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

0,5

Comments

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

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

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

Example

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

Prog

(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)

Crossrefs

Cf. A110076, A110077.

Sequence in context: A278977 A097433 A308758 * A257064 A085800 A155190

Adjacent sequences: A110075 A110076 A110077 * A110079 A110080 A110081

Keyword

nonn

Author

Farideh Firoozbakht, Aug 01 2005

Extensions

More terms from Max Alekseyev, Aug 08 2005

Terms a(44) onward from Max Alekseyev, Mar 04 2014

Status

approved