A053087 -id:A053087 - cp's OEIS frontend

A053092 2^A053087 kara n.

1, 1, 3, 1, 7, 3, 5, 1, 15, 7, 3, 11, 5, 19, 9, 1, 31, 15, 7, 13, 25, 3, 23, 11, 21, 5, 19, 37, 9, 35, 17, 1, 63, 31, 15, 29, 7, 27, 53, 13, 25, 49, 3, 47, 23, 45, 11, 43, 21, 41

Offset: 1

Robert Lozyniak (11(AT)onna.com), Feb 27 2000

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

a053092 n = f 1 where
   f x = case x `kara` n of
              Nothing -> f $ 2 * x
              Just y  -> y
   kara a b = if null ks then Nothing else Just $ head ks
              where ks = [c | c <- [1..a], a <= c * b, a > c * (b - 1)]
-- Reinhard Zumkeller, Mar 30 2013

A053405 Definition: A kara B = C, where C is the least nonnegative integer such that C * B >= A and C * (B-1) < A. a(n) is the smallest k such that n kara k is undefined.

Original entry on oeis.org

3, 4, 4, 5, 5, 4, 6, 5, 5, 6, 6, 6, 5, 7, 7, 6, 6, 8, 7, 7, 7, 6, 8, 8, 8, 7, 7, 9, 9, 8, 8, 8, 7, 9, 9, 9, 9, 8, 8, 10, 10, 10, 9, 9, 9, 8, 11, 10, 10, 10, 10, 9, 9, 11, 11, 11, 11, 10, 10, 10, 9, 12, 12, 11, 11, 11, 11, 10, 10, 12, 12, 12, 12, 12, 11, 11, 11, 10, 13, 13, 13, 12, 12, 12, 12

Offset: 4

Robert Lozyniak (11(AT)onna.com), Jan 08 2000

			7 kara 4 = 2 because 2 * 4 > 7 and 2 * 3 < 7.
7 kara 5 is undefined: 7 kara 5 != 2 because 2 * (5-1) > 7 and 7 kara 5 != 1 because 1 * 5 < 7.
4 kara 3 is undefined: 4 kara 2 = 2; 4 kara 4 = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 4..10000

Cf. A053087, A257213.

a053405 n = head [a | a <- [1..], n `kara` a == Nothing] where
   kara a b = if null ks then Nothing else Just $ head ks
              where ks = [c | c <- [1..a], a <= c * b, a > c * (b - 1)]
-- Reinhard Zumkeller, Mar 30 2013

kara[a_, b_] := Module[{r, c}, r = Reduce[c*b >= a && c*(b-1) < a, c, Integers]; If[r === False, Null, c /. {ToRules[r]} // First]]; a[n_] := For[k = 2, True, k++, If[!IntegerQ[n ~kara~ k], Return[k]]]; Table[a[n], {n, 4, 88}] (* Jean-François Alcover, Sep 13 2013 *)

a(n) = A257213(n-1) + 1. - Ridouane Oudra, Apr 19 2025

Corrected and extended by James Sellers, Jan 10 2000

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A053092 2^A053087 kara n.

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A053405 Definition: A kara B = C, where C is the least nonnegative integer such that C * B >= A and C * (B-1) < A. a(n) is the smallest k such that n kara k is undefined.

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