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.

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

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..23.

Example

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.

Prog

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

Crossrefs

Cf. A013929, A258211, A258332.

Sequence in context: A268290 A183791 A115533 * A187315 A083636 A124069

Adjacent sequences: A256010 A256011 A256012 * A256014 A256015 A256016

Keyword

nonn

Author

Juri-Stepan Gerasimov, May 31 2015

Extensions

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

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

Status

approved