A321504 -id:A321504 - cp's OEIS frontend

A140078 Numbers k such that k and k+1 have 4 distinct prime factors.

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

Offset: 1

Artur Jasinski, May 07 2008

nonn

Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite. - Charles R Greathouse IV, Jun 02 2016

The subsequence of terms where k and k+1 are also squarefree is A318896. - R. J. Mathar, Jul 15 2023

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 161 (entry for 7314).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz and 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.

Similar sequences with k distinct prime factors: A074851 (k=2), A140077 (k=3), this sequence (k=4), A140079 (k=5).
Cf. A093548.
Equals A321504 \ A321494.

a = {}; Do[If[Length[FactorInteger[n]] == 4 && Length[FactorInteger[n + 1]] == 4, AppendTo[a, n]], {n, 1, 100000}]; a (* Artur Jasinski, May 07 2008 *)
Transpose[Position[Partition[PrimeNu[Range[20000]],2,1],?(#[[1]] == #[[2]] == 4&),{1},Heads->False]][[1]] (* _Harvey P. Dale, Jun 21 2013 *)
SequencePosition[PrimeNu[Range[22000]],{4,4}][[;;,1]] (* Harvey P. Dale, Jun 20 2024 *)

isok(n) = (omega(n)==4) && (omega(n+1)==4); \\ Michel Marcus, Sep 04 2015

{k: k in A033993 and k+1 in A033993}. - R. J. Mathar, Jul 19 2023

Link provided by Harvey P. Dale, Jun 21 2013

A140077 Numbers n such that n and n+1 have 3 distinct prime factors.

Original entry on oeis.org

230, 285, 429, 434, 455, 494, 560, 594, 609, 615, 644, 645, 650, 665, 740, 741, 759, 804, 805, 819, 825, 854, 860, 884, 902, 935, 945, 969, 986, 987, 1001, 1014, 1022, 1034, 1035, 1044, 1064, 1065, 1070, 1085, 1104, 1105, 1130, 1196, 1209, 1220, 1221

Offset: 1

Artur Jasinski, May 07 2008

nonn

Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite. - Charles R Greathouse IV, Sep 14 2015

See A321503 for numbers n such that n & n+1 have at least 3 prime divisors, disjoint union of this and A321493, the terms of A321503 which are not in this sequence. A321493 has A140078 as a subsequence, which in turn is subsequence of A321504, and so on. Since n and n+1 can't share a prime factor, we have a(1) > sqrt(p(3+3)#) > A000196(A002110(3+3)). Note that A000196(A002110(3+4)) = A321493(1) exactly! - M. F. Hasler, Nov 13 2018

Harvey P. Dale, 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, A140078, A140079.

Equals A321503 \ A321493.

a = {}; Do[If[Length[FactorInteger[n]] == 3 && Length[FactorInteger[n + 1]] == 3, AppendTo[a, n]], {n, 1, 100000}]; a (*Artur Jasinski*)
SequencePosition[PrimeNu[Range[1250]],{3,3}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 27 2017 *)

is(n)=omega(n)==3&&omega(n+1)==3 \\ Charles R Greathouse IV, Sep 14 2015

{k: k in A033992 and k+1 in A033992}. - R. J. Mathar, Jul 19 2023

A321494 Numbers k such that k and k+1 have at least 4 but not both exactly 4 distinct prime factors.

Original entry on oeis.org

38570, 40754, 51414, 51765, 58695, 60605, 62985, 66044, 68585, 70889, 71070, 73185, 73814, 74865, 77349, 82004, 83265, 83720, 83979, 85085, 87009, 90804, 90915, 91805, 91884, 92378, 94094, 94829, 96459, 97565, 98769, 98889, 100814, 101269, 101660, 104005, 104754, 105468, 107184, 108030, 108185, 108965

Offset: 1

M. F. Hasler, Nov 12 2018

nonn

A321504 lists numbers n such that k and k+1 both have at least 4 distinct prime factors, while A140078 lists numbers such that k and k+1 have exactly 4 distinct prime factors. This sequence is the complement of the latter in the former, it consists of terms with indices (124, 214, 219, 276, 321, 415, ...) of the former.

Cf. A140078, A321504; A321493, A321496 (analog for 3 & 5 factors).

aQ[n_]:=Module[{v={PrimeNu[n],PrimeNu[n+1]}},Min[v]>3 && v!={4,4}]; Select[Range[120000], aQ] (* Amiram Eldar, Nov 12 2018 *)

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

A321504 \ A140078.

A321503 Numbers m such that m and m+1 both have at least 3 distinct prime factors.

Original entry on oeis.org

230, 285, 429, 434, 455, 494, 560, 594, 609, 615, 644, 645, 650, 665, 714, 740, 741, 759, 804, 805, 819, 825, 854, 860, 884, 902, 935, 945, 969, 986, 987, 1001, 1014, 1022, 1034, 1035, 1044, 1064, 1065, 1070, 1085, 1104, 1105, 1130, 1196, 1209, 1220, 1221, 1235, 1239, 1245, 1265

Offset: 1

M. F. Hasler, Nov 13 2018

nonn

Disjoint union of A140077 (omega({m, m+1}) = {3}) and A321493 (not both have exactly 3 prime divisors). The latter contains terms with indices {15, 60, 82, 98, 99, 104, ...} of this sequence.

Numbers m and m+1 can never have a common prime factor (consider them mod p), therefore the terms are > sqrt(A002110(3+3)), A002110 = primorial.

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

Subsequence of A000977.
Cf. A255346, A321504 .. A321506, A321489 (analog for k = 2, ..., 7 prime divisors).
Cf. A321493, A321494 .. A321497 (subsequences of the above: m or m+1 has more than k prime divisors).
Cf. A074851, A140077, A140078, A140079 (complementary subsequences: m and m+1 have exactly k = 2, 3, 4, 5 prime divisors).

aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>2]; Select[Range[1300], aQ] (* Amiram Eldar, Nov 12 2018 *)

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

a(n) ~ n. - Charles R Greathouse IV, Jan 25 2025

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)

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A140078 Numbers k such that k and k+1 have 4 distinct prime factors.

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A140077 Numbers n such that n and n+1 have 3 distinct prime factors.

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A321494 Numbers k such that k and k+1 have at least 4 but not both exactly 4 distinct prime factors.

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A321503 Numbers m such that m and m+1 both have at least 3 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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