A323288 Largest number that can be obtained from the "Choix de Bruxelles", version 2 (A323460) operation applied to n.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 110, 112, 114, 116, 118, 40, 42, 44, 46, 48, 210, 212, 214, 216, 218, 60, 62, 64, 66, 68, 310, 312, 314, 316, 318, 80, 82, 84, 86, 88, 410, 412, 414, 416, 418, 100, 102, 104, 106, 108, 510, 512, 514, 516, 518

Offset

1,1

Comments

Equally, this is the largest number that can be obtained from the "Choix de Bruxelles", version 1 (A323286) operation applied to n.

Maximal element in row n of irregular triangle in A323460 (or, equally, A323286).

Conjecture: If n contains no digit >= 5, then a(n) = 2*n; otherwise, a(n) is obtained from n by doubling the substring from the last digit >= 5 to the last digit. - Charlie Neder, Jan 19 2019. (This is true. - N. J. A. Sloane, Jan 22 2019)

Corollary: a(n)/n < 10 for all n, and a(n) = 10 - 1/k + O(1/k^2) for n = 10*k+5. - N. J. A. Sloane, Jan 23 2019

The high-water marks for a(n)/n occur at n = 1,15,25,35,45,..., cf. A017329. - N. J. A. Sloane, Jan 23 2019

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane, "Choix de Bruxelles": A New Operation on Positive Integers, arXiv:1902.01444 [math.NT], Feb 2019; Fib. Quart. 57:3 (2019), 195-200.

Formula

a(n) >= 2*n. - Rémy Sigrist, Jan 15 2019

Prog

(PARI) a(n, base=10) = { my (d=digits(n, base), v=2*n); for (w=1, #d, for (l=0, #d-w, if (d[l+1], my (h=d[1..l], m=fromdigits(d[l+1..l+w], base), t=d[l+w+1..#d]); v = max(v, fromdigits(concat([h, digits(m*2, base), t]), base))))); v } \\ Rémy Sigrist, Jan 15 2019
(Python)
def a(n):
    s, out = str(n), {n}
    for l in range(1, len(s)+1):
        for i in range(len(s)+1-l):
            if s[i] == '0': continue
            t = int(s[i:i+l])
            out.add(int(s[:i] + str(2*t) + s[i+l:]))
            if t&1 == 0: out.add(int(s[:i] + str(t//2) + s[i+l:]))
    return max(out)
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Jul 24 2022

Crossrefs

Cf. A323286, A323460, A017329.

Sequence in context: A194396 A321193 A284788 * A322131 A209211 A064676

Adjacent sequences: A323285 A323286 A323287 * A323289 A323290 A323291

Keyword

nonn,base

Author

N. J. A. Sloane, Jan 15 2019

Extensions

More terms from Rémy Sigrist, Jan 15 2019

Status

approved