A245435 - OEIS

%I #22 Jan 12 2025 11:11:09

%S -36,-288,-36,0,-972,-288,0,-2304,-36,-500,-33750,-7776,-2304,0,

%T -12348,-36,-288,-4500,-18432,-108,-4096,-26244,-7776,-972,-5000,

%U -47916,-1372,-36,-36,-972,-79092,-1728,-26244,500,-98784,-4500,-43904,-36000,-16875,-2304,-8000

%N For any composite number n with more than a single prime factor, take the polynomial defined by the product of the terms (x-pi)^ei, where pi are the prime factors of n with multiplicities ei. Integrate this polynomial from the minimum to the maximum value of pi. This sequence lists the values of the integrals that are integer.

%C Corresponding values of the integrals generated by the terms of A245284.

%H Paolo P. Lava, <a href="/A245435/b245435.txt">Table of n, a(n) for n = 1..1000</a>

%e n=55 is the first number for which the integral is an integer. In fact its prime factors are 5 and 11: min(pi)=5, max(pi)=11. Polynomial: (x-5)*(x-11)= x^2-16*x+55. Integral: x^3/3-8*x^2+55*x. The value of the integral from x=5 to x=11 is -36.

%p with(numtheory): P:=proc(i) local a, b, c, d, k, m, m1, m2, n,t;

%p for k from 1 to i do a:=ifactors(k)[2]; b:=nops(a); c:=op(a); d:=1;

%p if b>1 then m1:=c[1, 1]; m2:=0; for n from 1 to b do

%p for m from 1 to c[n, 2] do d:=d*(x-c[n, 1]); od;

%p if c[n, 1]<m1 then m1:=c[n, 1]; fi;

%p if c[n, 1]>m2 then m2:=c[n, 1]; fi; od;

%p t:=int(d, x=m1..m2); if type(t,integer) then print(t); fi; fi; od; end:

%p P(10^4);

%Y Cf. A203612, A203613, A203614, A245284.

%K easy,sign,changed

%O 1,1

%A _Paolo P. Lava_, Aug 22 2014