A259379 Numbers k of the form a - b + c, such that k^3 equals the decimal concatenation a//b//c and numbers k, b, and c have the same number of digits.

155, 209, 274, 286, 287, 351, 364, 428, 573, 637, 715, 727, 846, 923, 1095, 1096, 2191, 8905, 18182, 18183, 81818, 81819, 326734, 336634, 663367, 673267, 2727273, 2727274, 4545454, 5454547, 7272727, 23529411, 23529412, 76470589

Offset

1,1

Comments

This sequence is infinite because it has several infinite subsequences. For example:

274, 326734, 332667334, 3..326..673..34 etc.;

364, 336634, 333666334, 3..36..63..34 etc.;

637, 663367, 666333667, 6..63..36..67 etc.;

727, 673267, 667332667, 6..673..326..67 etc.

Note that: 274 + 727 = 364 + 637 = 1001 and 326734 + 673267 = 336634 + 663367 = 1000001.

Many numbers come in pairs, like: (286, 287), (1095, 1096), (18182, 18183) but also bigger number (140017877, 140017878) and (859982123, 859982124).

140017877 + 859982124 = 140017878 + 859982123 = 1000000001.

Links

Pieter Post, Table of n, a(n) for n = 1..189

Example

155^3 = 3723875 and 155 = 3 - 723 + 875.
715^3 = 365525875 and 715 = 365 - 525 + 875.

Prog

(Python)
def modb(n, m):
    kk = 0
    l = 1
    while n > 0:
        na = n % m
        l += 1
        kk += ((-1)**l) * na
        n //= m
    return kk
for n in range (100, 10**9):
    ll = len(str(n))
    if modb(n**3, 10**ll) == n:
        print(n, end=', ') # corrected by David Radcliffe, May 09 2025
(PARI) isok(n)=nb = #digits(n, 10); if (a = n^3 \ 10^(2*nb), c = n^3 % 10^nb; b = (n^3 - a*10^(2*nb))\10^nb; n^3 == (a-b+c)^3; ); \\ Michel Marcus, Aug 05 2015

Crossrefs

Cf. A056733, A101311, A118937, A118938, A228381, A257796, A258482.

Sequence in context: A381381 A159258 A214472 * A122475 A383328 A252805

Adjacent sequences: A259376 A259377 A259378 * A259380 A259381 A259382

Keyword

nonn,base

Author

Pieter Post, Jul 22 2015

Status

approved