A379931 -id:A379931 - cp's OEIS frontend

A379929 Numbers that have the same number of prime factors, counted with multiplicity, as there are runs in their base-10 representation.

2, 3, 5, 7, 10, 11, 14, 15, 21, 25, 26, 34, 35, 38, 39, 46, 49, 51, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 115, 118, 119, 122, 124, 125, 130, 133, 138, 147, 148, 153, 154, 155, 164, 165, 166, 170, 171, 172, 174, 175, 177, 182, 186, 190, 195, 207, 212, 221, 226, 230, 231

Offset: 1

Robert Israel, Jan 06 2025

Numbers k such that A001222(k) = A043562(k).

			a(5) = 10 is a term because 10 has two runs (1 and 0) and two prime factors, 2 and 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001222, A043562, A379930, A379931.

filter:= proc(n) local L; L:= convert(n,base,10); nops(L) - numboccur(0, L[2..-1]-L[1..-2]) = numtheory:-bigomega(n) end proc:
select(filter, [$1..1000]);

A379929Q[n_] := PrimeOmega[n] == Length[Split[IntegerDigits[n]]];
Select[Range[300], A379929Q] (* Paolo Xausa, Jan 08 2025 *)

from sympy import primeomega
def ok(n): return primeomega(n) == len(s:=str(n)) - sum(1 for i in range(1, len(s)) if s[i-1] == s[i])
print([k for k in range(1, 232) if ok(k)]) # Michael S. Branicky, Jan 08 2025

A379930 Numbers that have the same number of divisors as there are runs in their base-10 representation.

Original entry on oeis.org

1, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 121, 169, 199, 211, 223, 227, 229, 233, 277, 289, 311, 331, 337, 361, 433, 443, 449, 499, 529, 557, 577, 599, 661, 677, 733, 773, 811, 841, 877, 881, 883, 887, 911, 961, 977, 991, 997, 1018, 1027, 1037, 1041

Offset: 1

Robert Israel, Jan 06 2025

Numbers k such that A000005(k) = A043562(k).

			a(5) = 23 is a term because it has two runs, 2 and 3, and two divisors, 1 and 23.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A043562, A379929, A379931.

filter:= proc(n) local L; L:= convert(n, base, 10); nops(L) - numboccur(0, L[2..-1]-L[1..-2]) = numtheory:-tau(n) end proc:
select(filter, [$1..1000]);

A379930Q[n_] := DivisorSigma[0, n] == Length[Split[IntegerDigits[n]]];
Select[Range[1000], A379930Q] (* Paolo Xausa, Jan 08 2025 *)

cp's OEIS Frontend

A379929 Numbers that have the same number of prime factors, counted with multiplicity, as there are runs in their base-10 representation.

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