A177430 Least monotonically increasing logarithmic derivative consisting of only squares.

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

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..40.

Formula

a(n) = A177431(n)^2.

Example

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.

Prog

(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)}

Crossrefs

Cf. A177431, A177432.

Sequence in context: A085585 A026062 A108945 * A256705 A092941 A075393

Adjacent sequences: A177427 A177428 A177429 * A177431 A177432 A177433

Keyword

nonn

Author

Paul D. Hanna, Sep 06 2010

Extensions

Edited to prevent horizontal scrolling -- Charles R Greathouse IV, Sep 08 2010

Status

approved