A068088 -id:A068088 - cp's OEIS frontend

A242804 Integers k such that each of k, k+1, k+2, k+4, k+5, k+6 is the product of two distinct primes.

213, 143097, 194757, 206133, 273417, 684897, 807657, 1373937, 1391757, 1516533, 1591593, 1610997, 1774797, 1882977, 1891761, 2046453, 2051493, 2163417, 2163957, 2338053, 2359977, 2522517, 2913837, 3108201, 4221753

Offset: 1

Daniel Constantin Mayer, May 23 2014

nonn

A remarkable gap occurs between the initial two members, and the sequence seems to be rather sparse compared to the related A242805.
Here, the first member k of the sextet is the reference, whereas in A068088 the center k+3 is selected as reference. Observe that k+3 must be divisible by the square 4.
All terms are congruent to 9 (mod 12). - Zak Seidov, Apr 14 2015
From Robert Israel, Apr 15 2015: (Start)
All terms are congruent to 33 (mod 36).
Numbers k in A039833 such that k+4 is in A039833. (End)
From Robert G. Wilson v, Apr 15 2015: (Start)
k is congruent to 33 (mod 36) so one of its factors is 3 and the other is == 11 (mod 12);
k+1 is congruent to 34 (mod 36) so one of its factors is 2 and the other is == 17 (mod 18);
k+2 is congruent to 35 (mod 36) so its factors are == +-1 (mod 6);
k+4 is congruent to 1 (mod 36) so its factors are == +-1 (mod 6);
k+5 is congruent to 2 (mod 36) so one of its factors is 2 and the other is == 1 (mod 18);
k+6 is congruent to 3 (mod 36) so one of its factors is 3 and the other is == 1 (mod 12). (End).
Number of terms < 10^m: 0, 0, 1, 1, 1, 7, 39, 169, 882, 4852, 27479, ...,. - Robert G. Wilson v, Apr 15 2015
Or, numbers k such that k, k+1 and k+2 are terms in A175648. - Zak Seidov, Dec 08 2015

			213=3*71, 214=2*107, 215=5*43, 217=7*31, 218=2*109, 219=3*73.

Zak Seidov and Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A242793 (minima for two, three and more prime divisors) and A068088 (arbitrary squarefree integers).

Cf. A242805, A039833, A175648, A202319.

f:= t -> numtheory:-issqrfree(t) and (numtheory:-bigomega(t) = 2):
select(t -> andmap(f, [t,t+1,t+2,t+4,t+5,t+6]), [seq(36*k+33,k=0..10^6)]); # Robert Israel, Apr 15 2015

fQ[n_] := PrimeQ[n/3] && PrimeQ[(n + 1)/2] && PrimeQ[(n + 5)/2] && PrimeQ[(n + 6)/3] && PrimeNu[{n + 2, n + 4}] == {2, 2} == PrimeOmega[{n + 2, n + 4}]; k = 33; lst = {}; While[k < 10^8, If[fQ@ k, AppendTo[lst, k]]; k += 36]; lst (* Robert G. Wilson v, Apr 14 2015 and revised Apr 15 2015 after Zak Seidov and Robert Israel *)

default(primelimit, 1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=2; for(n=lb, ub, if(issquarefree(n)&&(o==omega(n)), loc=loc+1; if(1==loc, i=n; ); if(2==loc, if(i+1==n, j=n; ); if(i+1

forstep(x=213,4221753,12, if( isprime(x/3) && isprime((x+1)/2) && 2==omega(x+2) && 2==bigomega(x+2) && 2==omega(x+4) && 2==bigomega(x+4) && isprime((x+5)/2) && isprime((x+6)/3), print1(x", "))) \\ Zak Seidov, Apr 14 2015

a(n) = A202319(n) - 1. - Jon Maiga, Jul 10 2021

A242793 The minimal integer x such that each of the six integers x, x+1, x+2, x+4, x+5, x+6 is squarefree with exactly n prime divisors.

Original entry on oeis.org

213, 73293, 9743613, 6639266409

Offset: 2

Daniel Constantin Mayer, May 22 2014

This is the next step in my project to study the distribution of increasingly extensive clusters of squarefree integers with fixed number of prime divisors: triples x,x+1,x+2 were investigated in A242492 and here we study sextets x,x+1,x+2,x+4,x+5,x+6 with a central gap x+3, since x+3 must be divisible by the square 4.

The term 6639266409 required 30 hours of CPU time on an iMac with Intel i7 Quadcore CPU running OS X Lion.

			213=3*71, 214=2*107, 215=5*43, 217=7*31, 218=2*109, 219=3*73;
73293=3*11*2221, 73294=2*13*2819, 73295=5*107*137,
73297=7*37*283, 73298=2*67*547, 73299=3*53*461;
9743613=3*11*503*587, 9743614=2*59*71*1163, 9743615= 5*7*167*1667, 9743617=13*37*47*431, 9743618=2*17*19*15083, 9743619=3*83*109*359;
6639266409=3*29*109*421*1663, 6639266410=2*5*7*113*839351,
6639266411=17*23*89*101*1889, 6639266413=13*61*79*131*809,
6639266414=2*11*349*857*1009, 6639266415=3*5*73*149*40693.

