A145050 -id:A145050 - cp's OEIS frontend

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

Vladimir Shevelev, Oct 05 2008, Oct 07 2008

nonn

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

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

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

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

Extended by R. J. Mathar, Aug 02 2010

A145017 Squarefree positive integers k for which k-(floor(sqrt(k)))^2 is a perfect square.

Original entry on oeis.org

1, 2, 5, 10, 13, 17, 26, 29, 34, 37, 53, 58, 65, 73, 82, 85, 97, 101, 109, 122, 130, 137, 145, 170, 173, 178, 185, 194, 197, 205, 221, 226, 229, 241, 257, 265, 281, 290, 293, 298, 305, 314, 349, 362, 365, 370, 377, 386, 397, 401, 409, 442, 445, 457, 466, 485, 493

Offset: 1

Vladimir Shevelev, Sep 29 2008

nonn

If an odd prime p divides a(n) then it has the form 4k+1.

Conjecture. For every n>=1 there exist infinitely many primes p of the form 4k+1 for which for a(n) > 1 we have s*p-(floor(sqrt(s*p)))^2 is not a perfect square for s=1,...,a(n)-1 while a(n)*p-(floor(sqrt(a(n)*p)))^2 is a perfect square. (See A145016(s=1) and A145022, A145023, A145047, A145048, A145049, A145050 correspondingly for s=2, s=5, s=10, s=13, s=17, s=26.) - Vladimir Shevelev, Sep 30 2008

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A005117, A020893, A145016, A145022, A145023, A145047, A145048, A145049, A145050.

Select[Range@ 500, And[SquareFreeQ@ #, IntegerQ@ Sqrt[# - Floor[Sqrt@ #]^2]] &] (* Michael De Vlieger, Jan 12 2020 *)

is(n)={issquarefree(n) && issquare(n-sqrtint(n)^2)} \\ Andrew Howroyd, Jan 12 2020

Missing a(40) inserted and terms a(42) and beyond from Andrew Howroyd, Jan 12 2020

A145215 a(n) is the minimal prime of the form 4k+1 for which s=A008784(n) is the minimal positive integer such that sa(n)-floor(sqrt(sa(n)))^2 is a square.

Original entry on oeis.org

5, 41, 353, 1237, 2749, 3037, 10369, 6569, 27253, 38561, 14897, 33289, 27917, 171629, 143513, 76081, 37649, 373273, 399181, 63029, 133157, 637601, 425197, 94261, 499321, 910853, 229849, 149837

Offset: 1

Vladimir Shevelev, Oct 05 2008

nonn

See the conjecture in the comment at A145047. In addition, I conjecture that for every such s there exist infinitely many primes of the form 4k+1.

Cf. A145016, A145022, A145023, A145047-A145050, A008487.

f(s)=forprime(p=2,,if(p%4>1 || !issquare(s*p-sqrtint(s*p)^2),next);for(i=1,s-1,if(issquare(i*p-sqrtint(i*p)^2), next(2)));return(p))
S=select(n->if(n%2==0, if(n%4, n/=2, return(0))); n==1||vecmax(factor(n)[, 1]%4)==1, vector(150,i,i));
apply(f, S) \\ Charles R Greathouse IV, Feb 07 2013

a(22) corrected by Charles R Greathouse IV, Feb 07 2013

A249298 Smallest positive integer k, such that s-kn is a square where s is the smallest square >= kn.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 5, 1, 5, 6, 1, 1, 9, 2, 9, 2, 1, 12, 13, 1, 1, 14, 1, 3, 17, 2, 19, 1, 3, 20, 1, 1, 23, 24, 3, 1, 25, 2, 27, 6, 1, 30, 31, 1, 1, 2, 3, 7, 35, 4, 1, 2, 3, 40, 41, 1, 41, 42, 1, 1, 1, 4, 47, 10, 5, 2, 51, 1, 51, 52, 3, 12, 1, 6, 57, 1, 1, 60, 61, 1, 3, 62, 7, 3, 65, 2

Offset: 1

Valtteri Raiko, Oct 24 2014

nonn

For any n>=3, there exists at least one positive integer k, 1 <= k <= n-1 such that the difference between the smallest square >= k*n and k*n is a square. To prove this, consider the multiplier k = n-2. Then (n-2)*n = (n-1)^2-1, thus the difference from the next square is 1, which is a square. If n = 1, k = 1 and if n = 2, k = 2.

Smallest positive integer k such that ceiling(sqrt(k*n))^2-k*n is a square.

			a(10) = 4, for ceiling(sqrt(10))^2-10 = 6, ceiling(sqrt(2*10))^2-2*10 = 5, ceiling(sqrt(3*10))^2-3*10 = 6 and ceiling(sqrt(4*10))^2-4*10 = 9 = 3^2.

Cf. A000290, A145236 (equals a(A000040)), A068527 (difference for k=1).

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

dif[n_] := Ceiling[Sqrt[n]]^2 - n;a[k_] := Module[{n = 1}, While[dif[dif[n*k]] != 0, n++]; Return[n]];Table[a[k], {k, 1, 90}]

a(n) = {k=1; while(!issquare(ceil(sqrt(k*n))^2-k*n), k++); k;} \\ Michel Marcus, Oct 24 2014

A145230 Numbers of different values of the minimal factors s for primes of the form 4k+1 not exceeding 10^n (see A145215).

Original entry on oeis.org

1, 2, 3, 7, 17, 38

Offset: 1

Vladimir Shevelev, Oct 05 2008

nonn

A145016 A145022 A145023 A145047 A145048 A145049 A145050 A145215

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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.

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A145017 Squarefree positive integers k for which k-(floor(sqrt(k)))^2 is a perfect square.

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A145215 a(n) is the minimal prime of the form 4k+1 for which s=A008784(n) is the minimal positive integer such that s*a(n)-floor(sqrt(s*a(n)))^2 is a square.

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A249298 Smallest positive integer k, such that s-k*n is a square where s is the smallest square >= k*n.

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A145230 Numbers of different values of the minimal factors s for primes of the form 4k+1 not exceeding 10^n (see A145215).

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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.

A145215 a(n) is the minimal prime of the form 4k+1 for which s=A008784(n) is the minimal positive integer such that sa(n)-floor(sqrt(sa(n)))^2 is a square.

A249298 Smallest positive integer k, such that s-kn is a square where s is the smallest square >= kn.