A080569 -id:A080569 - cp's OEIS frontend

A087977 a(n) is the first term in the first chain of at least n consecutive numbers each having exactly four distinct prime factors.

210, 7314, 37960, 134043, 357642, 1217250, 1217250, 14273478, 44939642, 76067298, 163459742, 547163235, 2081479430, 2771263512, 11715712410, 17911205580, 56608713884, 118968284928, 118968284928, 585927201062, 585927201062, 585927201062, 585927201062

Offset: 1

Labos Elemer, Sep 26 2003

Eggleton and MacDougall show that there are no more than 419 terms in this sequence. - T. D. Noe, Oct 13 2008

a(28) > 2 * 10^15. - Toshitaka Suzuki, Jun 22 2025

			a(6) = a(7) = 1217250 because the relevant 7 successive numbers have 4 distinct prime factors:
  1217250 = 2   *  3^2 *   5^3 * 541;
  1217251 = 7   * 17   *  53   * 193;
  1217252 = 2^2 * 23   * 101   * 131;
  1217253 = 3   * 47   *  89   *  97;
  1217254 = 2   * 19   * 103   * 311;
  1217255 = 5   * 13   *  61   * 307;
  1217256 = 2^3 *  3   *  67   * 757.

Toshitaka Suzuki, Table of n, a(n) for n = 1..27 (terms 1..23 from Donovan Johnson).
Roger B. Eggleton and James A. MacDougall, Consecutive integers with equally many principal divisors, Math. Mag. 81 (2008), 235-248.

Cf. A080569 (m=3), A064708 (m=2).

Cf. A087978, A138206, A138207, A154573.

k=1; Do[While[Union[Table[Length[FactorInteger[i]], {i, k, k+n-1}]]!={4}, k++ ]; Print[k], {n, 1, 8}]
Module[{d4=Table[If[PrimeNu[n]==4,1,0],{n,143*10^5}]},Flatten[Table[ SequencePosition[d4,PadRight[{},n,1],1],{n,8}],1][[All,1]]] (* Requires Mathematica version 10 or later *) (* This generates the first 8 terms of the sequence *) (* Harvey P. Dale, Aug 25 2017 *)

More terms from Don Reble, Sep 29 2003
a(13)-a(19) from Donovan Johnson, Mar 06 2008
a(20)-a(23) from Donovan Johnson, Jan 15 2009

A087978 a(n) is the first term in a chain of at least n consecutive numbers, each having exactly m = 5 distinct prime factors.

Original entry on oeis.org

2310, 254540, 1042404, 21871365, 129963314, 830692265, 4617927894, 18297409143, 41268813542, 287980277114, 1182325618032, 6455097761454, 14207465691240, 54049709480208, 90987640183352, 546525829796442, 546525829796442

Offset: 1

Labos Elemer, Sep 26 2003

Every chain of 30030 consecutive numbers has exactly one number divisible by 30030 = 2 * 3 * 5 * 7 * 11 * 13 hence is divisible by more than five distinct primes. Therefore the sequence is finite. - David A. Corneth, Jul 19 2023

a(18) > 2 * 10^15. - Toshitaka Suzuki, Jun 23 2025

Roger B. Eggleton and James A. MacDougall, Consecutive Integers with Equally Many Principal Divisors, Math. Mag. 81 (2008), 235-248. [_T. D. Noe_, Oct 13 2008]

Cf. A064708 (m=2), A080569 (m=3), A087977 (m=4).
Cf. A138206, A138207, A154573. - Donovan Johnson, Jan 15 2009
Cf. A046387.

k=1; Do[While[Union[Table[Length[FactorInteger[i]], {i, k, k+n-1}]]!={5}, k++ ]; Print[k], {n, 1, 8}]

More terms from Don Reble, Sep 29 2003
a(7)-a(10) from Donovan Johnson, Mar 06 2008
a(11)-a(12) from Donovan Johnson, Jan 15 2009
a(13)-a(15) from Toshitaka Suzuki, Apr 06 2025
a(16)-a(17) from Toshitaka Suzuki, Jun 23 2025

A364308 Numbers k such that k, k+1 and k+2 have exactly 3 distinct prime factors.

