A092487 -id:A092487 - cp's OEIS frontend

A006255 R. L. Graham's sequence: a(n) = smallest m for which there is a sequence n = b_1 < b_2 < ... < b_t = m such that b_1b_2...*b_t is a perfect square.

1, 6, 8, 4, 10, 12, 14, 15, 9, 18, 22, 20, 26, 21, 24, 16, 34, 27, 38, 30, 28, 33, 46, 32, 25, 39, 35, 40, 58, 42, 62, 45, 44, 51, 48, 36, 74, 57, 52, 50, 82, 56, 86, 55, 60, 69, 94, 54, 49, 63, 68, 65, 106, 70, 66, 72, 76, 87, 118, 75, 122, 93, 77, 64, 78, 80, 134, 85, 92, 84

Offset: 1

N. J. A. Sloane and Robert G. Wilson v

Every nonprime appears exactly once in this sequence.
If n is a square we can take t=1 and a(n) = n. If n is a prime > 3, then a(n) = 2n and t=3. If n is twice a prime, say p, then a(n) = 3p most of the time. The sequence b_1 < b_2 < ... < b_t will not contain either perfect squares or primes for they bring nothing to the solution. Also I know of no n such that t = 2. - Robert G. Wilson v, Jan 30 2002
Let k be a fixed integer and p be a prime, then a(k*p) = (k+1)*p for sufficiently large p. - Peter Kagey, Feb 03 2015
From David A. Corneth, Oct 26 2016: (Start)
Is for all k*p in A277624, a(k*p) = (k+1) * p?
Conjecture: Let b(n) = A006530(A007913(n)). If b(n)^2 >= 2 * n then a(n) = n + b(n) except for n = 3, 10, and 171.
(End)
a(n) <= A072905(n).
a(n) <= 2*n for all n > 3.
a(n) >= n + A006530(A007913(n)) for all nonsquare n. - Peter Kagey, Feb 21 2015

			a(2) = 6 because the best such sequence is 2,3,6.
For n = 3 through 6 the {smallest m then smallest t then smallest product} solutions are 3,6,8; 4; 5,8,10; 6,8,12.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., Problem 4.39, pages 147, 616, 533. [Reference revised by N. J. A. Sloane, Jan 13 2014]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Peter Kagey, Table of n, a(n) for n = 1..10000
Sarosh Adenwalla, On a Generalisation of a Function of Ron Graham's, arXiv:2504.19196 [math.NT], 2025.
R. L. Graham, Bijection between integers and composites, Problem 1242, Math. Mag., 60 (1987), p. 180. [Note that unless you subscribe to JSTOR this link will only show page 178, which contains a different problem proposed by R. L. Graham. - _N. J. A. Sloane_, Jan 13 2014]
Peter Kagey and Krishna Rajesh, On a Conjecture about Ron Graham's Sequence, arXiv:2410.04728 [math.NT], 2024.

Having minimized m, next minimize t, then minimize product: A066400 and A066401 give values of t and square root of b_1*...*b_t.
If squares are omitted we get A233421.
A067565 is the inverse of R. L. Graham's sequence.
Cf. A245499, A070229, A072905, A092487.

Table[k = 0; Which[IntegerQ@ Sqrt@ n, k, And[PrimeQ@ n, n > 3], k = n, True, While[Length@ Select[n Map[Times @@ # &, n + Rest@ Subsets@ Range@ k], IntegerQ@ Sqrt@ # &] == 0, k++]]; k + n, {n, 40}] (* Michael De Vlieger, Oct 26 2016 *)

If n is a square we can take t=1 and a(n)=n.
a(n) = A245499(n,A066400(n)). - Reinhard Zumkeller, Jul 25 2014
a(n) = A092487(n) + n. - Peter Kagey, Oct 22 2016

More terms from Robert G. Wilson v, Jan 30 2002

Erroneous program (pointed out by Peter Kagey) removed by Reinhard Zumkeller, Nov 28 2014

A092488 a(n) = least k such that {n+0, n+1, n+2, n+3, ... n+k} has a nonempty subset the product of whose members is a square.

Original entry on oeis.org

0, 2, 1, 0, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 14, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17

Offset: 1

R. K. Guy, Apr 02 2004

			Often a(n) is the distance from n to the next square. But, e.g., a(26)=9 (not 10) because 27*28*30*32*35 is a square.

R. K. Guy, Unsolved Problems in Number Theory, B30.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A068527, A068568, A092487.

(* This program is not suitable to compute a large number of terms *)
a[n_] := Module[{n0, t}, n0 = Ceiling[Sqrt[n]]^2; If[n == n0, Return[0]]; Do[t = Table[n+j, {j, 0, k}]; If[AnyTrue[Subsets[t, {m}], IntegerQ[ Sqrt[ Times @@ #]]&], Return[k]], {k, 1, n0-n}, {m, 1, k+1}] ]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 200}] (* Jean-François Alcover, Nov 19 2016 *)

A277624 Composite numbers which have a dominant prime factor. A prime factor p of n is dominant if floor(sqrt(p)) > (n/p).

