A321495 -id:A321495 - 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

A321493 Numbers m such that m and m+1 both have at least 3, but m or m+1 has at least 4 distinct prime factors.

Original entry on oeis.org

714, 1364, 1595, 1770, 1785, 1869, 2001, 2090, 2145, 2184, 2210, 2261, 2345, 2379, 2414, 2639, 2805, 2820, 2849, 2870, 2925, 3002, 3009, 3059, 3080, 3219, 3255, 3289, 3354, 3366, 3444, 3450, 3485, 3534, 3654, 3689, 3705

Offset: 1

M. F. Hasler, Nov 13 2018

nonn

A321503 lists numbers m such that m and m+1 both have at least 3 distinct prime factors, while A140077 lists numbers such that m and m+1 have exactly 3 distinct prime factors. This sequence is the complement of the latter in the former, it consists of terms with indices (15, 60, 82, 98, 99, 104, ...) of the former.

Since m and m+1 can't share a prime factor, we have a(n)*(a(n)+1) >= p(3+4)# = A002110(7). Remarkably enough, a(1) = A000196(A002110(3+4)) exactly!

M. F. Hasler, Table of n, a(n) for n = 1..5000

Cf. A140077, A321503.

Cf. A321494, A321495, A321496, A321497 (analog for 4, 5, 6, 7 factors).

aQ[n_]:=Module[{v={PrimeNu[n],PrimeNu[n+1]}},Min[v]>2 && v!={3,3}]; Select[Range[120000], aQ] (* Amiram Eldar, Nov 12 2018 *)
dpfQ[{a_,b_}]:=a>2&&b>2&&(a>3||b>3); Position[Partition[PrimeNu[Range[4000]],2,1],?dpfQ]//Flatten (* _Harvey P. Dale, Apr 21 2025 *)

select( is(n)=omega(n)>2&&omega(n+1)>2&&(omega(n)>3||omega(n+1)>3), [1..1300])

A321503 \ A140077.

A321496 Numbers k such that k and k+1 have at least 6 but not both exactly 6 distinct prime factors.

Original entry on oeis.org

58524465, 64822394, 130578734, 133595384, 143489709, 153124685, 155197965, 156532089, 157955720, 159653409, 165706904, 169075829, 170118234, 175458920, 184867605, 188377475, 193042394, 194236679, 195719810, 199150665, 202429590, 208727805, 209801514, 211588454

Offset: 1

M. F. Hasler, Nov 12 2018

nonn

Complement of A273879 (k and k+1 have exactly 6 distinct prime factors) in A321506 (k and k+1 have at least 6 distinct prime factors).

Cf. A273879, A321506; A321495 (analog for 5 factors).

aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>5 && v!={6, 6}]; Select[Range[10^9], aQ] (* Amiram Eldar, Nov 12 2018 *)

is(n)=omega(n)>5&&omega(n+1)>5&&(omega(n)>6||omega(n+1)>6)

A321506 \ A273879.

More terms from Amiram Eldar, Nov 12 2018

A321505 Numbers k such that k and k+1 each have at least 5 distinct prime factors.

Original entry on oeis.org

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, 707420, 712425, 720005, 720290, 728364, 743589

Offset: 1

M. F. Hasler, Nov 12 2018

nonn

Equals A140079 up to a(34) but a(35) = 728364 is not in A140079, see A321495.

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

Cf. A140079 (variant with "exactly 5"), A321495 (terms not in A140079).

Cf. A321504 (analog for k=4 prime factors), A321506 (analog for k=6).

Select[Range[750000], PrimeNu[#] > 4 && PrimeNu[# + 1] > 4 &] (* Amiram Eldar, Nov 12 2018 *)

PARI
```
is(n)=omega(n)>=5&&omega(n+1)>=5
```

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)

A321502 Numbers m such that m and m+1 have at least 2, but m or m+1 has at least 3 prime divisors.

Original entry on oeis.org

65, 69, 77, 84, 90, 104, 105, 110, 114, 119, 129, 132, 140, 153, 154, 155, 164, 165, 170, 174, 182, 185, 186, 189, 194, 195, 203, 204, 209, 219, 220, 221, 230, 231, 234, 237, 245, 246, 252, 254, 258, 259, 260, 264, 265, 266, 272, 273, 275, 279, 284, 285, 286, 290, 294, 299, 300, 305

Offset: 1

M. F. Hasler, Nov 27 2018

nonn

Since m and m+1 cannot have a common factor, m(m+1) has at least 2+3 prime divisors (= distinct prime factors), whence m+1 > sqrt(primorial(5)) ~ 48. It turns out that a(1)*(a(1)+1) = 2*3*5*11*13, i.e., the prime factor 7 is not present.

Cf. A321493, A321494, A321495, A321496, A321497 (analog for k = 3, ..., 7 prime divisors).
Cf. A074851, A140077, A140078, A140079 (m and m+1 have exactly k = 2, 3, 4, 5 prime divisors).
Cf. A255346, A321503 .. A321506, A321489 (m and m+1 have at least 2, ..., 7 prime divisors).
Cf. A006049, A006549, A093548.

select( is_A321502(n)=vecmax(n=[omega(n), omega(n+1)])>2&&vecmin(n)>1, [1..500])

Equals A255346 \ A074851.

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

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A321493 Numbers m such that m and m+1 both have at least 3, but m or m+1 has at least 4 distinct prime factors.

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A321496 Numbers k such that k and k+1 have at least 6 but not both exactly 6 distinct prime factors.

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A321505 Numbers k such that k and k+1 each have at least 5 distinct prime factors.

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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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A321502 Numbers m such that m and m+1 have at least 2, but m or m+1 has at least 3 prime divisors.

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