A110078 a(n) is number of solutions of the equation sigma(x)=10^n.
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
Keywords
Examples
a(4)=2 because 8743 & 9481 are all solutions of the equation sigma(x)=10^4.
Links
- 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
Programs
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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
Comments