A110078 -id:A110078 - cp's OEIS frontend

A110077 a(n) is the smallest number m such that sigma(m)=10^n and if there is no such m, a(n)=0.

1, 0, 0, 0, 8743, 71193, 640737, 5906061, 65624979, 590624811, 5498542791, 55995364341, 549871699041, 5582882097891, 55828827410391, 542546715730761, 5469955867029591, 53226216007355979, 532262221390168479, 5300249369031696429, 52602977416561263909, 531074469279114815229

Offset: 0

Farideh Firoozbakht, Aug 01 2005

nonn

A110078(n) gives number of solutions of the equation sigma(x)=10^n.

Conjecture: For n>3 a(n) is positive.

			a(9)=590624811 because sigma(590624811)=sigma(3^3*7*3124999) sigma(3^3)*sigma(7)*sigma(3124999)=40*8*3125000=10^9 and 590624811 is the smallest number m with this property (sigma(m)=10^9).

Max Alekseyev, Table of n, a(n) for n = 0..1000
Max Alekseyev, PARI/GP scripts for miscellaneous math problems
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. A110076, A110078.

{ a(n) = invsigma(10^n)[1] } \\ Max Alekseyev, Apr 26 2010

a(10)-a(11) from Donovan Johnson and Farideh Firoozbakht, Nov 22 2008

a(12) onward from Max Alekseyev, Apr 26 2010, Mar 06 2014

A110076 a(n) is the largest number m such that sigma(m)=10^n, or if there is no such m a(n)=0.

Original entry on oeis.org

1, 0, 0, 0, 9481, 99301, 997501, 9993001, 99948001, 999795001, 9999750001, 99998670001, 999997950001, 9999986700001, 99999975000001, 999999198750001, 9999999187500001, 99999995096707501, 999999919987500001, 9999999986700000001, 99999499999999800001, 999999999907500000001, 9999999999796009687501

Offset: 0

Farideh Firoozbakht, Jul 31 2005

nonn

Conjecture: For n>3 a(n) is positive.

For 4 <= n <= 102, a(n) is the product of two distinct primes, but a(103) = a(49)*a(54) and is the product of four distinct primes: 1862645149230957031249999 * 5368709119999999999999999 * 79999999999999999999999999 * 12499999999999999999999999999. - David Wasserman, Nov 18 2008

			a(12)=999997950001 because sigma(999997950001)=sigma(799999*1249999) =800000*1250000=10^12 and 999997950001 is the largest number with this property(sigma(m)=10^12).

Max Alekseyev, 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. A110077, A110078.

a[0] = 1; a[1] = a[2] = a[3] = 0; a[n_] := (For[m = 1, DivisorSigma[ 1, 10^n - m] != 10^n, m++ ];10^n - m); Do[Print[a[n]], {n, 0, 12}]

More terms from David Wasserman, Nov 18 2008

Terms a(19) onward from Max Alekseyev, Mar 06 2014

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

cp's OEIS Frontend

A110077 a(n) is the smallest number m such that sigma(m)=10^n and if there is no such m, a(n)=0.

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A110076 a(n) is the largest number m such that sigma(m)=10^n, or if there is no such m a(n)=0.

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Mathematica

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

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Programs

Maple

PARI

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