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.

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A247827 Least prime factor of A247676.

Original entry on oeis.org

5, 7, 5, 11, 5, 13, 5, 7, 11, 5, 7, 5, 13, 19, 5, 7, 5, 23, 17, 7, 13, 5, 11, 5, 19, 5, 7, 17, 5, 29, 7, 5, 11, 5, 7, 31, 5, 17, 7, 11, 5, 13, 5, 23, 19, 5, 7, 17, 5, 13, 37, 7, 5, 11, 31, 5, 7, 19, 13, 5, 11, 7, 5, 29, 5, 13, 11, 5, 7
Offset: 1

Views

Author

Odimar Fabeny, Sep 24 2014

Keywords

Links

Crossrefs

Cf. A056608, A247676.

A247870 Least prime factor of odd numbers congruent to 4 modulo 9.

Original entry on oeis.org

7, 5, 11, 5, 13, 5, 7, 11, 5, 17, 7, 5, 13, 11, 5, 7, 19, 5, 7, 17, 5, 5, 29, 5, 11, 7, 13, 5, 17, 7, 5, 11, 31, 5, 7, 23, 5, 19, 7, 5, 29, 13, 5, 11, 5, 7, 5, 23, 13, 7, 5, 5, 7, 43, 5, 11, 17, 7, 19, 5, 23, 5, 11, 13, 5, 41, 7, 47, 17, 5, 31, 11, 7, 5, 13, 29, 5, 7, 23, 37, 11, 5, 17, 7, 13, 5, 43, 19, 5, 5, 7, 17, 5, 11
Offset: 1

Views

Author

Odimar Fabeny, Sep 25 2014

Keywords

Links

Crossrefs

Cf. A056608, A247678.

A247871 Least prime factor of A247679.

Original entry on oeis.org

5, 5, 11, 7, 5, 7, 5, 17, 11, 13, 5, 7, 5, 7, 5, 13, 17, 5, 11, 5, 7, 5, 29, 7, 5, 23, 19, 5, 7, 13, 5, 11, 7, 5, 17, 5, 13, 11, 19, 5, 23, 7, 31, 5, 11, 7, 5, 5, 7, 19, 11, 5, 41, 13, 7, 23, 5, 17, 5, 29, 37, 5, 19, 7, 5, 11, 17, 7, 5, 23, 13, 5, 7, 11, 5, 7, 5, 13, 41, 11, 43, 5, 31, 37, 5, 7, 11, 5, 17, 7, 5, 5, 7, 13, 29, 5
Offset: 1

Views

Author

Odimar Fabeny, Sep 25 2014

Keywords

Links

Crossrefs

Cf. A056608, A247679.

Programs

  • Mathematica
    FactorInteger[#][[1,1]]&/@Select[Range[17,4000,18],CompositeQ] (* Harvey P. Dale, Jun 19 2022 *)

A247872 Least prime factor of A247681(n).

Original entry on oeis.org

5, 7, 5, 7, 5, 11, 17, 5, 7, 19, 5, 11, 7, 5, 13, 5, 11, 23, 5, 19, 7, 5, 13, 7, 5, 17, 5, 7, 13, 5, 23, 7, 5, 29, 17, 5, 11, 13, 5, 31, 7, 37, 19, 5, 11, 7, 5, 17, 5, 7, 11, 5, 29, 7, 5, 17, 11, 5, 31, 23, 41, 5, 13, 7, 5, 19, 7, 5, 13, 5, 7, 5, 23, 7, 5, 19, 11, 31, 5, 5, 7, 11, 5, 37, 7, 5, 47, 53, 11, 5, 7, 43, 13, 5, 7
Offset: 2

Views

Author

Odimar Fabeny, Sep 25 2014

Keywords

Links

Crossrefs

Cf. A056608, A247681.

Programs

  • Maple
    count:= 0:
for n from 1 while count < 100 do
   m:= 1+18*n;
   if not isprime(m) then
      count:= count+1;
      A[count]:= min(numtheory:-factorset(m))
   fi
od:
seq(A[i],i=1..count); # Robert Israel, Sep 30 2014

A247876 Least prime factor of A247682.

Original entry on oeis.org

7, 5, 5, 7, 13, 5, 7, 5, 19, 5, 11, 17, 5, 7, 5, 11, 13, 7, 5, 19, 5, 7, 23, 11, 5, 13, 7, 5, 11, 5, 19, 17, 13, 5, 7, 29, 5, 7, 5, 13, 5, 7, 37, 5, 7, 5, 31, 11, 23, 5, 17, 29, 5, 7, 11, 5, 7, 5, 43, 11, 5, 7, 19, 5, 37, 31, 7, 5, 29, 13, 5, 17, 5, 11, 7, 19, 23, 5, 13, 7, 5, 11, 17, 5, 7, 31, 5, 29, 19, 7, 47, 5, 23, 5, 41, 13
Offset: 1

Views

Author

Odimar Fabeny, Sep 25 2014

Keywords

Links

Crossrefs

Cf. A056608, A247682.

