A245435 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.

-36, -288, -36, 0, -972, -288, 0, -2304, -36, -500, -33750, -7776, -2304, 0, -12348, -36, -288, -4500, -18432, -108, -4096, -26244, -7776, -972, -5000, -47916, -1372, -36, -36, -972, -79092, -1728, -26244, 500, -98784, -4500, -43904, -36000, -16875, -2304, -8000

Offset

1,1

Comments

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

Links

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Example

n=55 is the first number for which the integral is 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.

Maple

with(numtheory): P:=proc(i) local a, b, c, d, k, m, m1, m2, n, t;
for k from 1 to i do a:=ifactors(k)[2]; b:=nops(a); c:=op(a); d:=1;
if b>1 then m1:=c[1, 1]; m2:=0; for n from 1 to b do
for m from 1 to c[n, 2] do d:=d*(x-c[n, 1]); od;
if c[n, 1]<m1 then m1:=c[n, 1]; fi;
if c[n, 1]>m2 then m2:=c[n, 1]; fi; od;
t:=int(d, x=m1..m2); if type(t, integer) then print(t); fi; fi; od; end:
P(10^4);

Crossrefs

Cf. A203612, A203613, A203614, A245284.

Sequence in context: A187511 A185243 A014136 * A288963 A091081 A017462

Adjacent sequences: A245432 A245433 A245434 * A245436 A245437 A245438

Keyword

easy,sign

Author

Paolo P. Lava, Aug 22 2014

Status

approved