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 A203612 For any number n take the polynomial formed by the product of the terms (x-pi), where pi’s are the prime factors of n. Then calculate the area between the minimum and the maximum value of the prime factors. This sequence lists the numbers for which the area is a positive integer. 7
 429, 605, 663, 969, 1001, 1105, 1183, 1311, 1445, 1653, 1955, 2139, 2185, 2261, 2527, 2553, 2645, 2697, 2755, 3179, 3219, 3335, 3741, 3813, 4199, 4205, 4371, 4551, 4693, 4807, 4929, 4991, 5217, 5289, 5819, 5865, 5883, 5945, 5957, 6063, 6293, 6355, 6549, 6630 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Paolo P. Lava, Table of n, a(n) for n = 1..10000 EXAMPLE n=1445. Prime factors: 5, 17, 17: min(pi)=5, max(pi)=17. Polynomial: (x-5)*(x-17)^2=x^3-39*x^2+459*x-1445. Integral: x^4/4-13*x^3+459/2*x^2-1445*x. The area from x=5 to x=17 is 1728. n=999187. Prime factors: 7, 349, 409: min(pi)=7, max(pi)=409. Polynomial: (x-7)*(x-349)*(x-409)=x^3-765*x^2+148047*x-999187. Integral: x^4/4-255*x^3+148047/2*x^2-999187*x. The area from x=7 to x=409 is 1526672988. MAPLE with(numtheory); P:=proc(i) local a, b, c, d, k, m, m1, m2, n, p; 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]m2 then m2:=c[n, 1]; fi;    od;    p:=int(d, x=m1..m2); if (trunc(p)=p and p>0) then print(k); fi; fi; od; end: P(500000); MATHEMATICA apiQ[n_]:=Module[{f=Flatten[Table[#[[1]], #[[2]]]&/@FactorInteger[ n]], in}, in = Integrate[Times@@(x-f), {x, f[[1]], f[[-1]]}]; Positive[in] && IntegerQ[ in]]; Select[Range[7000], apiQ] (* Harvey P. Dale, May 27 2016 *) CROSSREFS Cf. A203613, A203614. Sequence in context: A334558 A320712 A338344 * A250330 A034278 A116870 Adjacent sequences:  A203609 A203610 A203611 * A203613 A203614 A203615 KEYWORD nonn AUTHOR Paolo P. Lava, Jan 05 2012 STATUS approved

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Last modified October 29 12:50 EDT 2020. Contains 338066 sequences. (Running on oeis4.)