cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Farideh Firoozbakht, Aug 01 2005

Keywords

Comments

A110078(n) gives number of solutions of the equation sigma(x)=10^n.
Conjecture: For n>3 a(n) is positive.

Examples

			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).

Links

Crossrefs

Cf. A110076, A110078.

Programs

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

Extensions

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

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

Original entry on oeis.org

1, 0, 0, 0, 2, 4, 7, 9, 15, 23, 36, 53, 85, 124, 202, 289, 425, 603, 864, 1209, 1699, 2397, 3386, 4665, 6440, 8801, 12101, 16338, 22078, 29565, 39557, 52615, 69823, 92338, 121622, 159435, 208513, 271775, 353436, 457759, 591191, 760763, 976412, 1250011, 1596723, 2034474, 2585159, 3277192, 4145341, 5232888, 6591553
Offset: 0

Views

Author

Farideh Firoozbakht, Aug 01 2005

Keywords

Comments

Conjecture: For n>2, a(n+1)>a(n).

Examples

			a(4)=2 because 8743 & 9481 are all solutions of the equation sigma(x)=10^4.

Links

Crossrefs

Cf. A110076, A110077.

Programs

  • PARI
    { a(d) = local(X,Y,P,L,n,f,p,m,l); X=Pol([1,0],x); Y=Pol([1,0],y); P=Set(); L=listcreate(10000); for(i=0,d, for(j=0,d, n=2^i*5^j; if(n==1,next); f=factorint(n-1)[,1]; for(k=1,length(f), p=f[k]; m=n*(p-1)+1; while(m%p==0,m\=p); if(m==1, l=setsearch(P,p); if(l==0,l=setsearch(P,p,1); P=setunion(P,[p]); listinsert(L,1,l)); L[l]+=X^i*Y^j ) ) )); R=1+O(x^(d+1))+O(y^(d+1)); for(l=1,length(L),R*=L[l]); listkill(L); vector(d+1,n,polcoeff(polcoeff(R,n-1),n-1)) } (Alekseyev)

Formula

a(n) = coefficient of x^n*y^n in Prod_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 = sigma(p^k) for some nonnegative integer k. - Max Alekseyev, Aug 08 2005

Extensions

More terms from Max Alekseyev, Aug 08 2005
Terms a(44) onward from Max Alekseyev, Mar 04 2014

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

Views

Author

Labos Elemer, Jun 13 2002

Keywords

Examples

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

Links

Crossrefs

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

Formula

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

Extensions

More terms from Max Alekseyev, Apr 26 2010
Showing 1-3 of 3 results.

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