A247877 Least prime factor of A247683.

Original entry on oeis.org

5, 5, 7, 13, 11, 5, 7, 5, 5, 13, 5, 17, 7, 23, 5, 11, 7, 5, 5, 7, 11, 17, 19, 5, 13, 7, 5, 23, 31, 11, 5, 5, 7, 19, 11, 5, 7, 5, 13, 23, 5, 7, 17, 5, 19, 7, 29, 5, 11, 37, 5, 41, 17, 5, 7, 13, 5, 19, 7, 5, 11, 5, 7, 13, 29, 31, 5, 7, 11, 5, 37, 5, 17, 5, 7, 41, 5, 47, 13, 7, 5, 11, 29, 23, 5, 7, 5, 13, 11, 7, 5, 19, 5, 37
Offset: 1

Views

Author

Odimar Fabeny, Sep 25 2014

Keywords

Links

Crossrefs

Cf. A056608, A247683.

A306353 Number of composites among the first n composite numbers whose least prime factor p is that of the n-th composite number.

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 6, 2, 7, 8, 9, 3, 10, 11, 1, 12, 4, 13, 14, 15, 5, 16, 2, 17, 18, 6, 19, 20, 21, 7, 22, 23, 1, 24, 8, 25, 26, 3, 27, 9, 28, 29, 30, 10, 31, 4, 32, 33, 11, 34, 35, 36, 12, 37, 2, 38, 39, 13, 40, 41, 5, 42, 14, 43, 44, 3, 45, 15, 46, 6, 47, 48, 16, 49, 50, 51, 17, 52, 53, 54, 18, 55, 56, 7
Offset: 1

Views

Author

Jamie Morken and Vincenzo Librandi, Feb 09 2019

Keywords

Comments

Composites with least prime factor p are on that row of A083140 which begins with p
Sequence with similar values: A122005.
Sequence written as a jagged array A with new row when a(n) > a(n+1):
1, 2, 3,
1, 4, 5, 6,
2, 7, 8, 9,
3, 10, 11,
1, 12,
4, 13, 14, 15,
5, 16,
2, 17, 18,
6, 19, 20, 21,
7, 22, 23,
1, 24,
8, 25, 26,
3, 27,
9, 28, 29, 30.
A153196 is the list B of the first values in successive rows with length 4.
B is given by the formula for A002808(x)=A256388(n+3), an(x)=A153196(n+2)
For example: A002808(26)=A256388(3+3), an(26)=A153196(3+2).
A243811 is the list of the second values in successive rows with length 4.
A047845 is the list of values in the second column and A104279 is the list of values in the third column of the jagged array starting on the second row.
Sequence written as an irregular triangle C with new row when a(n)=1:
1,2,3,
1,4,5,6,2,7,8,9,3,10,11,
1,12,4,13,14,15,5,16,2,17,18,6,19,20,21,7,22,23,
1,24,8,25,26,3,27,9,28,29,30,10,31,4,32,33,11,34,35,36,12,37,2,38,39,13,40,41,5,42,14,43,44,3,45,15,46,6,47,48,16,49,50,51,17,52,53,54,18,55,56,7,57,19,58,4,59.
A243887 is the last value in each row of C.
The second value D on the row n > 1 of the irregular triangle C is a(A053683(n)) or equivalently A084921(n). For example for row 3 of the irregular triangle:
D = a(A053683(3)) = a(16) = 12 or D = A084921(3) = 12. This is the number of composites < A066872(3) with the same least prime factor p as the A053683(3) = 16th composite, A066872(3) = 26.
The number of values in each row of the irregular triangle C begins: 3,11,18,57,39,98,61,141,265,104,351,268,...
The second row of the irregular triangle C is A117385(b) for 3 < b < 15.
The third row of the irregular triangle C has similar values as A117385 in different order.

Examples

			First composite 4, least prime factor is 2, first case for 2 so a(1)=1.
Next composite 6, least prime factor is 2, second case for 2 so a(2)=2.
Next composite 8, least prime factor is 2, third case for 2 so a(3)=3.
Next composite 9, least prime factor is 3, first case for 3 so a(4)=1.
Next composite 10, least prime factor is 2, fourth case for 2 so a(5)=4.

Links

Crossrefs

Cf. A002808, A256388, A056608, A083140, A122005, A153196, A243811, A047845, A104279, A243887, A117385, A216244, A084921, A066872, A053683, A038110, A038111, A020639.

