A048760 -id:A048760 - cp's OEIS frontend

A256098 Denominators for the numerators A256097.

1, 2, 1, 1, 4, 2, 4, 1, 1, 6, 3, 2, 3, 6, 1, 1, 8, 4, 8, 2, 8, 4, 8, 1, 1, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 1, 12, 6, 4, 3, 12, 2, 12, 3, 4, 6, 12, 1, 1, 14, 7, 14, 7, 14, 7, 2, 7, 14, 7, 14, 7, 14, 1, 1, 16, 8, 16, 4, 16, 8, 16, 2, 16, 8, 16, 4, 16, 8, 16, 1

Offset: 1

Wolfdieter Lang, Mar 24 2015

The corresponding numerators are given in A256097.

See A256097 for comments, references and links.

			See A256097.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A000196, A048760, A048761, A256097.

A000196[n_]:=Floor[Sqrt[n]]; a[n_]:=Denominator[(A000196[n]+n/A000196[n])/2]; Array[a,80](* Stefano Spezia, Feb 15 2025 *)

a(n) = denominator(R(n)), with the rational (in lowest terms) R(n) = f(n) + (n - f(n)^2)/(2*f(n)) = (f(n) + n/f(n))/2 with f(n) := floor(sqrt(n)) = A000196(n), for n >= 1.

a(61)-a(80) from Stefano Spezia, Feb 15 2025

A381054 a(n) is the least k such that floor(sqrt(nk/d(nk))) - floor(sqrt(d(n*k))) = 1, where d(k) is the largest divisor of k which is <= sqrt(k).

Original entry on oeis.org

5, 4, 4, 2, 1, 2, 1, 1, 2, 1, 4, 1, 4, 1, 1, 5, 9, 1, 9, 3, 1, 2, 9, 3, 2, 2, 2, 3, 16, 2, 16, 3, 2, 5, 2, 2, 25, 5, 2, 2, 25, 2, 25, 1, 1, 5, 25, 2, 2, 1, 3, 1, 36, 1, 1, 2, 3, 8, 36, 1, 36, 8, 1, 3, 1, 1, 49, 3, 3, 1, 49, 1, 49, 13, 1, 3, 1, 1, 49, 1

Offset: 1

Hassan Baloui, Apr 14 2025

In the case of semiprime numbers n=p*q, the sequence a(n) choses the first two multiples of the prime factors p and q such that there is exactly one square between these multiples (say k*p, m*q) then a(n)=m*k.
In the case of any composite number n, for each pair of conjugate divisors (d_i, n/d_i) we chose the first two multiples say (k_i*d_i, m_i*n/d_i) which are exactly one square apart, then a(n) is the smallest product k_i*m_i and k_i*d_i is the largest divisor of k_i*m_i*n which is <= sqrt(k_i*m_i*n), and i runs over half the number of divisors of n (since only pairs are considered).
a(n) can also factor numbers as follow :(to be proven)
One pair of the roots of the five quadratic equations X^2-(2*A000196(a(n)*n)+{1,2,3,4 or 5})*X+a(n)*n =0 is a pair of positive integers say (X_1,X_2) then gcd(X_1,n) and gcd(X_2,n) are nontrivial divisors of n.
A382286 is the first sequence of an infinite sequence of sequences and the current sequence a(n) is the second one. These sequences can be defined as:
For each positive integer m, the corresponding sequence evaluated at a positive integer n is the least k such that floor(sqrt(n*k/d(n*k))) - floor(sqrt(d(n*k))) = m, where d(k) is the largest divisor of k which is <= sqrt(k).
If we denote for each m the corresponding sequence a_m(n), it is conjectured that (a_m(n)=(A000196(n)-m)^2 iff n is a prime > m^2+1) the cases m=0:A382286,1 are answered affirmatively

			Let C(k)=floor(sqrt(k/d(k))) - floor(sqrt(d(k))), where d(k) is the largest divisor of k which is <= sqrt(k):
a(1)=5 since C(k) > 1 for k=1 to 4 and C(5)=1
a(2)=4 since C(2*k) > 1 for k=1 to 3 and C(2*4)=1
a(3)=4 since C(3*k) > 1 for k=1 to 3 and C(3*4)=1
a(13*113)=7 because the first multiples of the prime factors which are exactly one square apart are 7*13 and 113.
a(13*114)=3 because from the pairs of conjugate divisors which are (1,1482), (2,741), (3,494), (6,247), (13,114), (19,78),(26,57) and (38,39) the first pair that have multiples exactly one square apart is (26,57) and these multiples are (3*26,57) and 57 is the largest divisor of 3*13*114 <= sqrt(3*13*114)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000196, A033676, A033677, A048760, A382286, A383115.

