A384819 Nonnegative numbers a(n) < n for n >= 1 such that exp( Sum_{n>=1} (n^2 - a(n))*x^n/n ) is a power series with integral coefficients.

0, 1, 2, 1, 4, 3, 6, 1, 2, 7, 10, 3, 12, 11, 3, 1, 16, 3, 18, 15, 14, 19, 22, 3, 4, 23, 2, 27, 28, 12, 30, 1, 15, 31, 7, 3, 36, 35, 32, 7, 40, 37, 42, 7, 21, 43, 46, 3, 6, 7, 27, 11, 52, 3, 34, 35, 50, 55, 58, 44, 60, 59, 5, 1, 18, 0, 66, 19, 39, 40, 70, 3, 72, 71, 3, 23, 1, 19, 78, 55, 2, 79, 82, 41, 47, 83, 51, 47, 88, 84, 74, 31, 86, 91, 17, 3, 96, 11, 42, 15, 100

Offset: 1

Author

Paul D. Hanna, Jun 18 2025

Keywords

• nonn

Comments

Conjecture: a(p^n) = p - 1 when p is prime for n >= 1.

Examples

			L.g.f.: A(x) = 0*x + 1*x^2/2 + 2*x^3/3 + 1*x^4/4 + 4*x^5/5 + 3*x^6/6 + 6*x^7/7 + 1*x^8/8 + 2*x^9/9 + 7*x^10/10 + 10*x^11/11 + 3*x^12/12 + 12*x^13/13 + 11*x^14/14 + 3*x^15/15 + 1*x^16/16 + ...
where the following is a power series with integral coefficients
exp( x/(1-x)^2 - A(x) ) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 14*x^5 + 25*x^6 + 43*x^7 + 74*x^8 + 124*x^9 + 205*x^10 + 335*x^11 + 543*x^12 + 869*x^13 + 1379*x^14 + 2170*x^15 + 3388*x^16 + ... + A384820(n)*x^n + ...
which is equivalent to
exp( Sum_{n>=1} (n^2 - a(n))*x^n/n ) = exp(x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 21*x^5/5 + 33*x^6/6 + 43*x^7/7 + 63*x^8/8 + 79*x^9/9 + 93*x^10/10 + 111*x^11/11 + 141*x^12/12 + 157*x^13/13 + 185*x^14/14 + 222*x^15/15 + 255*x^16/16 + ...).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..520

Crossrefs

Cf. A384820, A082579 (exp(x/(1-x)^2)).

Programs

PARI

{a(n) = my(A=[1]); for(i=2,n, A = concat(A,t);
for(t=1,(#A)^2+1, if( denominator( eval(polcoef( exp( intformal(Ser(A)) ),#A)) )==1, A[#A] = t + (#A)*(#A-1); break)) ); n^2 - A[n]}
for(n=1,101, print1(a(n),", "))