A318964 -id:A318964 - 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

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

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

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).

cp's OEIS Frontend

A140079 Numbers n such that n and n+1 have 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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A318896 Numbers k such that k and k+1 are the product of exactly four distinct primes.

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

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Mathematica

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