A072075 -id:A072075 - cp's OEIS frontend

A097649 Duplicate of A072075.

1, 11, 101, 1111, 10291, 100651, 1004251, 10165751, 100064101, 1000078501, 10000222501, 100062501601, 1000062516001, 10000062660001, 100002441447211, 1003922328562757, 10000390625025601, 100000002482366251

Offset: 0

dead

Name was: a(n) is the smallest number m such that phi(m)=10^n.

A051445 Smallest k such that phi(k) = 2n, or 0 if there is no such k.

Original entry on oeis.org

3, 5, 7, 15, 11, 13, 0, 17, 19, 25, 23, 35, 0, 29, 31, 51, 0, 37, 0, 41, 43, 69, 47, 65, 0, 53, 81, 87, 59, 61, 0, 85, 67, 0, 71, 73, 0, 0, 79, 123, 83, 129, 0, 89, 0, 141, 0, 97, 0, 101, 103, 159, 107, 109, 121, 113, 0, 177, 0, 143, 0, 0, 127, 255, 131, 161, 0, 137

Offset: 1

Jud McCranie

nonn

The zero values are easy to prove because of the bounds on the phi function.

			a(4) = 15 as phi(15) = 2*4 and no k < 15 has phi(k) = 2*4.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A002181, A072075, A079695. For records see A132012, A132115.

a(n)=n+=n;for(k=n+1, solve(x=n,if(n<20,99,5*n*log(log(n))), x/(exp(Euler)*log(log(x))+3/log(log(x)))-n), if(eulerphi(k)==n,return(k))); 0 \\ Charles R Greathouse IV, Dec 19 2011

a(10^n/2) = A072075(n). - R. J. Mathar, Dec 12 2024

a(A079695(n)) = 0. - David A. Corneth, Dec 12 2024

A072074 Number of integers k such that phi(k) = 10^n.

Original entry on oeis.org

2, 2, 4, 11, 16, 24, 43, 63, 94, 152, 224, 324, 464, 644, 897, 1271, 1790, 2521, 3501, 4814, 6535, 8779, 11739, 15585, 20625, 27166, 35588, 46363, 60065, 77424, 99337, 127020, 161930, 205847, 260929, 329782, 415533, 522173, 654548, 818278, 1020391

Offset: 0

Labos Elemer, Jun 13 2002

nonn

a(n) is the coefficient of x^n*y^n in Product_p Sum_{u, v} x^u*y^v, where the product is taken over all primes p and the sum is taken over such u, v that 2^u*5^v = phi(p^k) for some nonnegative integer k. - Max Alekseyev, Apr 26 2010

Elaborating on above comment, primes p must be in A077497 and k must be 1 for primes other than 2 and 5. - Ray Chandler, Feb 12 2012

			n=3: a(3)=11 because InvPhi(1000) = {1111, 1255, 1375, 1875, 2008, 2222, 2500, 2510, 2750, 3012, 3750}.

Ray Chandler, Table of n, a(n) for n = 0..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.

Cf. A000010, A014197, A014573, A110078, A072075, A072076.

Maple
```
[seq(nops(invphi(10^i)),i=1..8)];
```

a(n) = #invphi(10^n); \\ for invphi see Alekseyev link \\ Michel Marcus, May 14 2020

a(n) = Card{x : A000010(x)=10^n}.

More terms from Max Alekseyev, Apr 26 2010

A072076 Largest k such that EulerPhi(k) = 10^n.

Original entry on oeis.org

2, 22, 250, 3750, 41250, 414150, 4166250, 42281250, 438281250, 4400343750, 44266406250, 449238281250, 4510352343750, 45373066406250, 455545586718750, 4555455867187500, 45555287544813750, 455552875448137500, 4566844506855468750, 45668445068554687500

Offset: 0

Labos Elemer, Jun 13 2002

nonn

			n=3: a(3)=3750 because InvPhi(1000) = {1111, 1255, 1375, 1875, 2008, 2222, 2500, 2510, 2750, 3012, 3750}.

Ray Chandler, Table of n, a(n) for n = 0..1000
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2

Cf. A000010, A014197, A014573, A072074, A072075, A110076.

a(n) = Max{k; A000010(k) = 10^n}.

More terms from Max Alekseyev, Apr 26 2010

cp's OEIS Frontend

A097649 Duplicate of A072075.

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A051445 Smallest k such that phi(k) = 2n, or 0 if there is no such k.

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A072074 Number of integers k such that phi(k) = 10^n.

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A072076 Largest k such that EulerPhi(k) = 10^n.

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