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.

A253235 Numbers n such that the n-th cyclotomic polynomial has no root mod p for all primes p <= n.

Original entry on oeis.org

1, 12, 15, 24, 28, 30, 33, 35, 36, 40, 44, 45, 48, 51, 56, 60, 63, 65, 66, 69, 70, 72, 75, 76, 77, 80, 84, 85, 87, 88, 90, 91, 92, 95, 96, 99, 102, 104, 105, 108, 112, 115, 117, 119, 120, 123, 124, 126, 130, 132, 133, 135, 138, 140, 141, 143, 144, 145, 150, 152, 153, 154
Offset: 1

Views

Author

Eric Chen, Apr 19 2015

Keywords

Comments

Numbers n such that A253236(n) = 0.
Numbers n such that all divisors of Phi_n(b) are congruent to 1 (mod n) for every natural number b.
If p is prime, k, r are natural numbers, then:
Every n = p^r is not in this sequence.
Every n = 2p^r is not in this sequence.
n = 3p^r (p>3) is in this sequence iff p != 1 (mod 3).
n = 4p^r (p>4) is in this sequence iff p != 1 (mod 4).
n = 5p^r (p>5) is in this sequence iff p != 1 (mod 5).
...
n = kp^r (p>k) is in this sequence iff p != 1 (mod k).

Links

Crossrefs

For A253236(n) = 2, 3, 5, 7, 11, 13, see A000079, A038754, A245478, A245479, A245480, A245481.

Programs

  • PARI
    is(n)=my(P=polcyclo(n), f=factor(n)[, 1]); for(i=1, #f, if(#polrootsmod(P, f[i]), return(0))); 1 \\ Charles R Greathouse IV, Apr 20 2015

A245478 Numbers k such that the k-th cyclotomic polynomial has a root mod 5.

Original entry on oeis.org

1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500, 625, 1250, 2500, 3125, 6250, 12500, 15625, 31250, 62500, 78125, 156250, 312500, 390625, 781250, 1562500, 1953125, 3906250, 7812500, 9765625, 19531250, 39062500, 48828125, 97656250, 195312500, 244140625, 488281250
Offset: 1

Views

Author

Eric M. Schmidt, Jul 23 2014

Keywords

Comments

Numbers of the form 2^i * 5^j for 0 <= i <= 2 and j >= 0.

Examples

			The 4th cyclotomic polynomial x^2 + 1 considered modulo 5 has a root x = 2, so 4 is in the sequence.

References

  • Trygve Nagell, Introduction to Number Theory. New York: Wiley, 1951, pp. 164-168.

Links

Crossrefs

Cf. A000079, A038754, A245479, A245480, A245481.

Programs

  • PARI
    for(n=1,10^6,if(#polrootsmod(polcyclo(n),5),print1(n,", "))) /* by definition; rather inefficient. - Joerg Arndt, Jul 28 2014 */
  • PARI
    is(n)=n%8 && 2^valuation(n,2)*5^valuation(n,5)==n \\ Charles R Greathouse IV, Jul 29 2014
  • PARI
    Vec(-x*(4*x^2+2*x+1)/(5*x^3-1) + O(x^100)) \\ Colin Barker, Aug 01 2014
  • Sage
    def A245478(n) : return 2^((n-1)%3)*5^((n-1)//3)

Formula

a(3j + i) = 2^(i-1)*5^j for i = 1,2,3 and j >= 0.
a(n) = 5*a(n-3). G.f.: -x*(4*x^2+2*x+1) / (5*x^3-1). - Colin Barker, Aug 01 2014

A245479 Numbers n such that the n-th cyclotomic polynomial has a root mod 7.

Original entry on oeis.org

1, 2, 3, 6, 7, 14, 21, 42, 49, 98, 147, 294, 343, 686, 1029, 2058, 2401, 4802, 7203, 14406, 16807, 33614, 50421, 100842, 117649, 235298, 352947, 705894, 823543, 1647086, 2470629, 4941258, 5764801, 11529602, 17294403, 34588806, 40353607, 80707214, 121060821
Offset: 1

Views

Author

Eric M. Schmidt, Jul 23 2014

Keywords

Comments

Numbers of the form d*7^j for d = 1,2,3,6.

Examples

			The 3rd cyclotomic polynomial x^2 + x + 1 considered modulo 7 has a root x = 2, so 3 is in the sequence.

References

  • Trygve Nagell, Introduction to Number Theory. New York: Wiley, 1951, pp. 164-168.

Links

Crossrefs

Cf. A018379, A038754, A245478, A245480, A245481.

