cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A203614 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 equal to zero.

Original entry on oeis.org

105, 140, 231, 627, 748, 750, 897, 935, 1470, 1581, 1729, 2205, 2465, 2625, 2967, 3404, 3549, 4123, 4301, 4715, 5452, 5487, 6256, 7623, 7685, 7881, 9009, 9717, 10707, 10829, 10988, 11319, 11339, 13310, 14993, 15470, 16377, 17353, 17457, 17901, 20213, 20915
Offset: 1

Views

Author

Paolo P. Lava, Jan 05 2012

Keywords

Comments

Prime numbers are excluded because are banal solutions: in fact for them min(pi)=max(pi)=pi and then the area is zero.
Any squarefree number with an odd number of prime factors which are symmetrically distributed around the central one is part of the sequence. For instance with n=53295 the prime factors are 3, 5, 11, 17, 19 and 3+8=11=19-8, 5+6=11=17-6.

Examples

			n=140. Prime factors: 2, 2, 5, 7: min(pi)=2, max(pi)=7. Polynomial to integrate from 2 to 7: (x-2)^2*(x-5)*(x-7)=x^4-16*x^3+87*x^2-188x+140. The resulting area is equal to zero.

Links

Crossrefs

Cf. A203612, A203613.

Programs

  • Maple
    with(numtheory);
P:=proc(i)
local a,b,c,d,k,m,m1,m2,n;
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;
   if int(d,x=m1..m2)=0 then print(k); fi;
fi;
od;
end:
P(500000);

A203613 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 negative integer.

Original entry on oeis.org

55, 85, 91, 115, 133, 145, 187, 195, 204, 205, 217, 235, 247, 253, 259, 265, 275, 285, 295, 301, 319, 351, 355, 357, 385, 391, 403, 415, 425, 427, 445, 451, 465, 469, 476, 481, 483, 493, 505, 511, 517, 535, 553, 555, 559, 565, 575, 583, 589, 595, 609, 621, 637
Offset: 1

Views

Author

Paolo P. Lava, Jan 05 2012

Keywords

Examples

			n=217. Prime factors: 7, 31: min(pi)=7, max(pi)=31. Polynomial: (x-7)*(x-31)=x^2-38*x+217. Integral: x^3/3-19*x^2+217*x. The area from x=7 to x=31 is -2304.
n=53151. Prime factors: 3, 7, 2531: min(pi)=3, max(pi)=2531. Polynomial: (x-7)*(x-349)*(x-409)=x^3-2541*x^2+25331*x-53151. Integral: x^4/4-847*x^3+25331/2*x^2-53151*x. The area from x=3 to x=2531 is - 3392739409920.

Links

Crossrefs

Cf. A203612, A203614.

Programs

  • 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);

A245284 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 numbers for which the integral is an integer.

Original entry on oeis.org

55, 85, 91, 105, 115, 133, 140, 145, 187, 195, 204, 205, 217, 231, 235, 247, 253, 259, 265, 275, 285, 295, 301, 319, 351, 355, 357, 385, 391, 403, 415, 425, 427, 429, 445, 451, 465, 469, 476, 481, 483, 493, 505, 511, 517, 535, 553, 555, 559, 565, 575, 583, 589
Offset: 1

Views

Author

Paolo P. Lava, Aug 22 2014

Keywords

Comments

The union of A203612 U A203613 U A203614.

Examples

			n=1001. Prime factors: 7, 11 and 13: min(pi)=7, max(pi)=13. Polynomial: (x-7)*(x-11)*(x-13)= x^3-31*x^2+311*x-1001. Integral: x^4/4-31/3*x^3+311/2*x^2-1001*x. The area from x=7 to x=13 is 36.
n=1005. Prime factors: 3, 5 and 67: min(pi)=3, max(pi)=67. Polynomial: (x-3)*(x-5)*(x-67)= x^3-75*x^2+551*x-1005. Integral: x^4/4-25*x^3+551/2*x^2-1005*x. The area from x=3 to x=67 is -1310720.
n=1470. Prime factors: 2, 3, 5 and 7^2: min(pi)=2, max(pi)=7. Polynomial: (x-2)*(x-3)*(x-5)*(x-7)^2= x^5-24*x^4+220*x^3-954*x^2+1939*x-1470. Integral: x^6/6-24/5*x^5+55*x^4-318*x^3+1939/2*x^2-1470*x. The area from x=3 to x=67 is 0.