Cf. A242492 (triples) and A242804, A242805, A242806, A242829, and also A068088.

{ default(primelimit,1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=1; while(o<5, o=o+1; for(n=lb,ub, if(issquarefree(n)&&(o==omega(n)), loc=loc+1; if(1==loc, i=n; ); if(2==loc, if(i+1==n, j=n; ); if(i+1

A270996 T(i, j) = k is the least squarefree number with a run of exactly i>=0 nonsquarefree numbers immediately preceding k and a run of exactly j>=0 nonsquarefree numbers immediately succeeding k.

Original entry on oeis.org

2, 1, 3, 10, 17, 7, 101, 149, 151, 47, 246, 51, 26, 97, 8474, 1685, 8479, 727, 1861, 241, 843, 22026, 849, 3178, 2526, 10826, 30247, 22019, 217077, 190453, 813251, 55779, 183553, 5045, 580847, 826823

Offset: 0

Hartmut F. W. Hoft, Mar 28 2016

The sequence a(n) = T(i, j) represents the traversal of this matrix by its successive rising antidiagonals.

a(2*i*(i+1)) = A270344(i), for all i >= 0.

			a(13) = T(1, 3) = 97 since 96, 98, 99 and 100 are nonsquarefree while 95, 97, and 101 are squarefree, and 97 is the smallest number surrounded by the 1,3 pattern.
The matrix T(i, j) with first 8 complete antidiagonals together with some additional elements including the first 7 elements on the diagonal which are A270344(0)..A270344(6):
-------------------------------------------------------------------------
i\j      0       1       2       3        4         5          6        7
-------------------------------------------------------------------------
0:       2       3       7      47     8474       843      22019   826823
1:       1      17     151      97      241     30247     580847   217069
2:      10     149      26    1861    10826      5045     204322 16825126
3:     101      51     727    2526   183553   1944347   28591923 43811049
4:     246    8479    3178   55779  5876126  19375679   67806346
5:    1685     849  813251  450553 29002021   8061827 2082929927
6:   22026  190453  200854 4100277 97447622 245990821 8996188226
7:  217077  826831 7507930 90557979
T(6, 5) = 245990821, T(5, 6) = 2082929927, and all numbers in antidiagonal 11 are larger than 10^8.

Hartmut F. W. Hoft, Numbers in the first 11 antidiagonals of the matrix.

Cf. A005117, A007675, A068088, A073247, A073248, A073251, A228649, A268330-A268334, A270344.

(* The function computes the least number in the specified interval *)
nsfRun[n_] := Module[{i=n}, While[!SquareFreeQ[i], i++]; i-n]
a270996[{low_, high_},{widthL_, widthR_}] := Module[{i=low, r, s, first=0}, While[i<=high, r=nsfRun[i]; If[r != widthL, i+=r+1, s=nsfRun[i+r+1]; If[s != widthR, If[s != widthL, i+=r+s+2, i+=r+1], first=i+r; i=high+1]]]; first]
a270996[{0, 5000},{2, 3}] (* computes a(18) = T(2, 3) *)

A242492 For any integer m > 1, the m-th term of the sequence is the minimal squarefree integer x with exactly m prime divisors such that x+1 and x+2 are also squarefree integers with exactly m prime divisors.

Original entry on oeis.org

33, 1309, 203433, 16467033, 1990586013, 41704979953, 102099792179229

Offset: 2

Daniel Constantin Mayer, May 16 2014

The five terms for m = 2,3,4,5,6 were computed with the aid of PARI/GP. But it seems to be rather difficult to compute higher terms, if they exist at all.

The distribution of squarefree integers with exactly m prime factors is given in the book by Montgomery and Vaughan, Multiplicative Number Theory, but I do not have access to it and do not know whether it also addresses the problem of three consecutive numbers of this kind.

			33 = 3*11, 34 = 2*17, 35 = 5*7;
1309 = 7*11*17, 1310 = 2*5*131, 1311 = 3*19*23;
203433 = 3*19*43*83, 203434 = 2*7*11*1321, 203435 = 5*23*29*61;
16467033 = 3*11*17*149*197, 16467034 = 2*19*23*83*227, 16467035 = 5*13*37*41*167; (CPU time 48 seconds)
1990586013 = 3*13*29*67*109*241, 1990586014 = 2*23*37*43*59*461, 1990586015 = 5*11*17*19*89*1259. (CPU time 2 hours and 34 minutes)

Hugh L. Montgomery and Robert C. Vaughan: "Multiplicative Number Theory: 1. Classical Theory", Cambridge studies in advanced mathematics, vol. 97, Cambridge University Press (2007)

Math Overflow, Asymptotics of special square-free numbers, Mar 20 2014
Daniel Constantin Mayer, PARI/GP script "AdjacentSquareFree.gp"
Daniel Constantin Mayer, PARI/GP script "SquareFreeTriplets.gp"
Prime puzzles & problems connection, Square-free triples

Cf. A007675 (any m), A039833 (m=2), A066509 (m=3), A176167 (m=4), A192203 (m=5), A068088 (sextets with gap).

