A140079 -id:A140079 - 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

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

A364266 The first term in a chain of at least 3 consecutive numbers each with exactly 5 distinct prime factors.

Original entry on oeis.org

1042404, 3460280, 3818828, 3998664, 4638984, 4991964, 5540248, 5701254, 5715500, 5964958, 6772050, 6794084, 7237384, 7453964, 7459088, 7745318, 7757034, 7993194, 8083634, 8153430, 8168194, 8273628, 8340834, 8340980, 8414756, 8486994, 8698898, 8722634, 8758904

Offset: 1

R. J. Mathar, Jul 16 2023

nonn

			1042404 = 2^2*3*11*53*149, 1042405 = 5*6*143*29*79 and 1042406 = 2*17*23*31*43 each have 5 distinct prime factors, so 1042404 is in the sequence.

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

Cf. A192203 (subsequence for squarefree triples). Subsequence of A140079 (2 consec.) and of A006073.

Cf. A364308 (3 dist. factors), A364309 (4 dist. factors), A364265 (6 dist. factors), A001221, A087978.

omega := proc(n)
    nops(numtheory[factorset](n)) ;
end proc:
for k from 1 do
    if omega(k) = 5 then
        if omega(k+1) = 5 then
            if omega(k+2) = 5 then
                print(k) ;
            end if;
        end if;
    end if;
end do:

seq[lim_] := Module[{s  = {}, q1 = False, q2 = False, q3}, Do[q3 = PrimeNu[k] == 5; If[q1 && q2 && q3, AppendTo[s, k-2]]; q1 = q2; q2 = q3, {k, 3, lim}]; s]; seq[10^7] (* Amiram Eldar, Oct 01 2024 *)

a(1) = A087978(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 5}. - R. J. Mathar, Jul 18 2023

A192203 Numbers k such that k, k+1, and k+2 are each the product of exactly 5 distinct primes.

Original entry on oeis.org

16467033, 18185869, 21134553, 21374353, 21871365, 22247553, 22412533, 22721585, 24845313, 25118093, 25228929, 25345333, 25596933, 26217245, 27140113, 29218629, 29752345, 30323733, 30563245, 31943065, 32663265, 33367893, 36055045, 38269021, 39738061, 40547065

Offset: 1

Gil Broussard, Jun 25 2011

nonn

Numbers k such that k, k+1, and k+2 are all members of A046387. - N. J. A. Sloane, Jul 17 2024
A subsequence of A242608 intersect A016813. - M. F. Hasler, May 19 2014
All terms are congruent to 1 mod 4. - Zak Seidov, Dec 22 2014

			a(1)=16467033 because it is the product of 5 distinct primes (3,11,17,149,197), and so are a(1)+1: 16467034 (2,19,23,83,227), and a(1)+2: 16467035 (5,13,37,41,167).

Zak Seidov, Table of n, a(n) for n = 1..154

Cf. A046387, A140079. Subsequence of A318964 and of A364266.

Cf. A039833, A066509, A176167.

SequencePosition[Table[If[PrimeNu[n]==PrimeOmega[n]==5,1,0],{n,164*10^5,406*10^5}],{1,1,1}][[;;,1]]+164*10^5-1 (* Harvey P. Dale, Jul 17 2024 *)

forstep(n=1+10^7,1e8,4, for(k=n,n+2,issquarefree(k)||next(2)); for(k=n,n+2,omega(k)==5||next(2));print1((n)", ")) \\ M. F. Hasler, May 19 2014

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

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.

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
```

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

A318964 Numbers k such that both k and k+1 are the product of exactly five distinct primes.

Original entry on oeis.org

378014, 421134, 483405, 486590, 486794, 489345, 507129, 545258, 549185, 558789, 558830, 634809, 637329, 663585, 667029, 690234, 720290, 776985, 782690, 823745, 824109, 853005, 853034, 855645, 873885, 883245, 892905, 935714, 945230, 968253, 987734, 999005, 1005081, 1013726

Offset: 1

Seiichi Manyama, Sep 06 2018

nonn

			n | a(n)                           | a(n)+1
--+--------------------------------+--------------------------------
1 | 378014 = 2 * 7 * 13 * 31 *  67 | 378015 = 3 *  5 * 11 * 29 * 79
2 | 421134 = 2 * 3 *  7 * 37 * 271 | 421135 = 5 * 11 * 13 * 19 * 31
3 | 483405 = 3 * 5 * 13 * 37 *  67 | 483406 = 2 *  7 * 11 * 43 * 73

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Subsequence of A140079.

Cf. A046387, A052215, A215217, A263990, A318896.

is(n) = omega(n)==5 && omega(n+1)==5 && bigomega(n)==5 && bigomega(n+1)==5 \\ Felix Fröhlich, Sep 06 2018

cp's OEIS Frontend

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

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A364266 The first term in a chain of at least 3 consecutive numbers each with exactly 5 distinct prime factors.

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A192203 Numbers k such that k, k+1, and k+2 are each the product of exactly 5 distinct primes.

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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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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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