Links

Crossrefs

Cf. A203612, A203613, A203614, A245435. Subsequence of A024619.

Programs

  • Maple
    isA245284 := proc(n)
    local pfs,x1,x2,po,x ;
    if isprime(n) then
        false;
    else
        pfs := ifactors(n)[2] ;
        if nops(pfs) > 1 then
            x1 := A020639(n) ;
            x2 := A006530(n) ;
            po := mul((x-op(1,p))^op(2,p),p=pfs) ;
            int(po,x=x1..x2) ;
            type(%,'integer') ;
        else
            false;
        end if;
    end if;
end proc:
for n from 4 to 600 do
    if isA245284(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 07 2014

Extensions

Definition and example corrected by R. J. Mathar, Sep 07 2014

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.

Original entry on oeis.org

-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

Views

Author

Paolo P. Lava, Aug 22 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A203612, A203613, A203614, A245284.

Programs

  • 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]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);

A324072 For any composite number k take the polynomial defined by the product of the terms (x-d_i), where d_i are the aliquot parts of k. Integrate this polynomial from the minimum to the maximum value of d_i. Sequence lists the numbers k for which the integral is a positive integer.

Original entry on oeis.org

35, 143, 209, 247, 323, 527, 589, 713, 851, 899, 989, 1073, 1147, 1247, 1333, 1591, 1763, 2257, 2479, 2501, 2623, 2747, 2867, 2881, 2993, 3139, 3149, 3233, 3239, 3397, 3431, 3551, 3599, 3713, 3869, 3953, 4087, 4187, 4307, 4453, 4661, 4693, 4819, 4891, 5141, 5183
Offset: 1

Views

Author

Paolo P. Lava, Feb 14 2019

Keywords

Comments

Composites with an integral equal to zero are listed in A129521.
Similar to A203612 where prime factors are taken into account.

Examples

			Aliquot parts of 35 are 1, 5, 7. Polynomial: (x-1)*(x-5)*(x-7) = x^3 - 13*x^2 + 47*x - 35. Integral: x^4/4 - (13/3)*x^3 + (47/2)*x^2 - 35*x. The area from x=1 to x=7 is 36.

Crossrefs

Cf. A129521, A203612, A203613, A203614, A245284, A324073.

Programs

  • Maple
    with(numtheory): P:=proc(n) local a,k,x,y;
a:=sort([op(divisors(n) minus {n})]);
y:=int(mul((x-k),k=a),x=1..a[nops(a)]);
if frac(y)=0 and y>0 then n; fi; end: seq(P(i),i=2..5183);

A324073 For any composite number n take the polynomial defined by the product of the terms (x-d_i), where d_i are the aliquot parts of n. Integrate this polynomial from the minimum to the maximum value of d_i. Sequence lists the numbers for which the integral is a negative integer.

Original entry on oeis.org

14, 21, 26, 32, 33, 38, 39, 49, 51, 57, 62, 65, 69, 74, 86, 87, 93, 95, 111, 122, 123, 125, 129, 133, 134, 141, 146, 155, 158, 159, 169, 177, 182, 183, 185, 194, 201, 206, 213, 215, 217, 218, 219, 237, 242, 249, 254, 259, 267, 273, 278, 291, 301, 302, 303, 305
Offset: 1

Views

Author

Paolo P. Lava, Feb 14 2019

Keywords

Comments

Composite with an integral equal to zero are listed in A129521.
Similar to A203613 where prime factors are taken into account.
If all the divisors were considered, then prime numbers with an integral with a negative integer would be those listed in A002476.

Examples

			Aliquot parts of 32 are 1, 2, 4, 8, 16. Polynomial: (x-1)*(x-2)*(x-4)*(x-8)*(x-16) = x^5-31*x^4+310*x^3-1240*x^2+1984*x-1024. Integral: x^6/6-31/5*x^5+155/2*x^4-1240*x^3/3+992*x^2-1024*x. The area from x=1 to x=16 is -81000.

Crossrefs

Cf. A002476, A129521, A203612, A203613, A203614, A245284, A324072.

Programs

  • Maple
    with(numtheory): P:=proc(n) local a,k,x,y;
a:=sort([op(divisors(n) minus {n})]);
y:=int(mul((x-k),k=a),x=1..a[nops(a)]);
if frac(y)=0 and y<0 then n; fi; end: seq(P(i),i=2..305);
Showing 1-6 of 6 results.

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