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A110078 a(n) is number of solutions of the equation sigma(x)=10^n. 4


%S 1,0,0,0,2,4,7,9,15,23,36,53,85,124,202,289,425,603,864,1209,1699,

%T 2397,3386,4665,6440,8801,12101,16338,22078,29565,39557,52615,69823,

%U 92338,121622,159435,208513,271775,353436,457759,591191,760763,976412,1250011,1596723,2034474,2585159,3277192,4145341,5232888,6591553

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

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

%H Max Alekseyev, <a href="/A110078/b110078.txt">Table of n, a(n) for n = 0..1000</a>

%H Max A. Alekseyev, <a href="http://arxiv.org/abs/1401.6054">Computing the (number of) inverses of Euler's totient and other multiplicative functions</a>, arXiv preprint arXiv:1401.6054, 2014

%F 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

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

%o (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)

%Y Cf. A110076, A110077.

%K nonn

%O 0,5

%A _Farideh Firoozbakht_, Aug 01 2005

%E More terms from _Max Alekseyev_, Aug 08 2005

%E Terms a(44) onward from _Max Alekseyev_, Mar 04 2014

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Last modified November 25 12:53 EST 2015. Contains 264417 sequences.