) 1)))))

a(n) = Product_{i=1..n} prime(i)^(prime(i)-1).
a(0) = 1; and for n >= 1, a(n) = a(n-1) * A000040(n)^(A000040(n)-1).
a(n) = A276086(A057588(n)).

A180934 Numbers m such that m^k has m divisors for some k >= 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 28, 29, 31, 37, 40, 41, 43, 45, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 153, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 225, 227

Offset: 1

David W. Wilson, Sep 26 2010

nonn

All primes p are in this sequence, since p^(p-1) has p divisors.

For all odd semiprimes s, s^2 is in this sequence, since s^((s-1)/2) has s divisors.

			11^10 has 11 divisors, so 11 is in the sequence.
225^7 has 225 divisors, so 225 is in the sequence.

David W. Wilson, Table of n, a(n) for n=1..10000

A000005(m^k) = m for some k >= 1.
A180935 gives the corresponding k.
Cf. A036878, A046315.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]], k = 1}, While[n > Times @@ (k*e + 1), k++]; n == Times @@ (k*e + 1)]; q[1] = True; Select[Range[250], q] (* Amiram Eldar, Apr 09 2024 *)

A381215 Numbers k such that the difference between the largest and smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of k is 1.

Original entry on oeis.org

2, 8, 9, 36, 72, 81, 108, 216, 625, 15625, 117649, 5764801, 25937424601, 3138428376721, 23298085122481, 3937376385699289, 48661191875666868481, 14063084452067724991009, 104127350297911241532841, 37589973457545958193355601, 907846434775996175406740561329

Offset: 1

Paolo Xausa, Feb 19 2025

			72 is a term because 72 = 2^3*3^2, the set of these bases and exponents is {2, 3} and 3 - 2 = 1.

Paolo Xausa, Table of n, a(n) for n = 1..155

Positions of ones in A381214.

Cf. A036878, A081812, A104126, A381212, A381317.

Join[{2, 8, 9, 36, 72, 81, 108, 216}, Flatten[Map[#^{# - 1, # + 1} &, Prime[Range[3, 10]]]]]

For n >= 9, a(n) = A381317(n-4).

A111134 Numbers of the form p^(p^k-1) (p prime, k>=0).

Original entry on oeis.org

1, 2, 8, 9, 128, 625, 6561, 32768, 117649, 2147483648, 25937424601, 2541865828329, 23298085122481, 59604644775390625, 9223372036854775808, 48661191875666868481, 104127350297911241532841, 907846434775996175406740561329, 147808829414345923316083210206383297601

Offset: 1

Franz Vrabec, Oct 17 2005

nonn

The refactorable prime powers A000961.

Amiram Eldar, Table of n, a(n) for n = 1..102

Intersection of A000961 and A033950.

The sequence forms a subsequence of the refactorable numbers A033950; on the other hand A036878 is a subsequence.

f[p_, max_] := Module[{k = 1, s = {}, r}, While[(r = p^(p^k - 1)) < max, AppendTo[s, r]; k++]; s]; seq[max_] := Module[{s = {1}, p = 2, s1}, While[(s1 = f[p, max]) != {}, s = Join[s, s1]; p = NextPrime[p]]; Union[s]]; seq[10^30] (* Amiram Eldar, Aug 01 2024 *)

More terms from Robert G. Wilson v, Oct 21 2005

a(18)-a(19) from Amiram Eldar, Aug 01 2024

A130614 a(n) = p^(p-2), where p = prime(n).

Original entry on oeis.org

1, 3, 125, 16807, 2357947691, 1792160394037, 2862423051509815793, 5480386857784802185939, 39471584120695485887249589623, 3053134545970524535745336759489912159909

Offset: 1

Jonathan Vos Post, Jun 18 2007

Number of labeled trees on p(n) nodes, where p(n) is the n-th prime.

Let p = prime(n). For n >= 2, (-1)^((p-1)/2) * a(n) is the discriminant of the p-th cyclotomic polynomial. - Jianing Song, May 10 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..77

Cf. A000040, A000272, A004124, A036878.

