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.

A080569 a(n) is the first number in the first run of at least n successive numbers, all having exactly 3 distinct prime factors.

Original entry on oeis.org

30, 230, 644, 1308, 2664, 6850, 10280, 39693, 44360, 48919, 218972, 526095, 526095, 526095, 17233173, 127890362, 29138958036, 146216247221, 23671413563491, 36966736685739
Offset: 1

Views

Author

Randy L. Ekl, Feb 21 2003

Keywords

Comments

The 19th term, if it exists, is at least 1.1 * 10^12. - Fred Schneider, Jan 05 2008
There can be at most 209 terms in this sequence. Any list of 210 consecutive numbers must contain a number n which is multiple of 2*3*5*7 = 210. So omega(n) would be >3. - Fred Schneider, Jan 05 2008
Eggleton and MacDougall show that there are no more than 59 terms in this sequence. [From T. D. Noe, Oct 13 2008]
a(19) > 10^13. - Donovan Johnson, Jun 11 2013
a(19) <= 7523987244435061. - Donovan Johnson, Jul 08 2013
a(21) > 2 * 10^15, if it exists. - Toshitaka Suzuki, Jun 23 2025

Examples

			a(3) = 644 because 644 = 2^2 * 7 * 23, so omega(644) = 3, 645 = 3*5*43, so omega(645) = 3 and 646 = 2*17*19, so omega(646) = 3 and no other number n < 644 has omega(n)=omega(n+1)=omega(n+2)=3.

Links

Crossrefs

Cf. A064708, A064709, A185032, A048932.

Programs

  • Mathematica
    k = 1; Do[ While[ Union[ Table[ Length[ FactorInteger[i]], {i, k, k + n - 1}]] != {3}, k++ ]; Print[k], {n, 1, 16}]
  • PARI
    k=1; for(i=1,600000,s=1; for(j=1,k,if(omega(i+j-1)!=3,s=0,)); if(s==1,print1(i,", "); k++; i--,) )

Extensions

Edited and extended by Robert G. Wilson v, Feb 22 2003
More terms from Don Reble, Mar 02 2003
a(19)-a(20) from Toshitaka Suzuki, Apr 01 2025

A064708 Initial term of run of (at least) n consecutive numbers with just 2 distinct prime factors.

Original entry on oeis.org

6, 14, 20, 33, 54, 91, 141, 141
Offset: 1

Views

Author

Robert G. Wilson v, Oct 13 2001

Keywords

Comments

It can be shown by an application of Mihailescu's theorem that a(12) does not exist, since then there would be two 3-smooth numbers close together (it suffices to check up to 2*3^3).
If a(9) exists, it is greater than 10^30. - Don Reble, Mar 02 2003
If a(9) exists, it is greater than 10^3000. - Charles R Greathouse IV, Apr 22 2009
Eggleton and MacDougall prove that no terms exist beyond a(9) and conjecture that a(9) does not exist. - Jason Kimberley, Jul 08 2017

Examples

			6 = 2*3; 14 = 2*7 and 15 = 3*5; 20 = 2^2*5, 21 = 3*7 and 22 = 2*11; 33 = 3*11, 34 = 2*17, 35 = 5*7 and 36 = (2*3)^2; etc.

Links

Crossrefs

Cf. A064709.

Programs

  • Mathematica
    Function[s, Function[t, Reverse@ FoldList[If[#2 > #1, #1, #2] &, Reverse[#]] &@ Map[t[[First@ FirstPosition[t[[All, -1]], k_ /; k == #] ]] &, Range[0, Max@ t[[All, -1]] ] ][[All, 1]] ]@ Join[{{First@ s, 0}, {#[[1, 1, 1]], 1}}, Rest@ Map[{#[[1, 1]], Length@ # + 1} &, #, {1}]] &@ SplitBy[Partition[Select[#, Last@ # == 1 &][[All, 1]], 2, 1], Differences] &@ Map[{First@ #, First@ Differences@ #} &, Partition[s, 2, 1]]]@ Select[Range[10^5], PrimeNu[#] == 2 &] (* Michael De Vlieger, Jul 17 2017 *)

A185032 Initial term of first run of exactly n consecutive numbers with 3 distinct prime factors.