Programs

  • Mathematica
    counts = {}
values = {}
For[i = 2, i < 130, i = i + 1,
If[PrimeQ[i], ,
x = PrimePi[FactorInteger[i][[1, 1]]];
  If[Length[counts] >= x,
   counts[[x]] = counts[[x]] + 1;
   AppendTo[values, counts[[x]]], AppendTo[counts, 1];
   AppendTo[values, 1]]]]
   (* Print[counts] *)
   Print[values]
  • PARI
    c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808
a(n) = my(c=c(n), lpf = vecmin(factor(c)[,1]), nb=0); for(k=2, c, if (!isprime(k) && vecmin(factor(k)[,1])==lpf, nb++)); nb; \\ Michel Marcus, Feb 10 2019

Formula

a(n) is approximately equal to A002808(n)*(A038110(x)/A038111(x)), with A000040(x)=A020639(A002808(n)).
For example if n=325, a(325)~= A002808(325)*(A038110(2)/A038111(2)) with A000040(2)=A020639(A002808(325)).
This gives an estimate of 67.499... and the actual value of a(n)=67.

A079772 Let C(n) be the n-th composite number; then a(n) is the smallest number > C(n) and not coprime to C(n).

Original entry on oeis.org

6, 8, 10, 12, 12, 14, 16, 18, 18, 20, 22, 24, 24, 26, 30, 28, 30, 30, 32, 34, 36, 36, 40, 38, 40, 42, 42, 44, 46, 48, 48, 50, 56, 52, 54, 54, 56, 60, 58, 60, 60, 62, 64, 66, 66, 70, 68, 70, 72, 72, 74, 76, 78, 78, 84, 80, 82, 84, 84, 86, 90, 88, 90, 90, 92, 98, 94, 96, 96, 100
Offset: 1

Views

Author

Amarnath Murthy, Jan 31 2003

Keywords

Comments

a(n) = A002808(n) + A056608(n). - Vladeta Jovovic, Jan 31 2003

Examples

			C(4) = 9 and C(5) = 10 hence a(4) = a(5) = 12.
For 91=7*13 we have 91+7 = 98; for 92=2*2*23 we have 92-2 = 94

Programs

  • PARI
    cminusp31(n) = \sum c+min prime div { for(x=2,n, forprime(p=2,floor(sqrt(x)), if(x%p==0 & isprime(x)==0, print1(x+p,","); break); ) ) }

Formula

a(n) = C(n) + smallest prime divisor of C(n).

Extensions

More terms from Cino Hilliard, Aug 12 2003

A085428 Sum of the smallest and largest prime divisors of the n-th composite number.

Original entry on oeis.org

4, 5, 4, 6, 7, 5, 9, 8, 4, 5, 7, 10, 13, 5, 10, 15, 6, 9, 7, 4, 14, 19, 12, 5, 21, 16, 7, 9, 13, 8, 25, 5, 14, 7, 20, 15, 5, 16, 9, 22, 31, 7, 33, 10, 4, 18, 13, 19, 26, 9, 5, 39, 8, 21, 18, 15, 7, 6, 43, 9, 22, 45, 32, 13, 7, 20, 25, 34, 49, 24, 5, 9, 14, 7
Offset: 0

Views

Author

Cino Hilliard, Aug 13 2003

Keywords

Crossrefs

Equals A056608(n) + A052369(n).

Programs

  • PARI
    maxpmindivc(n) = { for(x=4,n, s=0; forstep(p=2,x-1,1, if(x%p==0 & isprime(p),s=p; break); ); forstep(p=x-1,2,-1, if(x%p==0 & isprime(p),print1(s+p,","); break); ) ) }

A109709 Triangle in which n-th row gives the prime factors of the n-th composite number with repetition.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 5, 2, 2, 3, 2, 7, 3, 5, 2, 2, 2, 2, 2, 3, 3, 2, 2, 5, 3, 7, 2, 11, 2, 2, 2, 3, 5, 5, 2, 13, 3, 3, 3, 2, 2, 7, 2, 3, 5, 2, 2, 2, 2, 2, 3, 11, 2, 17, 5, 7, 2, 2, 3, 3, 2, 19, 3, 13, 2, 2, 2, 5, 2, 3, 7, 2, 2, 11, 3, 3, 5, 2, 23, 2, 2, 2, 2, 3, 7, 7, 2, 5, 5, 3, 17, 2, 2, 13, 2, 3, 3, 3
Offset: 1

Views

Author

Lior Manor, Aug 08 2005

Keywords

Examples

			Irregular triangle begins:
  2, 2;
  2, 3;
  2, 2, 2;
  3, 3;
  2, 5;
  ...

Crossrefs

Column k=1 gives A056608.
Last terms in rows give A052369.
Row lengths give A062502(n+1).
Cf. A002808, A027746.

Programs

  • Mathematica
    fn[{a_,b_}]:=Table[a,b];Flatten[Table[fn/@FactorInteger[ ResourceFunction["Composite"][n]],{n,37}]] (* James C. McMahon, Mar 29 2024 *)
Previous Showing 21-30 of 38 results. Next

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