Original entry on oeis.org

22, 26, 34, 38, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 116, 118, 122, 123, 124, 129, 134, 141, 142, 146, 148, 158, 159, 164, 166, 172, 177, 178, 183, 185, 188, 194, 201, 202, 205, 206, 212, 213, 214, 215, 218, 219, 226, 235, 236, 237, 244

Offset: 1

Peter Luschny, Oct 24 2016

From David A. Corneth, Oct 26 2016 and Oct 27 2016: (Start)
Numbers of the form k * p where p > (k + 1)^2 and p prime and k > 1.
If n has a dominant prime factor, it's A006530(n).
All primes p > 4 have the property that floor(sqrt(A006530(p))) = floor(sqrt(p)) > (p/A006530(p)) = 1.
A063763 is a supersequence.
(End)

			133230 is in this sequence because 133230 = 2*3*5*4441 and 2*3*5 = 30 < 66 = floor(sqrt(4441)).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A006530, A063763.

is_a := proc(n) max(numtheory:-factorset(n)):
not isprime(n) and floor(sqrt(%)) > (n/%) end:
select(is_a, [$1..244]);

Select[Select[Range@ 244, CompositeQ], Function[n, Total@ Boole@ Map[Function[p, Floor@ Sqrt@ p > n/p], FactorInteger[n][[All, 1]]] > 0]] (* Michael De Vlieger, Oct 27 2016 *)

upto(n) = my(l=List()); for(k=2, sqrtnint(n, 3), forprime(p=(k+1)^2, n\k, listput(l,k*p))); listsort(l); l
is(n) = if(!isprime(n)&&n>1, f=factor(n)[, 1];sqrtint(f[#f]) > n/f[#f], 0) \\ David A. Corneth, Oct 26 2016

from sympy import primefactors
from gmpy2 import is_prime, isqrt
A277624_list = []
for n in range(2,10**3):
    if not is_prime(n):
        for p in primefactors(n):
            if isqrt(p)*p > n:
                A277624_list.append(n)
                break # Chai Wah Wu, Oct 25 2016

Conjecture: A092487(a(n)) = A006530(a(n)). - David A. Corneth, Oct 27 2016

A359507 a(n) is the least integer k such that there exists a strictly increasing integer sequence n = b_1 < b_2 < ... < b_t = n + k with the property that b_1 XOR b_2 XOR ... XOR b_t = 0.

Original entry on oeis.org

0, 2, 3, 3, 3, 5, 3, 5, 3, 5, 3, 9, 3, 5, 3, 9, 3, 5, 3, 9, 3, 5, 3, 17, 3, 5, 3, 9, 3, 5, 3, 17, 3, 5, 3, 9, 3, 5, 3, 17, 3, 5, 3, 9, 3, 5, 3, 33, 3, 5, 3, 9, 3, 5, 3, 17, 3, 5, 3, 9, 3, 5, 3, 33, 3, 5, 3, 9, 3, 5, 3, 17, 3, 5, 3, 9, 3, 5, 3, 33, 3, 5, 3, 9, 3, 5, 3, 17, 3, 5, 3, 9, 3, 5, 3, 65, 3, 5, 3, 9, 3, 5

Offset: 0

Peter Kagey, Jan 03 2023

nonn

Conjecture: a(n) is of the form 2^k + 1 for all n > 0.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A092487, A359506, A359508.

A359506(n) = if(n==0, return (0), my (x=[n], y); for (m=n+1, oo, if (vecmin(y=[bitxor(v, m) | v<-x])==0, return (m), x=setunion(x, Set(y))))); \\ From A359506.
A359507(n) = (A359506(n)-n); \\ Antti Karttunen, Nov 22 2024

a(n) = A359506(n) - n.

More terms from Antti Karttunen, Nov 22 2024

A277606 Frequency of n in A245499.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 5, 1, 3, 1, 4, 1, 2, 5, 1, 1, 7, 1, 4, 2, 2, 1, 5, 1, 2, 5, 3, 1, 3, 1, 9, 2, 2, 5, 1, 1, 2, 2, 5, 1, 4, 1, 2, 4, 2, 1, 6, 1, 8, 2, 3, 1, 5, 2, 5, 2, 2, 1, 6, 1, 2, 7, 1, 3, 4, 1, 2, 2, 5, 1, 13, 1, 2, 10, 2, 6, 2, 1, 7, 1, 2

Offset: 1

David A. Corneth, Oct 23 2016

nonn

From David A. Corneth, Jan 02 2018: (Start)
For every positive integer n, we can create a tuple b of t increasing positive integers with b_1 = n, the product of these elements a perfect square and the largest element as small as possible.
A006255 lists b_t, the largest element of these tuples, A245499 lists these tuples and this sequence lists the frequency of n occurring in such a tuple, i.e., the frequency of n in A245499. (End)
Records occur for n = 1, 3, 6, 8, 18, 32, 72, 200, ... where a(n) is 1, 2, 3, 5, 7, 9, 13, 19, ... respectively.

			8 occurs in rows 3, 5, 6, 7 and 8 being respectively [3, 6, 8], [5, 8, 10], [6, 8, 12], [7, 8, 14] and [8, 10, 12, 15]. These are 5 rows so a(8) = 5.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A006255, A092487, A245499.

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A006255 R. L. Graham's sequence: a(n) = smallest m for which there is a sequence n = b_1 < b_2 < ... < b_t = m such that b_1*b_2*...*b_t is a perfect square.

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A092488 a(n) = least k such that {n+0, n+1, n+2, n+3, ... n+k} has a nonempty subset the product of whose members is a square.

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A277624 Composite numbers which have a dominant prime factor. A prime factor p of n is dominant if floor(sqrt(p)) > (n/p).

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A359507 a(n) is the least integer k such that there exists a strictly increasing integer sequence n = b_1 < b_2 < ... < b_t = n + k with the property that b_1 XOR b_2 XOR ... XOR b_t = 0.

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PARI

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A277606 Frequency of n in A245499.

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A006255 R. L. Graham's sequence: a(n) = smallest m for which there is a sequence n = b_1 < b_2 < ... < b_t = m such that b_1b_2...*b_t is a perfect square.