A007675 -id:A007675 - cp's OEIS frontend

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.

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

A268641 Squarefree numbers k such that k^2 + 1 and k^2 - 1 are also squarefree.

Original entry on oeis.org

2, 6, 14, 22, 30, 34, 42, 58, 66, 78, 86, 94, 102, 106, 110, 114, 130, 138, 142, 158, 166, 178, 186, 194, 202, 210, 214, 222, 230, 238, 254, 258, 266, 286, 302, 310, 322, 330, 346, 354, 358, 366, 390, 394, 398, 402, 410, 430, 434, 438, 446, 454, 462, 466, 470, 498

Offset: 1

K. D. Bajpai, Feb 09 2016

nonn

All the listed terms are even squarefree numbers.

Subsequence of A039956.

			a(2) = 6 = 2 * 3: 6^2 + 1 = 37 = 1 * 37; 6^2 - 1 = 35 = 5 * 7; 6, 37, 35 are all squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A007675, A039956, A049532, A049533, A069987, A173186.

[n : n in [1..1000]  |  IsSquarefree(n) and IsSquarefree(n^2+1) and IsSquarefree(n^2-1) ];

select(n -> andmap(issqrfree, [n, n^2+1, n^2-1]), [seq(n, n=2.. 10^3)]);

Select[Range[1000], SquareFreeQ[#] && SquareFreeQ[#^2 + 1] && SquareFreeQ[#^2 - 1] &]

for(n=2, 1000, issquarefree(n) & issquarefree(n^2 + 1) & issquarefree(n^2 - 1) & print1(n,", "))

A367697 Starts of runs of 15 consecutive integers that are exponentially odious numbers (A270428).

Original entry on oeis.org

9, 73, 137, 169, 201, 393, 521, 553, 633, 649, 713, 761, 809, 841, 889, 1001, 1033, 1065, 1129, 1145, 1193, 1225, 1273, 1289, 1353, 1385, 1513, 1545, 1577, 1609, 1657, 1769, 1785, 1865, 1897, 1929, 2025, 2089, 2169, 2217, 2297, 2345, 2377, 2409, 2441, 2505, 2569

Offset: 1

Amiram Eldar, Nov 27 2023

The maximal length of a run of consecutive exponentially odious numbers is 15 since numbers of the form 16*k + 8 are not exponentially odious. Thus all the terms of this sequence are of the form 16*k + 9 with k = 0, 4, 8, 10, 12, 24, 32, 34, 39, 40, ... .

The numbers of terms not exceeding 10^k for k = 1, 2, ... , are 1, 2, 15, 176, 1821, 18120, 181277, 1812917, 18129256, 181290721, ... . Apparently, the asymptotic density of this sequence exists and equals 0.018129... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A270428 and A367696.

Similar sequences: A007675, A194002, A325058, A328016.

expOdQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ[DigitCount[#, 2, 1]] &]; q[n_] := AllTrue[16*n + Range[9, 23], expOdQ]; 16 * Select[Range[0, 160], q] + 9

isexpod(n) = {my(f = factor(n)); for(i=1, #f~, if (!(hammingweight(f[i, 2]) % 2), return (0))); 1;}
is(n) = {my(k = (n-9)/16); if(denominator(k) > 1, return(0)); for(i=9, 23, if(!isexpod(16*k + i), return(0))); 1;}

A370600 Numbers m such that 4m + k is squarefree for k = 1..3.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 9, 10, 14, 16, 17, 19, 21, 23, 25, 26, 27, 28, 32, 34, 35, 39, 41, 44, 45, 46, 48, 50, 52, 53, 54, 55, 57, 59, 63, 64, 66, 70, 71, 75, 77, 79, 80, 82, 86, 88, 89, 91, 95, 97, 98, 99, 100, 102, 104, 107, 108, 109, 111, 113, 115, 116, 117, 120

Offset: 1

Michael De Vlieger, Apr 10 2024

Numbers m such that A008966(4m+1) + A008966(4m+2) + A008966(4m+3) = 3.
The number p^2*m is never squarefree, hence, 4*m is likewise never squarefree. Since 2 is the smallest prime, we have at most 3 consecutive squarefree numbers.
The asymptotic density of this sequence is 4 * Product_{p prime} (1 - 3/p^2) = 4 * A206256 = 0.501947... . - Amiram Eldar, Apr 16 2024

			For m = 0, all of {4(0)+1, 4(0)+2, 4(0)+3} = {1, 2, 3} are squarefree and composite; these are all squarefree semiprimes. Hence, 0 is in the sequence.
For m = 2, {4(2)+1, 4(2)+2, 4(2)+3} = {9, 10, 11} only the latter 2 numbers are squarefree. Therefore, 2 is not in the sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot f(m) at (x,y) = (m mod 361, -floor(m/361)), m = 0..130320, 4X exaggeration, where f(m) = A008966(4m + 1), A008966(4m + 2), A008966(4m + 3), the first term assigned red, second green, and third blue channel. Hence m in this sequence appear white, while those in A258332 appear black.

Cf. A005117, A007675, A008966, A206256, A258332.

Reap[Do[If[AllTrue[4 n + {1, 2, 3}, SquareFreeQ], Sow[n]], {n, 0, 120}] ][[-1, 1]]
Select[Range[0,150],AllTrue[4#+{1,2,3},SquareFreeQ]&] (* Harvey P. Dale, Aug 19 2025 *)

is(m) = issquarefree(4*m+1) && issquarefree(4*m+2) && issquarefree(4*m+3); \\ Amiram Eldar, Apr 16 2024

a(n) = (A007675(n)-1)/4.

A378458 Squarefree numbers k such that k + 1 is squarefree but k + 2 is not.

