A393113 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n > 2, a(n) shares a factor with digsum(a(n-1)), where digsum(k) is the sum of the digits of k.

1, 2, 4, 6, 3, 9, 12, 15, 8, 14, 5, 20, 16, 7, 21, 18, 24, 22, 26, 28, 25, 35, 30, 27, 33, 32, 40, 34, 42, 36, 39, 38, 11, 44, 46, 45, 48, 50, 55, 52, 49, 13, 54, 51, 56, 66, 57, 58, 65, 77, 60, 62, 64, 68, 63, 69, 70, 84, 72, 75, 74, 88, 76, 78, 80, 82, 85, 91

Offset

1,2

Comments

As the sequence is infinite no term, except a(1), can be a power of 10.

The majority of terms are concentrated along a line with gradient ~1.14, see the attached image of the first 1 million terms. However also present are scalloped lines each of which is composed of multiples of a different prime - these initially start off lower than the line of concentration but many eventually move above it. It is unclear what causes the scalloping behavior.

In the first 1 million terms there are no fixed points other than 1 and 2, but due to the pattern formed by the primes it is plausible more may exist.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..10000.

Scott R. Shannon, Image of the first 1 million terms. The green line is a(n) = n.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple represents powerful numbers that are not prime powers.

Example

a(8) = 15 as a(7) = 12 and digsum(12) = 3, and 15 is the smallest unused number that shares a factor with 3.

Mathematica

Block[{c, i, j, k, m, d, nn, u}, nn = 120; c[_] := False; m[_] := 1; j = d = 1; u = 2; Array[Set[c[10^#], True] &, 21, 0]; {1}~Join~Reap[Do[If[PrimePowerQ[d], d = FactorInteger[d][[1, 1]]; While[c[Set[k, m[d]*d]], m[d]++], k = u; While[Or[c[k], CoprimeQ[k, d]], k++]]; Sow[k]; Set[{c[k], j, d}, {True, k, Total@ IntegerDigits[k]}]; If[k == u, While[c[u], u++]], {n, 2, nn}] ][[-1, 1]] ] (* Michael De Vlieger, Feb 03 2026 *)

Prog

(Python)
from math import gcd
from itertools import count, islice
def is_pow10(k): return str(k).strip("0") == "1"
def agen(): # generator of terms
    yield from [1, 2]
    aset, an, m = {1, 2}, 2, 3
    while True:
        s = sum(map(int, str(an)))
        an = next(i for i in count(m) if i not in aset and gcd(i, s) > 1 and not is_pow10(i))
        yield an
        aset.add(an)
        while m in aset or is_pow10(m): m += 1
print(list(islice(agen(), 68))) # Michael S. Branicky, Feb 03 2026

Crossrefs

Cf. A007953, A064413, A391057, A028834, A037123.

Sequence in context: A365259 A356851 A291577 * A395077 A354434 A362842

Adjacent sequences: A393110 A393111 A393112 * A393114 A393115 A393116

Keyword

nonn,base,look

Author

Scott R. Shannon, Feb 02 2026

Status

approved