A145236 a(n) is the least positive integer such that if p_n is the n-th prime then (ceiling(sqrt(a(n)*p_n)))^2 - a(n)*p_n is a perfect square.

2, 1, 1, 3, 5, 5, 9, 9, 13, 17, 19, 23, 25, 27, 31, 35, 41, 41, 47, 51, 51, 57, 61, 65, 73, 75, 77, 81, 83, 85, 99, 101, 107, 109, 117, 119, 125, 129, 133, 139, 145, 145, 155, 157, 161, 163, 173, 183, 187, 189, 193, 199, 201, 209, 215, 221, 225, 227, 233, 237, 239, 247

Offset

1,1

Comments

Conjectures: 1) for n >= 2, the sequence does not decrease; 2) for n > 1, a(n) is odd; 3) a(n) can be equal to a(n+1) only for twins: p_(n+1) - p_n = 2 (although there also exist twins for which a(n) < a(n+1)).

All these conjectures are proved using the formula a(n) = p_n - 2*floor(sqrt(2p_n)) + 2, n > 1. See also A145701 and A145714. - Vladimir Shevelev, Oct 18 2008

Links

Table of n, a(n) for n=1..62.

Maple

A145236 := proc(n) local p, k, a ; p := ithprime(n) ; for k from 1 do ceil(sqrt(ceil(k*p))) ; a := %^2-k*p ; if issqr(a) then return k ; end if; end do: end proc:
for n from 1 do printf("%d, \n", A145236(n)) ; end do: # R. J. Mathar, Aug 02 2010

Crossrefs

Cf. A145016, A145022, A145023, A145047, A145048, A145049, A145050, A145215.

Sequence in context: A174620 A210098 A241188 * A162206 A075248 A359140

Adjacent sequences: A145233 A145234 A145235 * A145237 A145238 A145239

Keyword

nonn

Author

Vladimir Shevelev, Oct 05 2008, Oct 07 2008

Extensions

a(12)=23 (not 21). - Vladimir Shevelev, Oct 16 2008

Extended by R. J. Mathar, Aug 02 2010

Status

approved