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A110078
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a(n) is number of solutions of the equation sigma(x)=10^n.
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4
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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
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OFFSET
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0,5
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COMMENTS
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Conjecture: For n>2, a(n+1)>a(n).
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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a(4)=2 because 8743 & 9481 are all solutions of the equation sigma(x)=10^4.
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PROG
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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)
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CROSSREFS
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Cf. A110076, A110077.
Sequence in context: A278977 A097433 A308758 * A257064 A085800 A155190
Adjacent sequences: A110075 A110076 A110077 * A110079 A110080 A110081
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KEYWORD
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nonn
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AUTHOR
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Farideh Firoozbakht, Aug 01 2005
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EXTENSIONS
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More terms from Max Alekseyev, Aug 08 2005
Terms a(44) onward from Max Alekseyev, Mar 04 2014
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STATUS
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approved
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