Original entry on oeis.org

30, 230, 644, 1308, 2664, 6850, 10280, 39693, 44360, 48919, 218972, 534078, 2699915, 526095, 17233173, 127890362, 29138958036, 146216247221, 23671413563491, 36966736685739
Offset: 1

Views

Author

Roger B. Eggleton, Jason Kimberley, and James A. MacDougall, Apr 12 2011

Keywords

Comments

The number of distinct prime factors is A001221.
If any following terms exist, they are greater than 10^13.
Eggleton and MacDougall show that there are no more than 59 terms in this sequence.
a(19) <= 7523987244435061. - Donovan Johnson, Jul 08 2013
a(21) > 2 * 10^15, if it exists. - Toshitaka Suzuki, Jun 23 2025

Examples

			a(14) < a(13) because the first run of 13 consecutive integers i with A001221(i)=3 is not a maximal run.

Links

Crossrefs

Cf. A080569, A064709, and A185042.

Extensions

a(19)-a(20) from Toshitaka Suzuki, Mar 24 2025

A088983 Numbers n such that each of the 6 consecutive numbers n through n+5 has exactly two distinct prime factors.

Original entry on oeis.org

91, 141, 142, 143, 212, 213, 214, 323, 324, 2302, 2303
Offset: 1

Views

Author

Labos Elemer, Sep 30 2003

Keywords

Comments

Initial segment of A045934 is identical to this sequence but in A045934 the 12th term is divisible by 3 prime factors. Is the present sequence complete?
No more terms < 3*10^8. - David Wasserman, Aug 29 2005
a(12) > 10^40, if it exists. - Giovanni Resta, May 10 2017
From David A. Corneth, May 14 2017: (Start)
We're looking for at least 6 consecutive positive integers that each have exactly two distinct prime divisors. I.e. 6 consecutive positive integers m with omega(m) = 2. Now of exactly 6 consecutive integers, exactly one of them is divisible by 6, i.e. m is of the form 2*3*k. However m has exactly 2 distinct prime divisors, so k can only have prime divisors 2 or 3. Now, suppose m ends in 6 or higher. Then one of the consecutive integers is divisible by 10 = 2*5. I.e. it's of the form 2*5*t. Then t can only have prime divisors 2 and 5. (End)
This sequence has no run of four consecutive integers, since Eggleton and MacDougall prove that there are no more than 9 consecutive integers with A001221(k) = 2. They conjecture that A007774 contains no runs of 9 consecutive integers, and has only two runs of size 8 (at 141 and 212) and two maximal runs of size 7 (at 323 and 2302); they add that the maximal run of size 6 at 91 might be the only such run, so A088983 might be complete. - Roger Eggleton via Jason Kimberley, Jul 12 2017

Links

Crossrefs

Cf. A064709, A045932, A045933, A045934, A045935, A074851, A006073, A003586.

Programs

  • Mathematica
    Select[Range[3000], AllTrue[# + Range[0, 5], Length@FactorInteger[#] == 2 &] &] (* Giovanni Resta, May 09 2017 *)

Extensions

Definition simplified by Roger Eggleton via Jason Kimberley, Jul 12 2017

A185042 Initial term of first run of exactly n consecutive numbers with 4 distinct prime factors.

Original entry on oeis.org

210, 7314, 37960, 134043, 357642, 2713332, 1217250, 14273478, 44939642, 76067298, 163459742, 547163235, 2081479430, 2771263512, 11715712410, 17911205580, 56608713884, 203594236366, 118968284928, 2500769994070, 3157129230489, 22498525938216, 585927201062
Offset: 1

Views

Author

Roger B. Eggleton, Jason Kimberley, and James A. MacDougall, Apr 12 2011

Keywords

Comments

The number of distinct prime factors is A001221.
a(23) = 585927201062; a(n) > 10^13 for n = 20, 21, 22, and n >= 24, if they exist.
Eggleton and MacDougall show that there are no more than 419 terms in this sequence.
a(28) > 2 * 10^15. - Toshitaka Suzuki, Jun 22 2025

Examples

			a(6) > a(7) because the first run of 6 consecutive integers i with A001221(i)=4 is not maximal.

Links

Crossrefs

Cf. A064709, A185032, A087977.

Extensions

a(20)-a(22) from and a(23) added by Toshitaka Suzuki, Mar 24 2025

A136012 Numbers n in any run of two or more numbers each having exactly two distinct prime divisors.

Original entry on oeis.org

14, 15, 20, 21, 22, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 68, 69, 74, 75, 76, 77, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 133, 134, 135, 136, 141, 142, 143, 144
Offset: 1

Views

Author

Enoch Haga, Dec 09 2007

Keywords

Examples

			a(3) to a(5), 20 to 22, are a run of 3 consecutive numbers each having exactly 2 distinct prime divisors.

Crossrefs

Cf. A001221, A064709.

Extensions

Corrected and edited by R. J. Mathar, Dec 12 2007
Showing 1-6 of 6 results.

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