A256013 - OEIS

%I #35 Sep 08 2022 08:46:11

%S 24574158,29146163,156385858,173105316,246414308,404413338,553659041,

%T 556221794,745644336,760923063,789864069,794287963,893806805,

%U 983628183,1033093563,1134287383,1138839886,1418521141,1559578963,1702800491,1750142480,2080676083,2117324180

%N 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.

%C 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

%e 24574158 is in this sequence because

%e 9 * 24574158 + 1 = 221167423 = 230143 * 31^2,

%e 9 * 24674158 + 2 = 221167424 = 3455741 * 2^6,

%e 9 * 24674158 + 3 = 221167425 = 128213 * 23 * 5^3,

%e 9 * 24674158 + 4 = 221167426 = 80777 * 2 * 37^2,

%e 9 * 24674158 + 5 = 221167427 = 45127 * 29 * 13^2,

%e 9 * 24674158 + 6 = 221167428 = 18430619 * 3 * 2^2,

%e 9 * 24674158 + 7 = 221167429 = 33937 * 19 * 7^3,

%e 9 * 24674158 + 8 = 221167430 = 3889 * 47 * 5 * 2 * 11^2.

%o (Magma) [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)];

%o (PARI) is(n)=for(k=9*n+1,9*n+8,if(issquarefree(k),return(0))); 1 \\ _Charles R Greathouse IV_, Jun 02 2015

%Y Cf. A013929, A258211, A258332.

%K nonn

%O 1,1

%A _Juri-Stepan Gerasimov_, May 31 2015

%E a(3)-a(8) from _Charles R Greathouse IV_, Jun 02 2015

%E a(9)-a(23) from _Charles R Greathouse IV_, Jun 03 2015