Magma
```
[n^(n-2) : n in [2..40] | IsPrime(n)];
```

[p^(p-2): p in PrimesUpTo(50)]; // Vincenzo Librandi, Mar 27 2014

Table[Prime@n^(Prime@n - 2), {n, 20}] (* Vincenzo Librandi, Mar 27 2014 *)
#^(#-2)&/@Prime[Range[10]] (* Harvey P. Dale, Oct 18 2016 *)

a(n) = my(p=prime(n)); p^(p-2) \\ Felix Fröhlich, May 10 2021

a(n) = A000272(A000040(n)).

For n >= 2, (-1)^((p-1)/2) * a(n) = A004124(p), where p = prime(n). - Jianing Song, May 10 2021

Name edited by Felix Fröhlich, May 10 2021

A199768 Numbers whose greatest prime factor is less than their number of divisors.

Original entry on oeis.org

4, 6, 8, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 42, 45, 48, 50, 54, 56, 60, 64, 70, 72, 75, 80, 81, 84, 90, 96, 100, 105, 108, 112, 120, 126, 128, 132, 135, 140, 144, 150, 160, 162, 168, 180, 189, 192, 196, 198, 200, 210, 216, 220, 224, 225, 240, 243, 250, 252

Offset: 1

Franklin T. Adams-Watters, Nov 10 2011

The greatest prime factor equals the number of divisors only for 1 (as defined in A006530) and numbers of the form p^(p-1) for p a prime.

			4 has 2 as its greatest prime factor, and it has 3 factors (1, 2, 4), so it is in the sequence.
10 has 5 as its greatest prime factor, but it has only 4 factors (1, 2, 5, 10), so it is not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006530 (greatest prime factor), A000005 (number of divisors), A036878 (p^(p-1)).

Select[Range[300],FactorInteger[#][[-1,1]]Harvey P. Dale, Nov 19 2011 *)

A381317 Numbers of the form p^(p +- 1), where p is prime.

Original entry on oeis.org

2, 8, 9, 81, 625, 15625, 117649, 5764801, 25937424601, 3138428376721, 23298085122481, 3937376385699289, 48661191875666868481, 14063084452067724991009, 104127350297911241532841, 37589973457545958193355601, 907846434775996175406740561329, 480250763996501976790165756943041

Offset: 1

Paolo Xausa, Feb 20 2025

Paolo Xausa, Table of n, a(n) for n = 1..150

Union of A036878 and A104126.

Subsequence of A381215.

Flatten[Map[#^{# - 1, # + 1} &, Prime[Range[10]]]]

A104129 Integers of the form p^(p-1)+p where p is prime.

Original entry on oeis.org

4, 12, 630, 117656, 25937424612, 23298085122494, 48661191875666868498, 104127350297911241532860, 907846434775996175406740561352, 88540901833145211536614766025207452637390

Offset: 1

Cino Hilliard, Mar 06 2005

Sum of the reciprocals 1/a(n) rapidly converges to 0.3349291343132812638... - R. J. Mathar, Apr 27 2007

Cf. A000040, A005097, A036878, A085531.

#^(#-1)+#&/@Prime[Range[10]] (* Harvey P. Dale, Mar 23 2025 *)

From R. J. Mathar, Apr 27 2007: (Start)
a(n) = A036878(n)+p where p=A000040(n).
For p>2, a(n) = p*[1+A085531(A005097(n-1))]. (End)

Edited by R. J. Mathar, Apr 27 2007

cp's OEIS Frontend

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

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A380888 Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = -1.

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A276092 a(n) = Product_{i=1..n} prime(i)^(prime(i)-1), a(0)=1.

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A180934 Numbers m such that m^k has m divisors for some k >= 1.

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A381215 Numbers k such that the difference between the largest and smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of k is 1.

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A111134 Numbers of the form p^(p^k-1) (p prime, k>=0).

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A130614 a(n) = p^(p-2), where p = prime(n).

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