Programs

  • Mathematica
    m = 7; Function[d, Table[d[[k]] m^n, {n, 0, 9}, {k, Length@ d}]]@ Divisors[m - 1] // Flatten (* or *)
Rest@ CoefficientList[Series[-x (2 x + 1) (3 x^2 + 1)/(7 x^4 - 1), {x, 0, 40}], x] (* Michael De Vlieger, Jul 25 2016 *)
LinearRecurrence[{0,0,0,7},{1,2,3,6},50] (* Harvey P. Dale, Oct 10 2018 *)
  • PARI
    for(n=1,10^6,if(#polrootsmod(polcyclo(n),7),print1(n,", "))) /* by definition; rather inefficient. - Joerg Arndt, Jul 28 2014 */
  • PARI
    Vec(-x*(2*x+1)*(3*x^2+1)/(7*x^4-1) + O(x^100)) \\ Colin Barker, Jul 30 2014
  • Sage
    def A245479(n) : return [6,1,2,3][n%4]*7^((n-1)//4)

Formula

a(n) = 7*a(n-4). G.f.: -x*(2*x+1)*(3*x^2+1) / (7*x^4-1). - Colin Barker, Jul 30 2014
From Benedict W. J. Irwin, Jul 22 2016: (Start)
a(n) appears to satisfy x*Prod_{n>=0} (1 + a(2^n+1)x^(2^n)) = Sum_{n>=1} a(n)x^n.
Then a(n+1)=a(2^x+1)a(2^y+1)a(2^z+1)..., where n=2^x+2^y+2^z+... .
For example,
n=12=2^2+2^3, then a(12+1)=a(2^2+1)*a(2^3+1) i.e. 343=49*7.
n=31=2^0+2^1+2^2+2^3+2^4, then a(31+1)=a(2)*a(3)*a(5)*a(9)*a(17) i.e. 4941258=2*3*7*49*2401.
(End)

A245480 Numbers n such that the n-th cyclotomic polynomial has a root mod 11.

Original entry on oeis.org

1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605, 1210, 1331, 2662, 6655, 13310, 14641, 29282, 73205, 146410, 161051, 322102, 805255, 1610510, 1771561, 3543122, 8857805, 17715610, 19487171, 38974342, 97435855, 194871710, 214358881, 428717762, 1071794405, 2143588810
Offset: 1

Views

Author

Eric M. Schmidt, Jul 23 2014

Keywords

Comments

Numbers of the form d*11^j for d=1,2,5,10.

Examples

			The 5th cyclotomic polynomial x^4 + x^3 + x^2 + x + 1 considered modulo 11 has a root x = 3, so 5 is in the sequence.

References

  • Trygve Nagell, Introduction to Number Theory. New York: Wiley, 1951, pp. 164-168.

Links

Crossrefs

Cf. A000079, A038754, A245478, A245479, A245481.

Programs

  • Mathematica
    CoefficientList[Series[x(2x+1)(5x^2+1)/(1-11x^4), {x, 0, 20}], x] (* Benedict W. J. Irwin, Jul 24 2016 *)
LinearRecurrence[{0,0,0,11},{1,2,5,10},40] (* Harvey P. Dale, Aug 04 2021 *)
  • PARI
    for(n=1,10^6,if(#polrootsmod(polcyclo(n),11),print1(n,", "))) /* by definition; rather inefficient. - Joerg Arndt, Jul 28 2014 */
  • PARI
    a(n)=11^((n-1)\4)*[10,1,2,5][n%4+1] \\ Charles R Greathouse IV, Jun 11 2015
  • Sage
    def A245480(n) : return [10,1,2,5][n%4]*11^((n-1)//4)

Formula

From Benedict W. J. Irwin, Jul 29 2016: (Start)
a(n) = 11*a(n-4).
G.f.: x*(1 + 2*x)*(1 + 5*x^2)/(1 - 11*x^4).
a(n) appears to satisfy x*Prod_{n>=0} (1 + a(2^n+1)x^(2^n)) = Sum_{n>=1} a(n)*x^n.
Then a(n+1) = a(2^x+1)*a(2^y+1)*a(2^z+1)..., where n=2^x+2^y+2^z+... .
For example, n=31=2^0+2^1+2^2+2^3+2^4, then a(31+1)=a(2)*a(3)*a(5)*a(9)*a(17) i.e. 194871710=2*5*11*121*14641.
(End)

A246946 a(1)=6; for n > 1, a(n) is the least integer not occurring earlier such that a(n) shares exactly two distinct prime divisors with a(n-1).

