A321505 -id:A321505 - cp's OEIS frontend

A140079 Numbers n such that n and n+1 have 5 distinct prime factors.

254540, 310155, 378014, 421134, 432795, 483405, 486590, 486794, 488565, 489345, 507129, 522444, 545258, 549185, 558789, 558830, 567644, 577940, 584154, 591260, 598689, 627095, 634809, 637329, 663585, 666995, 667029, 678755, 687939, 690234

Offset: 1

Artur Jasinski, May 07 2008

nonn

For the smallest number r such that r and r+1 have n distinct prime factors, see A093548.
Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite. - Charles R Greathouse IV, Jun 02 2016
Subsequence of the variant A321505 defined with "at least 5" instead of "exactly 5" distinct prime factors. See A321495 for the differences. - M. F. Hasler, Nov 12 2018
The subset of numbers where n and n+1 are also squarefree gives A318964. - R. J. Mathar, Jul 15 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim., Small gaps between almost primes, the parity problem and some conjectures of Erdos on consecutive integers, arXiv:0803.2636 [math.NT], 2008.

Cf. A074851, A140077, A140078, A273897.
Cf. A093548.
Equals A321505 \ A321495.

a = {}; Do[If[Length[FactorInteger[n]] == 5 && Length[FactorInteger[n + 1]] == 5, AppendTo[a, n]], {n, 1, 100000}]; a (*Artur Jasinski*)
Transpose[SequencePosition[Table[If[PrimeNu[n]==5,1,0],{n,700000}],{1,1}]][[1]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Jul 25 2015 *)

is(n)=omega(n)==5 && omega(n+1)==5 \\ Charles R Greathouse IV, Jun 02 2016

{k: k in A051270 and k+1 in A051270}. - R. J. Mathar, Jul 19 2023

A321506 Numbers m such that m and m+1 each have at least 6 distinct prime factors.

Original entry on oeis.org

11243154, 13516580, 16473170, 16701684, 17348330, 19286805, 20333495, 21271964, 21849905, 22054515, 22527141, 22754589, 22875489, 24031370, 25348070, 25774329, 28098245, 28618394, 28625960, 30259229, 31846269, 32642805, 32734910, 33205029, 33631520, 33641894, 35023365

Offset: 1

M. F. Hasler, Nov 12 2018

nonn

Equals A273879 up to a(138) = 58524465, which is not in A273879: see A321496 for the complement.

M. F. Hasler, Table of n, a(n) for n = 1..150

Cf. A273879 (variant with "exactly 6"), A321496 (terms not in A273879).

Cf. A321505 (analog for k=5 prime factors).

Select[Range[36000000], PrimeNu[#] > 5 && PrimeNu[# + 1] > 5 &] (* Amiram Eldar, Nov 12 2018 *)
Position[Partition[PrimeNu[Range[3503*10^4]],2,1],?(#[[1]]>5&&#[[2]]> 5&), 1,Heads-> False]//Flatten (* _Harvey P. Dale, Jul 28 2020 *)

PARI
```
is(n)=omega(n)>=6&&omega(n+1)>=6
```

A321504 Numbers k such that k and k+1 each have at least 4 distinct prime factors.

Original entry on oeis.org

7314, 8294, 8645, 9009, 10659, 11570, 11780, 11934, 13299, 13629, 13845, 14420, 15105, 15554, 16554, 16835, 17204, 17390, 17654, 17765, 18095, 18290, 18444, 18920, 19005, 19019, 19095, 19227, 20349, 20405, 20769, 21164, 21489, 21735, 22010, 22154, 22659, 23001, 23114, 23484, 23529, 23540, 23919, 24395

Offset: 1

M. F. Hasler, Nov 12 2018

nonn

Equals A140078 up to a(123) but a({124, 214, 219, 276, 321, 415, ...}) = { 38570, 51414, 51765, 58695, 62985, 71070, ...} are not in A140078, see A321494.

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A140078, A321494.

Cf. A321505, A321506 (variant for k=5 & k=6 prime factors).

SequencePosition[Table[If[PrimeNu[n]>3,1,0],{n,25000}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 29 2019 *)

PARI
```
is(n)=omega(n)>=4&&omega(n+1)>=4
```

A321495 Numbers k such that k and k+1 have at least 5 but not both exactly 5 distinct prime factors.

Original entry on oeis.org

728364, 1565564, 1774409, 1817529, 1923635, 2162094, 2187185, 2199834, 2225894, 2369850, 2557190, 2594514, 2659734, 2671305, 2794154, 2944689, 2964884, 3126045, 3139730, 3170244, 3244955, 3273809, 3279639, 3382379, 3387054, 3506810, 3555110, 3585945, 3686969, 3711630

Offset: 1

M. F. Hasler, Nov 12 2018

nonn

Complement of A140079 (k and k+1 have exactly 5 distinct prime factors) in A321505 (k and k+1 have at least 5 distinct prime factors).

Cf. A140079, A321505; A321494, A321496 (analog for 4 & 6 factors).

aQ[n_]:=Module[{v={PrimeNu[n],PrimeNu[n+1]}},Min[v]>4 && v!={5,5}]; Select[Range[120000], aQ] (* Amiram Eldar, Nov 12 2018 *)

is(n)=vecmin(n=[omega(n), omega(n+1)])>4&&n!=[5,5]

A321505 \ A140079.

A321497 Numbers k such that both k and k+1 have at least 7 distinct prime factors and at least one has more than 7 distinct prime factors.

Original entry on oeis.org

5163068910, 5327923964, 6564937379, 6880516929, 7122669554, 8567026545, 8814635115, 9533531370, 9611079114, 10245081314, 10246336814, 10697507414, 10783550414, 10796559410, 11260076190, 11458770609, 11992960265, 12043540145, 12172828590, 12745759740, 12850545785, 12946979220

Offset: 1

Amiram Eldar and M. F. Hasler, Nov 13 2018

nonn

Terms of A321489 (k and k+1 have at least 7 distinct prime factors) which don't satisfy the definition with "exactly 7".

Cf. A321489, A321503, A321504, A321505, A321506, A321493, A321494, A321495, A321496 (analog for 3 .. 6 factors).

aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>6 && v!={7, 7}]; Select[Range[10^10], aQ]

is(n)=omega(n)>6&&omega(n+1)>6&&(omega(n)>7||omega(n+1)>7)

cp's OEIS Frontend

A140079 Numbers n such that n and n+1 have 5 distinct prime factors.

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A321506 Numbers m such that m and m+1 each have at least 6 distinct prime factors.

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A321504 Numbers k such that k and k+1 each have at least 4 distinct prime factors.

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A321495 Numbers k such that k and k+1 have at least 5 but not both exactly 5 distinct prime factors.

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Mathematica

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A321497 Numbers k such that both k and k+1 have at least 7 distinct prime factors and at least one has more than 7 distinct prime factors.

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Programs

Mathematica

PARI