A381135 Numbers of the form d_1 d_2 d_3 ... where the sum of their digits multiplied by their digit positions is equal to their number of digits.

1, 20, 110, 300, 1010, 2100, 4000, 10010, 12000, 20100, 31000, 50000, 100010, 111000, 200100, 220000, 301000, 410000, 600000, 1000010, 1020000, 1101000, 1300000, 2000100, 2110000, 3001000, 3200000, 4010000, 5100000, 7000000, 10000010, 10110000, 11001000, 12100000

Offset: 1

Author

Leo Crabbe, Feb 14 2025

Keywords

• nonn
• base

Comments

The digits for these numbers are 1-indexed.

Numbers k such that A156207(k) = A055642(k).

Examples

			301000 is a member of this sequence because it has 3*1 + 1*3 = 6 digits.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Michael S. Branicky, Python program for OEIS A381135

Crossrefs

Cf. A055642, A156207.

Programs

PARI

isok(k) = my(d=digits(k)); sum(i=1, #d, i*d[i]) == #d; \\ Michel Marcus, Feb 17 2025

Python

def ok(n): return sum(int(d)*(i+1) for i, d in enumerate(str(n))) == len(str(n))

Python

# see linked program for a different algorithm
from itertools import count, islice
from sympy.utilities.iterables import partitions
def agen(): # generator of terms
    for d in count(1): yield from sorted(int("".join(str(p[k]) if k in p else "0" for k in range(1, d+1))) for p in partitions(d, m=d, k=d) if max(p.values()) < 10 if 1 in p and p[1] != 0)
print(list(islice(agen(), 34))) # Michael S. Branicky, Feb 19 2025

Extensions

More terms from Michel Marcus, Feb 17 2025