A318896 -id:A318896 - 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

A052215 a(n) = smallest number m such that m and m+1 are the product of exactly n distinct primes.

Original entry on oeis.org

2, 14, 230, 7314, 378014, 11243154, 965009045, 65893166030, 5702759516090, 605247139068494, 78971815814237709, 22593106657425552170

Offset: 1

Erich Friedman, Jan 29 2000

Prime factors may not be repeated in m and m+1. The difference between this sequence and A093548 is that in the latter, prime factors may be repeated. So the present sequence imposes more stringent conditions than A093548, hence a(n) >= A093548(n). - N. J. A. Sloane, Nov 21 2015
A115186(n) <= A093548(n) <= a(n). - Zak Seidov, Jan 16 2015
2^63 < a(12) <= 22593106657425552170. [Donovan Johnson, Oct 23 2008]
a(12) confirmed to be the upper limit of the range above. - Bert Dobbelaere, Jun 27 2019

			14 and 15 are both the product of 2 primes.
230 is the 3rd entry because we have (230=2*5*23, 231=3*7*11).

Cf. A093548 (another version), A093549, A093550, A115186, A318896.

Subsequence of A005117.

More terms from Naohiro Nomoto, Jul 08 2001
a(7) from Farideh Firoozbakht, Apr 06 2004
a(8)-a(10) from Martin Fuller, Jan 17 2006
a(11) from Donovan Johnson, Oct 23 2008
a(12) from Bert Dobbelaere, Jun 27 2019

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

A086263 Smaller of two consecutive squarefree numbers having equal numbers of prime factors.

Original entry on oeis.org

2, 14, 21, 33, 34, 38, 57, 85, 86, 93, 94, 118, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 217, 218, 230, 253, 285, 298, 301, 302, 326, 334, 381, 393, 394, 429, 434, 445, 446, 453, 481, 501, 514, 526, 537, 542, 553, 565, 609, 622, 633, 634

Offset: 1

Reinhard Zumkeller, Jul 14 2003

nonn

a(k) is a term of A075039 iff a(k)+1 = a(k+1).

If a prime divides a(n) then it does not divide a(n) + 1. If a prime divides a(n) + 1, then it does not divide a(n). The sets of prime divisors of a(n) and a(n) + 1 are disjoint. - Torlach Rush, Jan 13 2018

			230 = 2*5*23 and 230+1 = 3*7*11, therefore 230 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A005117, A263990 (2 prime factors), A215217 (3 prime factors), A318896 (4 prime factors), A318964 (5 prime factors), A001221, A001222, A075039.

Select[Range[2, 634], SquareFreeQ[#] && SquareFreeQ[# + 1] && Length[FactorInteger[#]] == Length[FactorInteger[# + 1]] &] (* T. D. Noe, Jun 26 2013 *)
#[[1,1]]&/@Select[Partition[Table[{n,If[SquareFreeQ[n],1,0], PrimeOmega[ n]},{n,700}],2,1],#[[1,2]]==#[[2,2]]==1&&#[[1,3]]==#[[2,3]]&] (* Harvey P. Dale, Dec 13 2014 *)

for(n=1,10^3, if ( issquarefree(n) && issquarefree(n+1) && (omega(n)==omega(n+1)) , print1(n,", "))); \\ Joerg Arndt, Jun 26 2013

A001221(a(n)) = A001222(a(n)) = A001221(a(n)+1) = A001222(a(n)+1).

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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A052215 a(n) = smallest number m such that m and m+1 are the product of exactly n distinct primes.

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

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PARI

A086263 Smaller of two consecutive squarefree numbers having equal numbers of prime factors.

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Mathematica

PARI

Formula

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

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Author

Keywords

Examples

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Programs

PARI