A379930 -id:A379930 - 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

A379931 Numbers whose maximum exponent in their prime factorization is the number of runs in their base-10 representation.

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 12, 18, 20, 22, 25, 28, 33, 36, 45, 49, 50, 52, 55, 60, 63, 66, 68, 75, 76, 77, 84, 90, 92, 98, 100, 104, 108, 111, 116, 117, 120, 125, 135, 136, 152, 168, 184, 188, 189, 216, 220, 222, 225, 228, 232, 244, 248, 250, 264, 270, 280, 296, 297, 300, 312, 328, 332, 338, 343, 351

Offset: 1

Robert Israel, Jan 06 2025

Numbers k such that A051903(k) = A043562(k).

If k has r runs, maximum exponent m <= r, and is coprime to 10, then 10^(r+1) * k is a term. Therefore this sequence is infinite.

			a(10) = 22 is a term because 22 = 2 * 11 has maximum exponent 1, and one run in its base 10 representation.

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

Cf. A043562, A051903, A379929, A379930.

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

A379931Q[n_] := n > 1 && Max[FactorInteger[n][[All, 2]]] == Length[Split[IntegerDigits[n]]];
Select[Range[400], A379931Q] (* 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python