A140078 -id:A140078 - cp's OEIS frontend

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

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

A140079 Numbers n such that n and n+1 have 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

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

A074851 Numbers k such that k and k+1 both have exactly 2 distinct prime factors.

Original entry on oeis.org

14, 20, 21, 33, 34, 35, 38, 39, 44, 45, 50, 51, 54, 55, 56, 57, 62, 68, 74, 75, 76, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 111, 115, 116, 117, 118, 122, 123, 133, 134, 135, 141, 142, 143, 144, 145, 146, 147, 152, 158, 159, 160, 161, 171, 175, 176, 177, 183, 184

Offset: 1

Benoit Cloitre, Sep 10 2002

Subsequence of A006049. - Michel Marcus, May 06 2016

			20=2^2*5 21=3*7 hence 20 is in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A006049, A006549, A001221.
Analogous sequences for m distinct prime factors: this sequence (m=2), A140077 (m=3), A140078 (m=4), A140079 (m=5), A273879 (m=6).
Cf. A093548.
Equals A255346 \ A321502.

Filtered([1..200],n->[Size(Set(Factors(n))),Size(Set(Factors(n+1)))]=[2,2]); # Muniru A Asiru, Dec 05 2018

[n: n in [2..200] | #PrimeDivisors(n) eq 2 and #PrimeDivisors(n+1) eq 2]; // Vincenzo Librandi, Dec 05 2018

Flatten[Position[Partition[Table[If[PrimeNu[n]==2,1,0],{n,200}],2,1],{1,1}]] (* Harvey P. Dale, Mar 12 2015 *)

isok(n) = (omega(n) == 2) && (omega(n+1) == 2); \\ Michel Marcus, May 06 2016

import sympy
from sympy.ntheory.factor_ import primenu
for n in range(1,200):
    if primenu(n)==2 and primenu(n+1)==2:
        print(n, end=', '); # Stefano Spezia, Dec 05 2018

a(n) seems to be asymptotic to c*n*log(n)^2 with c=0.13...

{k: A001221(k) = A001221(k+1) = 2}. - R. J. Mathar, Jul 18 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

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
```

A176167 First of a triple of consecutive integers, each the product of 4 distinct primes.

Original entry on oeis.org

203433, 214489, 225069, 258013, 294593, 313053, 315721, 352885, 389389, 409353, 418845, 421629, 452353, 464385, 478905, 485133, 500905, 508045, 508989, 526029, 528409, 538745, 542269, 542793, 548301, 556869, 559689, 569065, 571233, 579885

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

nonn

A subsequence of A242607 and A016813. - M. F. Hasler, May 19 2014

			203433 is a term: 203433 = 3*19*43*83, 203434 = 2*7*11*1321, 203435 = 5*23*29*61.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A039833, A066509, A192203. Subsequence of A140078 and of A318896.

f1[n_]:=Last/@FactorInteger[n]=={1,1,1,1};f2[n_]:=Max[Last/@FactorInteger[n]];lst={};Do[If[f1[n]&&f1[n+1]&&f1[n+2],AppendTo[lst,n]],{n,5*8!,7*9!}];lst

forstep(n=1+10^5,10^7,4, for(k=n,n+2,issquarefree(k)||next(2)); for(k=n,n+2,omega(k)==4||next(2));print1((n)",")) \\ M. F. Hasler, May 19 2014

A318896 Numbers k such that k and k+1 are the product of exactly four distinct primes.

Original entry on oeis.org

7314, 8294, 8645, 11570, 13629, 13845, 15105, 15554, 16554, 17390, 17654, 18290, 19005, 20405, 20769, 21489, 22010, 22154, 23001, 23114, 23529, 24530, 24765, 24870, 24969, 25346, 26690, 26894, 26961, 27434, 27965, 28105, 29145, 29210, 29414, 29469, 29666, 30414

Offset: 1

Seiichi Manyama, Sep 05 2018

nonn

This sequence is different from A140078. For example, A140078(4) = 9009 = 3^2 * 7 * 11 * 13 is not a term.

			n | a(n)                    | a(n)+1
--+-------------------------+-------------------------
1 | 7314 = 2 *  3 * 23 * 53 | 7315 = 5 * 7 * 11 *  19
2 | 8294 = 2 * 11 * 13 * 29 | 8295 = 3 * 5 *  7 *  79
3 | 8645 = 5 *  7 * 13 * 19 | 8646 = 2 * 3 * 11 * 131

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Subsequence of A140078.

Cf. A046386, A052215, A215217, A263990, A318964.

is(n) = omega(n)==4 && omega(n+1)==4 && bigomega(n)==4 && bigomega(n+1)==4 \\ Felix Fröhlich, Sep 05 2018

is(n) = factor(n)[, 2]~ == [1, 1, 1, 1] && factor(n+1)[, 2]~ == [1, 1, 1, 1] \\ David A. Corneth, Sep 06 2018

A273879 Numbers k such that k and k+1 have 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

Offset: 1

Charles R Greathouse IV, Jun 02 2016

nonn

Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite (Theorem 2).

			13516580 = 2^2 * 5 * 7 * 11 * 67 * 131 and 13516581 = 3 * 13 * 17 * 19 * 29 * 37 so 13516580 is in this sequence.

Charles R Greathouse IV, 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 Erdős on consecutive integers, arXiv:0803.2636 [math.NT], 2008.
D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim, Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers, International Mathematics Research Notices 7 (2011), pp. 1439-1450.

Numbers k such that k and k+1 have j distinct prime factors: A006549 (j=1, apart from the first term), A074851 (j=2), A140077 (j=3), A140078 (j=4), A140079 (j=5).

Cf. A006049, A093548.

SequencePosition[PrimeNu[Range[3265*10^4]],{6,6}][[All,1]] (* Harvey P. Dale, Nov 20 2021 *)

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

a(1) = A138206(2). - R. J. Mathar, Jul 15 2023

{k: k in A074969 and k+1 in A074969.} - R. J. Mathar, Jul 19 2023

A364309 Numbers k such that k, k+1 and k+2 have exactly 4 distinct prime factors.

Original entry on oeis.org

37960, 44484, 45694, 50140, 51428, 55130, 55384, 61334, 63364, 64294, 67164, 68264, 68474, 70004, 70090, 71708, 72708, 76152, 80444, 81548, 81718, 82040, 84434, 85490, 86240, 90363, 95380, 97382, 98020, 99084, 99384, 99428, 99788, 100164, 100490, 100594, 102254, 102542, 104804, 105994, 108204

Offset: 1

R. J. Mathar, Jul 18 2023

nonn

			37960 = 2^3*5*13*73, 37961 = 7*11*17*29, and 37962 = 2*3^3*19*37 each have 4 distinct prime factors, so 37960 is in the sequence.

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

Subsequence of A006073 and of A140078.
A176167 is a subsequence.
Cf. A364307 (2 factors), A364308 (3 factors), A364266 (5 factors), A364265 (6 factors), A001221, A087966, A168628.

q[n_] := q[n] = PrimeNu[n] == 4; Select[Range[10^5], q[#] && q[#+1] && q[#+2] &] (* Amiram Eldar, Oct 01 2024 *)

a(1) = A087966(3).
a(n)+1 = A168628(n).
{k: A001221(k) = A001221(k+1) = A001221(k+2) = 4}.

cp's OEIS Frontend

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

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

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A074851 Numbers k such that k and k+1 both have exactly 2 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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A321504 Numbers k such that k and k+1 each have at least 4 distinct prime factors.

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A176167 First of a triple of consecutive integers, each the product of 4 distinct primes.

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