A347189 Positive integers that can be expressed as the sum of powers of their digits (from right to left) with consecutive natural exponents.

1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 332, 1676, 121374, 4975929, 134116265, 1086588775, 3492159897, 8652650287, 8652650482

Offset: 1

Author

Reiner Moewald, Aug 21 2021

Keywords

• nonn
• base
• more

Comments

Any number > 9 consisting of just one digit (A014181) can't be in the list. (Provable using the Carmichael function.)

Examples

			24 = 4^2 + 2^3 is a term.
332 = 2^3 + 3^4 + 3^5 is another term.
		

Crossrefs

Cf. A023052, A032799, A014181.

Programs

Python

liste = []
for ex in range(0, 20):
    for t in range(1, 10000):
        n = t
        pot = ex
        ergebnis = 0
        while n > 0:
            pot = pot + 1
            rest = n % 10
            n = (n - rest) // 10
            zw = 1
            for i in range(pot):
                zw = zw * rest
            ergebnis = ergebnis + zw
        if (int(ergebnis) == t) and (t not in liste):
            liste.append(t)
liste.sort()
print(liste)

Python

def powsum(digits, startexp):
    return sum(digits[i]**(startexp+i) for i in range(len(digits)))
def ok(n):
    if n < 10: return True
    s = str(n)
    if set(s) <= {'0', '1'}: return False
    digits, startexp = list(map(int, s))[::-1], 1
    while powsum(digits, startexp) < n: startexp += 1
    return n == powsum(digits, startexp)
print(list(filter(ok, range(2*10**5)))) # Michael S. Branicky, Aug 29 2021

Extensions

a(15)-a(19) from Michael S. Branicky, Aug 31 2021