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A177430
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Least monotonically increasing logarithmic derivative consisting of only squares.
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2
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1, 9, 16, 25, 36, 144, 169, 225, 484, 784, 1024, 1156, 1444, 1521, 2601, 3025, 4489, 5184, 5625, 6400, 7744, 9216, 12996, 15876, 17161, 19600, 24649, 25281, 29929, 34969, 46656, 47089, 50176, 55225, 56169, 62500, 66564, 69169, 72361, 78400
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OFFSET
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1,2
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COMMENTS
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Conjecture: the sequence {a(n)^(m/2), n>=1} forms a logarithmic derivative of an integer sequence only when m is a nonnegative even integer.
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LINKS
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FORMULA
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EXAMPLE
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L.g.f.: A(x) = x + 9/2*x^2 + 16/3*x^3 + 25/4*x^4 + 36/5*x^5 +...
exp(A(x)) = 1 + x + 5*x^2 + 10*x^3 + 24*x^4 + 51*x^5 + 122*x^6 + 244*x^7 +...+ A177432(n)*x^n +...
The square-root of each term gives the sequence A177431, which begins:
[1, 3, 4, 5, 6, ...];
A177431 does not form a logarithmic derivative of an integer sequence.
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PROG
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(PARI) {a(n)=local(V, A=[1], M); V=Vec(exp(x+sum(k=2, n-1, a(k)*x^k/k)+t*x^n/n+x*O(x^n))); if(n==1, M=1, M=a(n-1); for(m=sqrtint(a(n-1))+1, 9*a(n-1), if(denominator(subst(V[ #V], t, m^2))==1, M=m^2; break)); M)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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