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A249784
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Number of divisors of n^(n^n).
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2
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1, 5, 28, 513, 3126, 2176875649, 823544, 50331649, 774840979, 100000000020000000001, 285311670612, 158993694406808436568227841, 302875106592254, 123476695691247958050243432972289, 191751059232884087544279144287109376, 73786976294838206465
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OFFSET
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1,2
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COMMENTS
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An infinite number of squares are terms of this sequence.
Proof: for any n of the form (p*q)^k (with p and q distinct primes), a(n) = (k * n^n + 1)^2.
It seems likely that the only nontrivial palindromes in this sequence comprise a subset of these squares and occur at n = 10^(10^M) for M>=0; at such values of n, a(n) = (10^(10^(10^M + M) + M) + 1)^2 = A033934(10^(10^M + M) + M). The actual decimal expansion of each of these numbers is of the form 1000...0002000...0001, where the total number of zero digits on each side of the 2 is 10^(10^M + M) + M - 1.
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LINKS
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FORMULA
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a(n) = Product_{j=1..m} (e_j * n^n + 1)
where m = number of distinct prime factors of n
and e_j = multiplicity of the j-th prime factor.
If n is a prime p, then m=1 and e_1=1, so
If n=10^L, then m=2 and e_1=e_2=L, so
a(10^L) = (L * 10^(L * 10^L) + 1)^2.
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EXAMPLE
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12 = 2^2 * 3^1 (two distinct prime factors, with multiplicities e_1=2 and e_2=1), so a(12) = (2*k+1)*(1*k+1) = 2*k^2 + 3*k + 1 where k = 12^12, so a(12) = 158993694406808436568227841.
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PROG
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(Magma) // program to generate b-file
for n in [1..155] do
k:=n^n;
F:=Factorization(n);
prod:=1;
for j in [1..#F] do
prod*:=F[j, 2]*k + 1;
end for;
n, prod;
end for;
(Sage)
n_exp_n = n^n
return mul(exp[1]*n_exp_n + 1 for exp in factor(n))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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