A056737 -id:A056737 - cp's OEIS frontend

A382485 a(n) = ceiling(n/d^2) where d is the largest divisor of n which is not greater than the square root of n.

1, 2, 3, 1, 5, 2, 7, 2, 1, 3, 11, 2, 13, 4, 2, 1, 17, 2, 19, 2, 3, 6, 23, 2, 1, 7, 3, 2, 29, 2, 31, 2, 4, 9, 2, 1, 37, 10, 5, 2, 41, 2, 43, 3, 2, 12, 47, 2, 1, 2, 6, 4, 53, 2, 3, 2, 7, 15, 59, 2, 61, 16, 2, 1, 3, 2, 67, 5, 8, 2, 71, 2, 73, 19, 3, 5, 2, 3, 79, 2, 1, 21, 83, 2, 4, 22, 10, 2, 89

Offset: 1

Clive Tooth, Mar 30 2025

Conjecture: There exists some constant, k, approximately equal to 1.2, such that a(n) is of average order k*n/log(n). See Tooth Link for evidence.

			a(12)=2 because the largest factor of 12, which is not greater than sqrt(12), is 3; and ceiling(12/3^2)=2.

Robert Israel, Table of n, a(n) for n = 1..10000
Clive Tooth, [Sum_{i=2..n} a(i)]/[Sum_{i=2..n} i/log(i)] for n=10^7 to 10^9

Cf. A033676, A033677, A056737.

f:= proc(n) local d;
     d:= max(select(t -> t^2 <= n, numtheory:-divisors(n)));
     ceil(n/d^2)
end proc:
map(f, [$1..100]); # Robert Israel, Apr 30 2025

a[n_]:=Ceiling[n/(Select[Divisors[n],#<=Sqrt[n]&][[-1]])^2];Array[a,89] (* James C. McMahon, Apr 07 2025 *)

a(n) = my(d=divisors(n)); ceil(n/d[(length(d)+1)\2]^2); \\ Michel Marcus, Apr 07 2025

a(n) = ceiling(A033677(n)/A033676(n)).
a(n) = 1 iff n is a square.
a(n) = n iff n is prime, or 1.

A307034 a(n) is the smallest integer k of the form k = x*(x + a(n-1)), such that A324920(k) = n, for some positive integer x, with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 10, 11, 26, 87, 178, 179, 362, 1835, 22164, 155197, 620804, 5587317, 55873270, 167619819, 1340958616, 57661222337, 345967334058, 25255615391563, 858690923314298, 4293454616571515, 60108364632001406, 3185743325496077327, 178401626227780333448, 1605614636050023001113

Offset: 0

Daniel Suteu and Robert G. Wilson v, Mar 23 2019

nonn

Inspired by A324921.

This sequence provides upper bounds for A324921, i.e.: A324921(n) <= a(n).

Cf. A056737, A324920, A324921.

f(n) = my(d=divisors(n)); d[(2+#d)\2]-d[(1+#d)\2]; \\ A056737
g(n) = my(k=0); while(n!=0, k++; n=f(n)); k; \\ A324920
a(n) = if(n <= 0, return(0)); my(t=a(n-1)); for(k=1, oo, if(g(k*(k+t)) == n, return(k*(k+t))));

cp's OEIS Frontend

A382485 a(n) = ceiling(n/d^2) where d is the largest divisor of n which is not greater than the square root of n.

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