d:= proc(n) max(select(t -> t^2 <= n, numtheory:-divisors(n))) end proc:
f:= proc(n) local k,t;
     for k from 1 do
       t:= d(n*k);
       if floor(sqrt(n*k/t)) - floor(sqrt(t)) = 1 then return k fi
     od
end proc:
map(f, [$1..100]); # Robert Israel, Jun 04 2025

d(n) = if(n<2, 1, my(d=divisors(n)); d[(length(d)+1)\2]); \\ A033676
a(n) = my(k=1,dnk=d(n*k)); while (sqrtint(n*k/dnk) - sqrtint(dnk) != 1, k++;dnk=d(n*k)); k;

Conjecture: a(n) < (A000196(n)-1)^2 iff n>=9 is a composite number.

Conjecture: a(n) = (A000196(n)-1)^2 iff n>=9 is a prime number.

A065734 Largest square <= sigma(n).

Original entry on oeis.org

1, 1, 4, 4, 4, 9, 4, 9, 9, 16, 9, 25, 9, 16, 16, 25, 16, 36, 16, 36, 25, 36, 16, 49, 25, 36, 36, 49, 25, 64, 25, 49, 36, 49, 36, 81, 36, 49, 49, 81, 36, 81, 36, 81, 64, 64, 36, 121, 49, 81, 64, 81, 49, 100, 64, 100, 64, 81, 49, 144, 49, 81, 100, 121, 81, 144, 64, 121, 81, 144

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A048760, A000203, A065730-A065741.

Table[Floor[Sqrt[DivisorSigma[1,n]]]^2,{n,70}] (* Harvey P. Dale, Oct 05 2015 *)

A000203(n) := divsum(n) ;
A048760(n) := (floor(sqrt(n)))^2 ;
A065734(n) := A048760(A000203(n)) ;
for n from 1 thru 10 do print(A065734(n)) ; /* R. J. Mathar, Feb 13 2012 */

a(n) = { sqrtint(sigma(n))^2 } \\ Harry J. Smith, Oct 28 2009

a(n) = A048760(A000203(n)).

Previous Mathematica program replaced by Harvey P. Dale, Oct 05 2015

A065735 Largest square <= product of first n primes.

Original entry on oeis.org

1, 4, 25, 196, 2304, 29929, 509796, 9696996, 223084096, 6469628356, 200559769921, 7420737154609, 304250260158441, 13082761295250201, 614889781233144561, 32589158470911505225, 1922760350143213168900

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A048760, A002110, A065730-A065741.

Floor[Sqrt[#]]^2&/@FoldList[Times,Prime[Range[20]]] (* Harvey P. Dale, Jan 13 2024 *)

{ m=1; for (n=1, 100, m*=prime(n); a=sqrtint(m)^2; write("b065735.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 28 2009

a(n) = A048760(A002110(n)).

A065737 Largest square <= binomial(n,2).

Original entry on oeis.org

0, 1, 1, 4, 9, 9, 16, 25, 36, 36, 49, 64, 64, 81, 100, 100, 121, 144, 169, 169, 196, 225, 225, 256, 289, 324, 324, 361, 400, 400, 441, 484, 484, 529, 576, 625, 625, 676, 729, 729, 784, 841, 900, 900, 961, 1024, 1024, 1089, 1156, 1225, 1225, 1296, 1369, 1369, 1444

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A048760, A000217, A000290, A061288, A065730-A065741.

Floor[Sqrt[#]]^2&/@Binomial[Range[50],2] (* Harvey P. Dale, Feb 28 2020 *)

a(n) = { sqrtint(binomial(n, 2))^2 } \\ Harry J. Smith, Oct 28 2009

a(n) = A048760(binomial(n,2)).

a(n) = A061288(n-1)^2 = A000290(A061288(n-1)) for n>=2. - Alois P. Heinz, Nov 07 2024

A065738 Largest square <= binomial(2n,n).

Original entry on oeis.org

1, 4, 16, 64, 225, 900, 3364, 12769, 48400, 184041, 703921, 2702736, 10394176, 40106889, 155102116, 601034256, 2333566249, 9075039169, 35345128009, 137845868176, 538256995600, 2104098203601, 8233427666025, 32247599617809

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n = 1..200

Cf. A048760, A000984, A065730-A065741.

a(n) = { sqrtint(binomial(2*n, n))^2 } \\ Harry J. Smith, Oct 29 2009

a(n) = A048760(A000984(n)).

