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 1-8 of 8 results.

A262723 Products of three distinct primes that form an arithmetic progression.

Original entry on oeis.org

105, 231, 627, 897, 935, 1581, 1729, 2465, 2967, 4123, 4301, 4715, 5487, 7685, 7881, 9717, 10707, 11339, 14993, 16377, 17353, 20213, 20915, 23779, 25327, 26331, 26765, 29341, 29607, 32021, 33335, 40587, 40807, 42911, 48635, 49321, 54739, 55581, 55637, 59563, 60297, 63017
Offset: 1

Views

Author

Antonio Roldán, Sep 28 2015

Keywords

Comments

This sequence is subsequence of A046389, A088595, A187073, A203614 and A229094.
Obviously, the most repeated prime divisor for values of a(n) is 3. - Altug Alkan, Sep 30 2015
These are numbers 3(2k + 3)(4k + 3) where 2k + 3 and 4k + 3 are prime, together with numbers p(p - 6d)(p + 6d) where p, p - 6d, and p + 6d are prime. - Charles R Greathouse IV, Mar 16 2018

Examples

			627 is in this sequence because 627=3*11*19, and 3, 11, 19 form an arithmetic progression (11-3 = 19-11).

Links

Crossrefs

Cf. A046389, A088595, A187073, A203614, A229094.

Programs

  • Mathematica
    Select[Range@ 64000, And[SquareFreeQ@ #, PrimeOmega@ # == 3, Subtract @@ Differences[First /@ FactorInteger@ #] == 0] &] (* Michael De Vlieger, Sep 30 2015 *)
  • PARI
    for(i=2,10^5,if(issquarefree(i)&&omega(i)==3,f=factor(i);if(f[1, 1]+f[3, 1]==2*f[2,1],print1(i,", "))))
  • PARI
    list(lim)=my(v=List()); lim\=1; forstep(d=6,sqrtint(lim\10),6, forprime(p=d+5, solve(x=sqrtn(lim,3),d*sqrtn(lim,3), x^3-d^2*x-lim)+.5, if(isprime(p-d) && isprime(p+d), listput(v, p*(p-d)*(p+d))))); forprime(p=5,(sqrt(24*lim+81)-27)/12+3.5, if(isprime(2*p-3), listput(v,p*(2*p-3)*3))); Set(v) \\ Charles R Greathouse IV, Mar 16 2018

Extensions

New name from Peter Munn, Aug 27 2022

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.

Original entry on oeis.org

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
Offset: 1

Views

Author

Paolo P. Lava, Jan 05 2012

Keywords

Examples

			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.

Links

Crossrefs

Cf. A203613, 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);
  • 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 *)

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

A294752 Squarefree products of k primes that are symmetrically distributed around their average. Case k = 5.

Original entry on oeis.org

53295, 119301, 229245, 399993, 608235, 623645, 1462731, 2324495, 3696189, 3973145, 4482879, 5356445, 5920971, 6249633, 7588977, 8270385, 10160943, 10450121, 10505373, 13185969, 13630011, 13760929, 14935029, 19095395, 20280795, 22566271, 23131549, 23408259, 24778401
Offset: 1

Views

Author

Paolo P. Lava, Nov 08 2017

Keywords

Examples

			53295 = 3*5*11*17*19. Prime factors average is (3 + 5 + 11 + 17 + 19)/5 = 11 and 3 + 8 = 11 = 19 - 8, 5 + 6 = 11 = 17 - 6.

Links

Crossrefs

Subsequence of A046387, A203614.
Cf. A006881 (k=2), A262723 (k=3), A294751 (k=4), A294776 (k=6).

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local a,b,k,n,ok;
for n from 2*3*5*7*11 to q do if not isprime(n) and issqrfree(n) then a:=ifactors(n)[2];
if nops(a)=h then b:=2*add(a[k][1],k=1..nops(a))/nops(a); ok:=1;
for k from 1 to trunc(nops(a)/2) do if a[k][1]+a[nops(a)-k+1][1]<>b then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; fi; od; end: P(10^9,5);
# Alternative:
N:= 10^8: # to get all terms <= N
M:= floor((8*N/15)^(1/3)):
P:= select(isprime, [seq(i,i=3..M,2)]): nP:= nops(P):
Res:= NULL:
for i3 from 3 to nP-2 do
  p3:= P[i3];
  for i1 from 1 to i3-2 do
    if isprime(2*p3 - P[i1]) then
      for i2 from i1+1 to i3-1 do
        if isprime(2*p3 - P[i2]) then
          v:=P[i1]*P[i2]*p3*(2*p3-P[i2])*(2*p3-P[i1]);
          if v <= N then Res:= Res, v fi
        fi
      od
     fi
   od
od:
sort([Res]): # Robert Israel, Nov 10 2017
  • PARI
    isok(n, nb=5) = {if (issquarefree(n) && (omega(n)==nb), f = factor(n)[, 1]~; avg = vecsum(f)/#f; for (k=1, #f\2, if (f[k] + f[#f-k+1] != 2*avg, return(0));); return (1););} \\ Michel Marcus, Nov 10 2017

Extensions

More terms from Giovanni Resta, Nov 09 2017
Missing term 23131549 inserted by Robert Israel, Nov 10 2017

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 1-8 of 8 results.

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