Original entry on oeis.org

644, 740, 804, 986, 1034, 1064, 1104, 1220, 1274, 1308, 1309, 1462, 1494, 1580, 1748, 1884, 1885, 1924, 1988, 2013, 2014, 2108, 2134, 2254, 2288, 2294, 2330, 2354, 2364, 2408, 2464, 2484, 2540, 2583, 2584, 2664, 2665, 2666, 2678, 2684, 2714, 2715, 2716, 2754, 2793

Offset: 1

R. J. Mathar, Jul 18 2023

nonn

			644 = 2^2*7*23 has 3 distinct prime factors, 645 = 3*5*43 has 3 distinct prime factors, and 646 = 2*17*19 has 3 distinct prime factors, so 644 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Subsequence of A006073 and of A140077.

Cf. A364307 (2 factors), A364309 (4 factors), A364266 (5 factors), A364265 (6 factors), A001221, A080569.

q[n_] := q[n] = PrimeNu[n] == 3; Select[Range[3000], q[#] && q[#+1] && q[#+2] &] (* Amiram Eldar, Oct 01 2024 *)

a(1) = A080569(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 3}.

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

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

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

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

Roger B. Eggleton and James A. MacDougall, Consecutive integers with equally many principal divisors, Math. Mag. 81 (2008), 235-248.
Roger B. Eggleton, Jason S. Kimberley, and James A. MacDougall, Principal divisor ranks of the first trillion positive integers, NOVA: The University of Newcastle’s Digital Research Repository (2009).

Cf. A080569, A064709, and A185042.

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

A357075 Numbers sandwiched between numbers with exactly three distinct prime factors.

Original entry on oeis.org

131, 139, 155, 169, 181, 221, 229, 239, 259, 265, 281, 307, 309, 311, 341, 349, 365, 371, 373, 379, 407, 409, 439, 441, 443, 469, 475, 491, 493, 505, 517, 519, 521, 529, 531, 533, 551, 559, 573, 581, 589, 599, 601, 611, 617, 619, 637, 643, 645, 664, 671, 679, 681, 683

Offset: 1

Tanya Khovanova, Sep 10 2022

Number k such that both k-1 and k+1 are in A033992.

			131 is sandwiched between 130 = 2*5*13 and 132 = 2^2*3*11. Both 130 and 132 have exactly three prime factors. Thus, 131 is in this sequence.

Cf. A033992, A357074, A080569.

Select[Range[1000],Length[FactorInteger[# + 1]] == 3 && Length[FactorInteger[# - 1]] == 3 &]
Mean/@SequencePosition[Table[If[PrimeNu[n]==3,1,0],{n,700}],{1,,1}] (* _Harvey P. Dale, Jul 06 2025 *)

is(n)=omega(n-1)==3 && omega(n+1)==3 \\ Charles R Greathouse IV, Sep 11 2022

list(lim)=my(v=List(),a=3,b,c); forfactored(n=132,lim\1+1, c=#n[2]~; if(c==3 && a==3, listput(v,n[1]-1)); a=b; b=c); Vec(v) \\ Charles R Greathouse IV, Sep 28 2022

from sympy import factorint
def isA033992(n): return len(factorint(n)) == 3
def ok(n): return isA033992(n-1) and isA033992(n+1)
print([k for k in range(700) if ok(k)]) # Michael S. Branicky, Sep 10 2022

A384507 Initial term of first run of exactly n consecutive numbers with exactly 5 distinct prime factors.

Original entry on oeis.org

2310, 254540, 1042404, 21871365, 129963314, 830692265, 4617927894, 18297409143, 41268813542, 287980277114, 1182325618032, 6455097761454, 14207465691240, 54049709480208, 90987640183352, 1164838922284960, 546525829796442

Offset: 1

Toshitaka Suzuki, Jun 23 2025

See A087978 for further details and an explanation of why this sequence is finite.
First differs from A087978 at n=16.
a(18) > 2 * 10^15.

			a(16) > a(17) because the first run of 16 consecutive integers i with A001221(i)=5 is not a maximal run.

Cf. A001221, A046387, A080569, A087977, A087978, A185032, A185042.

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A087977 a(n) is the first term in the first chain of at least n consecutive numbers each having exactly four distinct prime factors.

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A087978 a(n) is the first term in a chain of at least n consecutive numbers, each having exactly m = 5 distinct prime factors.

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A364308 Numbers k such that k, k+1 and k+2 have exactly 3 distinct prime factors.

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

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A357075 Numbers sandwiched between numbers with exactly three distinct prime factors.

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A384507 Initial term of first run of exactly n consecutive numbers with exactly 5 distinct prime factors.

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