Original entry on oeis.org

2, 6, 10, 14, 22, 30, 34, 38, 42, 46, 58, 61, 66, 70, 73, 78, 82, 86, 94, 102, 106, 110, 114, 118, 122, 130, 133, 138, 142, 145, 154, 158, 166, 173, 178, 182, 186, 190, 194, 202, 205, 210, 214, 218, 222, 226, 230, 238, 246, 254, 258, 262, 266, 273, 277, 282

Offset: 1

Gus Wiseman, Dec 02 2024

nonn

These are the positions of 2 in A378369 (difference between n and the next nonsquarefree number).

The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = A065474 - A206256 = 0.19714711803343537224... . - Amiram Eldar, Dec 03 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A007675 within A007674.
The version for prime power instead of nonsquarefree is a subset of A006549.
Another variation is A073247.
The version for nonprime instead of squarefree is A179384.
Positions of 0 in A378369 are A013929.
Positions of 1 in A378369 are A373415.
Positions of 2 in A378369 are A378458 (this).
Positions of 3 in A378369 are A007675.
A000961 lists the powers of primes, differences A057820.
A120327 gives the least nonsquarefree number >= n.
A378373 counts composite numbers between nonsquarefree numbers.
Cf. A065474, A067535, A081221, A206256, A236575, A377046, A378033, A378039, A378086.

Select[Range[100],NestWhile[#+1&,#,SquareFreeQ[#]&]==#+2&]

list(lim) = my(q1 = 1, q2 = 1, q3); for(k = 3, lim, q3 = issquarefree(k); if(q1 && q2 &&!q3, print1(k-2, ", ")); q1 = q2; q2 = q3); \\ Amiram Eldar, Dec 03 2024

A063736 Patterns of possible squarefree triples of 3 consecutive numbers {4k+1, 4k+2, 4k+3} are coded as follows: compute A008966(x) getting one of {000, 001, 010, 011, 100, 101, 110, 111} and convert to decimal.

Original entry on oeis.org

7, 7, 3, 7, 5, 7, 2, 7, 7, 7, 7, 3, 1, 5, 7, 6, 7, 7, 6, 7, 3, 7, 5, 7, 4, 7, 7, 7, 7, 3, 3, 1, 7, 6, 7, 7, 6, 5, 3, 7, 5, 7, 2, 6, 7, 7, 7, 3, 7, 5, 7, 6, 7, 7, 7, 7, 3, 7, 5, 7, 4, 3, 5, 7, 7, 3, 7, 5, 6, 6, 7, 7, 3, 5, 3, 7, 5, 7, 6, 7, 7, 3, 7, 3, 5, 4, 7, 4, 7, 7, 2, 7, 3, 6, 5, 7, 6, 7, 7, 7, 7, 3, 7, 5, 7

Offset: 0

Labos Elemer, Aug 24 2001

nonn

All code values arise corresponding to 8 classes of patterns. E.g., the first nonsquarefree triple (000 pattern, code=0) appears at 844, [845, 846, 847], 848 as a middle part of a nonsquarefree 5-tuple. Start values of code=7 triples are listed in A063238.

			a(0) = 4*A008966(1)+2*A008966(2)+A008966(3) = 4+2+1 = 7.
a(11) = 4*A008966(45)+2*A008966(46)+A008966(47) = 0+2+1 = 3.
a(12) = 4*A008966(49)+2*A008966(50)+A008966(51) = 0+0+1 = 1.
a(13) = 4*A008966(53)+2*A008966(54)+A008966(55) = 4+0+1 = 5.
a(14) = 4*A008966(57)+2*A008966(58)+A008966(59) = 4+2+1 = 7.

Cf. A007675, A063838, A008966.

a(n) = 4*A008966(4n+1)+2*A008966(4n+2)+A008966(4n+3).

A084694 Squarefree numbers which are products of three consecutive numbers. I.e., squarefree numbers of the form k^3 - k.

Original entry on oeis.org

6, 210, 2730, 10626, 26970, 39270, 54834, 74046, 195054, 287430, 342930, 474474, 635970, 830490, 1061106, 1190910, 1330890, 1481430, 2196870, 2627934, 2863146, 3944154, 4574130, 5639574, 6028386, 6434670, 7301190, 8242206, 9260790

Offset: 1

Amarnath Murthy, Jun 04 2003

nonn

No product of 4 consecutive numbers is squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A007531, A007675.

f[n_] := n*(n + 1)*(n + 2); f /@ Select[Range[250], AllTrue[# + {0, 1, 2}, SquareFreeQ] &] (* Amiram Eldar, Feb 25 2021 *)

lista(nn) = {for (k=1, nn, if (issquarefree(x=k^3-k), print1(x, ", ")););} \\ Michel Marcus, Jul 29 2017

a(n) = A007531(A007675(n)+2). - Amiram Eldar, Feb 25 2021

Offset corrected to 1 by Michel Marcus, Jul 29 2017

cp's OEIS Frontend

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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A268641 Squarefree numbers k such that k^2 + 1 and k^2 - 1 are also squarefree.

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A367697 Starts of runs of 15 consecutive integers that are exponentially odious numbers (A270428).

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A370600 Numbers m such that 4m + k is squarefree for k = 1..3.

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A378458 Squarefree numbers k such that k + 1 is squarefree but k + 2 is not.

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A063736 Patterns of possible squarefree triples of 3 consecutive numbers {4k+1, 4k+2, 4k+3} are coded as follows: compute A008966(x) getting one of {000, 001, 010, 011, 100, 101, 110, 111} and convert to decimal.

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A084694 Squarefree numbers which are products of three consecutive numbers. I.e., squarefree numbers of the form k^3 - k.

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