Cf. A242605-A242608 for start of triples of consecutive squarefree numbers with m=2,...,5 prime factors, A242621 for the analog of the present sequence in that spirit.

{default(primelimit,2M); lb=2; ub=2*10^9; m=1; i=0; j=0; loc=0; while(m<6, m=m+1; for(n=lb,ub, if(issquarefree(n)&&(m==omega(n)), loc=loc+1; if(1==loc, i=n; ); if(2==loc, if(i+1==n, j=n; ); if(i+1

a(n) = A093550(n)-1. - M. F. Hasler, May 20 2014

A373689 Perfect powers such that the three numbers before it and the three numbers after it are squarefree.

Original entry on oeis.org

4, 32, 36, 216, 256, 400, 900, 1156, 1296, 1764, 2704, 2916, 3136, 3600, 4356, 5184, 6084, 7056, 8100, 8464, 9216, 11236, 12996, 16384, 19044, 20164, 20736, 22500, 25600, 26244, 26896, 31684, 32400, 36864, 38416, 39204, 40000, 41616, 44100, 46656, 49284, 51984, 54756, 55696, 57600

Offset: 1

Massimo Kofler, Jun 13 2024

nonn

All terms of this sequence are divisible by 4.

			4 = 2^2 (between 1, 2 which is a prime number, 3 which is a prime number and 5 which is a prime number, 6 = 2 * 3 and 7 which is a prime number ).
32 = 2^5 (between 29 which is a prime number, 30 = 2 * 3 * 5, 31 which is a prime number and 33 = 3 * 11, 34 = 2 * 17 and 35 = 5 * 7).
36 = 2^2 * 3^2 (between 33 = 3 * 11, 34 = 2 * 17, 35 = 5 * 7 and 37 which is a prime number, 38 = 2 * 19 and 39 = 3 * 13).

Intersection of A001597 (or A075090) and A068088.

Cf. A005117, A373077, A373287.

Select[Range[60000], GCD @@ FactorInteger[#][[;; , 2]] > 1 && And @@ SquareFreeQ /@ (# + {-3, -2, -1, 1, 2, 3}) &] (* Amiram Eldar, Jun 13 2024 *)

A374393 Triangular numbers such that the three numbers before it and the three numbers after it are squarefree.

Original entry on oeis.org

36, 2016, 2556, 3240, 9180, 10296, 23220, 23436, 25200, 39060, 41616, 67896, 93096, 97020, 122760, 126756, 170820, 215496, 253116, 313236, 320400, 365940, 437580, 438516, 446040, 499500, 508536, 574056, 592416, 653796, 673380, 738720, 749700, 839160, 850860, 924120, 936396, 1024596, 1036080

Offset: 1

Massimo Kofler, Jul 07 2024

nonn

All terms of this sequence are divisible by 36, so they can't be squarefree.

			36 = 2^2 * 3^2 (the 8th triangular number) between 33 = 3 * 11, 34 = 2 * 17, 35 = 5 * 7 and 37 which is a prime number, 38 = 2 * 19 and 39 = 3 * 13.
2016 = 2^5 * 3^2 * 7 (the 63rd triangular number) between 2013 = 3 * 11 * 61, 2014 = 2 * 19 * 53, 2015 = 5 * 13 * 31 and 2017 which is a prime number, 2018 = 2 * 1009, 2019 = 3 * 673.
2556 = 2^2 * 3^2 * 71 (the 71st triangular number) between 2553 = 3 * 23 * 37, 2554 = 2 * 1277, 2555 = 5 * 7 * 73 and 2557 which is a prime number, 2558 = 2 * 1279, 2559 = 3 * 853.

Intersection of A000217 and A068088.

Cf. A005117.

Select[Accumulate[Range[1500]], And @@ (SquareFreeQ /@ (# + {-3, -2, -1, 1, 2, 3})) &] (* Amiram Eldar, Jul 07 2024 *)

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A242804 Integers k such that each of k, k+1, k+2, k+4, k+5, k+6 is the product of two distinct primes.

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A242793 The minimal integer x such that each of the six integers x, x+1, x+2, x+4, x+5, x+6 is squarefree with exactly n prime divisors.

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A270996 T(i, j) = k is the least squarefree number with a run of exactly i>=0 nonsquarefree numbers immediately preceding k and a run of exactly j>=0 nonsquarefree numbers immediately succeeding k.

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A242492 For any integer m > 1, the m-th term of the sequence is the minimal squarefree integer x with exactly m prime divisors such that x+1 and x+2 are also squarefree integers with exactly m prime divisors.

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A373689 Perfect powers such that the three numbers before it and the three numbers after it are squarefree.

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A374393 Triangular numbers such that the three numbers before it and the three numbers after it are squarefree.

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