A258332 -id:A258332 - cp's OEIS frontend

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

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.

A256013 Numbers n such that none of 9n + 1, 9n + 2, 9n + 3, 9n + 4, 9n + 5, 9n + 6, 9n + 7 and 9n + 8 are squarefree.

Original entry on oeis.org

24574158, 29146163, 156385858, 173105316, 246414308, 404413338, 553659041, 556221794, 745644336, 760923063, 789864069, 794287963, 893806805, 983628183, 1033093563, 1134287383, 1138839886, 1418521141, 1559578963, 1702800491, 1750142480, 2080676083, 2117324180

Offset: 1

Juri-Stepan Gerasimov, May 31 2015

nonn

Two of 9n+1..9n+8 are multiples of 4, so concentrate on the other six. The probability that any k of these six are all squarefree is P(k) := Product {p prime > 3} (p^2-k)/p^2. By inclusion-exclusion, the probability that none of the six are squarefree is 1 - 6P(1) + 15P(2) - 20P(3) + 15P(4) - 6P(5) + P(6), or roughly one in 92600000. - Michael R Peake, Apr 04 2017

			24574158 is in this sequence because
9 * 24574158 + 1 = 221167423 = 230143 * 31^2,
9 * 24674158 + 2 = 221167424 = 3455741 * 2^6,
9 * 24674158 + 3 = 221167425 = 128213 * 23 * 5^3,
9 * 24674158 + 4 = 221167426 = 80777 * 2 * 37^2,
9 * 24674158 + 5 = 221167427 = 45127 * 29 * 13^2,
9 * 24674158 + 6 = 221167428 = 18430619 * 3 * 2^2,
9 * 24674158 + 7 = 221167429 = 33937 * 19 * 7^3,
9 * 24674158 + 8 = 221167430 = 3889 * 47 * 5 * 2 * 11^2.

Cf. A013929, A258211, A258332.

[n: n in [1..25000000] | not IsSquarefree(9*n+1) and not IsSquarefree(9*n+2) and not IsSquarefree(9*n+3) and not IsSquarefree(9*n+4) and not IsSquarefree(9*n+5) and not IsSquarefree(9*n+6) and not IsSquarefree(9*n+7) and not IsSquarefree(9*n+8)];

is(n)=for(k=9*n+1,9*n+8,if(issquarefree(k),return(0))); 1 \\ Charles R Greathouse IV, Jun 02 2015

a(3)-a(8) from Charles R Greathouse IV, Jun 02 2015

a(9)-a(23) from Charles R Greathouse IV, Jun 03 2015

cp's OEIS Frontend

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

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