Terms corrected and extended by Harry J. Smith, Oct 29 2009

A065739 Largest square <= sum of first n squares.

Original entry on oeis.org

1, 4, 9, 25, 49, 81, 121, 196, 256, 361, 484, 625, 784, 961, 1225, 1444, 1764, 2025, 2401, 2809, 3249, 3721, 4225, 4900, 5476, 6084, 6889, 7569, 8464, 9409, 10404, 11236, 12321, 13456, 14884, 16129, 17424, 18769, 20449, 21904, 23716, 25281

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A048760, A000330, A065730-A065741.

a(n) = { sqrtint(n*(n+1)*(2*n+1)/6)^2 } \\ Harry J. Smith, Oct 29 2009

a(n) = A048760(A000330(n)).

A065740 Largest square <= n^n.

Original entry on oeis.org

1, 4, 25, 256, 3025, 46656, 822649, 16777216, 387420489, 10000000000, 285310881025, 8916100448256, 302875094536249, 11112006825558016, 437893889165373169, 18446744073709551616, 827240261832761854009, 39346408075296537575424, 1978419655659032909463364, 104857600000000000000000000

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A048760, A000312, A065730-A065741.

a(n) = { sqrtint(n^n)^2 } \\ Harry J. Smith, Oct 29 2009

a(n) = A048760(A000312(n)).

A140596 Squares nearest to and < terms in A098562.

Original entry on oeis.org

9, 20449, 75625, 238144, 2208196, 3579664, 29192409, 39992976, 45441081, 55591936, 77281681, 108076816, 114982729, 155625625, 226201600, 302899216, 324648324, 519657616, 551310400, 618317956, 796989361, 943964176, 1039675536

Offset: 1

Enoch Haga, May 17 2008

			The first term of A098562 is 13, the prime sum of 2^2=4 and 3^2=9. The square just preceding 13 is 9, the first term of this sequence.

Carlos Rivera, Puzzle 128. Sum of consecutive squared primes a square, The Prime Puzzles & Problems Connection.
Carlos Rivera, Puzzle 443. Sum of cubes of consecutive primes, The Prime Puzzles and Problems Connection.

Cf. A048760, A140597, A066525, A098561, A098562.

a(n) = A048760(A098562(n)). - Jason Yuen, Sep 30 2024

A154332 Least positive integer m such that A087285(n) = A154333(m) = m^3 - next smaller square.

Original entry on oeis.org

3, 2, 32, 15, 17, 4, 7, 6, 35, 8, 11, 10, 14, 21, 12, 28, 65, 9, 56, 18, 136, 568, 23, 99, 101, 20, 13, 27, 34, 30, 143, 145, 38, 16, 19, 47, 195, 91, 197, 175, 26, 51, 59, 799, 69, 62, 163, 255, 257, 66, 31, 717, 2904, 33, 377, 79, 323, 325, 25

Offset: 1

M. F. Hasler, Jan 07 2009

nonn

The terms of this sequence constitute a "proof" for the terms listed in A087285. To prove that a number is NOT in A087285, one can check the finite number (A081120) of solutions to the corresponding Mordell equation, cf. references in A081121.

A154332(n) = { local(m); until(m++^3-sqrtint(m^3-1)^2==A087285[n],); m }

A087285(n) = A154333(a(n)) = a(n)^3 - [sqrt(a(n)^3 - 1)]^2 = A000578(a(n)) - A048760(a(n)^3-1).

cp's OEIS Frontend

A256098 Denominators for the numerators A256097.

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A381054 a(n) is the least k such that floor(sqrt(n*k/d(n*k))) - floor(sqrt(d(n*k))) = 1, where d(k) is the largest divisor of k which is <= sqrt(k).

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Maple

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A065734 Largest square <= sigma(n).

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Maxima

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A065735 Largest square <= product of first n primes.

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Mathematica

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A065737 Largest square <= binomial(n,2).

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Mathematica

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A065738 Largest square <= binomial(2n,n).

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PARI

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A065739 Largest square <= sum of first n squares.

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PARI

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A065740 Largest square <= n^n.

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A381054 a(n) is the least k such that floor(sqrt(nk/d(nk))) - floor(sqrt(d(n*k))) = 1, where d(k) is the largest divisor of k which is <= sqrt(k).