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 1-3 of 3 results.

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 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.