Original entry on oeis.org

6, 12, 18, 24, 30, 10, 20, 40, 50, 60, 15, 45, 75, 90, 36, 42, 14, 28, 56, 70, 35, 105, 21, 63, 84, 48, 54, 66, 22, 44, 88, 110, 55, 165, 33, 99, 132, 72, 78, 26, 52, 104, 130, 65, 195, 39, 117, 156, 96, 102, 34, 68, 136, 170, 80, 100, 120, 108, 114, 38, 76
Offset: 1

Views

Author

Michel Lagneau, Sep 08 2014

Keywords

Comments

All terms belong to A024619. Is this a permutation of A024619? - Michel Marcus, Nov 23 2015

Examples

			18 is in the sequence because the common prime distinct divisors between a(2)=12 and a(3)=18 are 2 and 3.

Links

Crossrefs

Cf. A184992, A245481, A246947.

Programs

  • Maple
    with(numtheory):a0:={2,3}:lst:={}:
for n from 6 to 100 do:
  ii:=0:
    for k from 3 to 50000 while(ii=0) do:
      y:=factorset(k):n0:=nops(y):lst1:={}:
        for j from 1 to n0 do:
        lst1:=lst1 union {y[j]}:
        od:
         a1:=a0 intersect lst1:
         if {k} intersect lst ={} and a1 <> {} and nops(a1)=2
          then
          printf(`%d, `,k):lst:=lst union {k}:a0:=lst1:ii:=1:
         else
         fi:
      od:
  od:
  • Mathematica
    f[s_List]:=Block[{m=s[[-1]],k=6},While[MemberQ[s,k]||Intersection[Transpose[FactorInteger[k]][[1]],Transpose[FactorInteger[m]][[1]]]=={}|| Length[Intersection[Transpose[FactorInteger[k]][[1]],Transpose[FactorInteger[m]][[1]]]]!=2,k++];Append[s,k]];Nest[f,{6},71]
  • PARI
    lista(nn) = {a = 6; print1(a, ", "); fa = (factor(a)[,1])~; va = [a]; k = 0; while (k != nn, k = 1; while (!((#setintersect(fa, (factor(k)[,1])~) == 2) && (! vecsearch(va, k))), k++); a = k; print1(a, ", "); fa = (factor(a)[,1])~; va = vecsort(concat(va, k)););} \\ Michel Marcus, Nov 23 2015

A253236 The unique prime p <= n such that n-th cyclotomic polynomial has a root mod p, or 0 if no such p exists.

Original entry on oeis.org

0, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 0, 13, 7, 0, 2, 17, 3, 19, 5, 7, 11, 23, 0, 5, 13, 3, 0, 29, 0, 31, 2, 0, 17, 0, 0, 37, 19, 13, 0, 41, 7, 43, 0, 0, 23, 47, 0, 7, 5, 0, 13, 53, 3, 11, 0, 19, 29, 59, 0, 61, 31, 0, 2, 0, 0, 67, 17, 0, 0, 71, 0
Offset: 1

Views

Author

Eric Chen, Apr 07 2015

Keywords

Comments

There is at most one prime p <= n such that n-th cyclotomic polynomial has a root mod p.
For prime n and every natural number k, a(n^k) = n.
If a(n) != 0, then a(n)|n.
a(n) is either 0 or A006530(n). See Corollary 23 of the Shevelev et al. link. - Robert Israel, Sep 07 2016

Links

Crossrefs

For a(n) = 0, see A253235.
For a(n) = 2, see A000079.
For a(n) = 3, see A038754.
For a(n) = 5, see A245478.
For a(n) = 7, see A245479.
For a(n) = 11, see A245480.
For a(n) = 13, see A245481.
Cf. A006530.

Programs

  • Maple
    N:= 1000: # to get a(1) to a(N)
f:= proc(n) local p,x,C,v;
    C:= numtheory:-cyclotomic(n,x);
    p:= max(numtheory:-factorset(n));
    for v from 0 to p-1 do
      if eval(C,x=v) mod p = 0 then return p fi
    od:
    0
end proc:
f(1):= 0:
seq(f(n),n=1..N); # Robert Israel, Sep 07 2016
  • Mathematica
    a[n_] := Module[{p, x, c, v}, c[x_] = Cyclotomic[n, x]; p = FactorInteger[ n][[-1, 1]]; For[v=0, vJean-François Alcover, Jul 27 2020, after Maple *)
  • PARI
    a(n) = forprime(p=2, n, if(#polrootsmod(polcyclo(n), p), return(p)))
  • PARI
    a(n)=my(P=polcyclo(n),f=factor(n)[,1]); for(i=1,#f, if(#polrootsmod(P, f[i]), return(f[i]))); 0 \\ Charles R Greathouse IV, Apr 07 2015
Showing 1-6 of 6 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.