A377012 Numbers k whose digits can be split into substrings so that the sum of these substrings raised to consecutive powers (1, 2, 3, ...) is the number k itself.

89, 135, 175, 518, 598, 1306, 1370, 1371, 1676, 2045, 2427, 3055, 5755, 5918, 9899, 11053, 24429, 88297, 135010, 234322, 255050, 255051, 360030, 360031, 494703, 512780, 517380, 568217, 767368, 779243, 785920, 785921, 788834, 819116, 931316, 986562, 998999, 1000100

Offset

1,1

Comments

At least two substrings are required and each substring must have at least one digit.

A subsequence is A032799 (except for 1-digit numbers).

Unlike A032799, which is finite, this sequence is infinite because; e.g., the pattern 89, 9899, 998999, ... can always be split into two equal-length substrings that generate a term as (10^i - 2)^1 + (10^i - 1)^2 = 10^i*(10^i - 2) + (10^i - 1) for all i > 0.

Leading zeros in strings are allowed here. The first such term generated by the author's program is 1010053 = 1^1 + (01005)^2 + 3^3. - Michael S. Branicky, Nov 30 2024

Another pattern is deduced from a(38) = 1000100 = 100^1 + 0^2 + 100^3 with formula (10^i)^1 + 0^2 + ... + 0^n + ([0...0]10^i)^(n+1) = (10^i)^(n+1) + 10^i with i > 1 and n > 1. - Francesco Di Matteo, Jan 15 2025

Links

Table of n, a(n) for n=1..38.

Example

175 = 1^1 + 7^2 + 5^3 is a term.
5755 = 5^1 + 75^2 + 5^3 is a term.
88297 = 88^1 + 297^2 is a term.
234322 = 23^1 + 4^2 + 3^3 + 22^4 is a term.

Prog

(Python)
import itertools
analys = range(1, 7) # increase this if you want
for limite in analys:
    numbers = range(pow(10, limite-1), pow(10, limite))
    r = range(1, limite+1)
    disp_temp = []
    for s in r:
        disp = list(itertools.product(r, repeat=s+1))
        disp_temp.extend(disp)
    disp_ok = [d for d in disp_temp if sum(d)==limite]
    for numero in numbers:
        str_numero = str(numero)
        for combo in disp_ok:
            k = limite
            totale = 0
            for c in range(len(combo), 0, -1):
                partenza = k-combo[c-1]
                porzione = str_numero[partenza:k]
                if c == 1:
                    totale = totale + int(porzione)
                else:
                    totale = totale + pow(int(porzione), c)
                k = k - combo[c-1]
            if totale == numero:
                print(numero)

Crossrefs

Cf. A005188, A208130, A347189, A362843.

Sequence in context: A132258 A257860 A256242 * A360422 A109560 A260557

Adjacent sequences: A377009 A377010 A377011 * A377013 A377014 A377015

Keyword

nonn,base

Author

Francesco Di Matteo, Oct 28 2024

Status

approved