A084891 -id:A084891 - cp's OEIS frontend

A162422 Numbers with at least 2 different numbers of digits among their prime factors.

22, 26, 33, 34, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 99, 102, 104, 106, 110, 111, 114, 115, 116, 117, 118, 119, 122, 123, 124, 129, 130, 132, 133, 134, 136, 138, 141, 142, 145, 146, 148, 152

Offset: 1

Cino Hilliard, Jul 03 2009

Complement of A162421. There are no prime numbers in this sequence.
These numbers can also be called factor rough numbers.
Basically, the number of digits of A020639(k) and of A006530(k) must differ to admit k into the sequence.

			1111 = 11*101 has factors with different digital lengths. Also it is the first occurrence that differs from A084891.

Cf. A084891, A162457

factorrough(m,n) =
{
local(x,a,j,f,ln);
for(x=m,n, f=0; a = ifactor(x); for(j=2,length(a), ln=length(Str(a[j-1])); if(length(Str(a[j]))!=ln,f=1;break);); if(f,print1(x",")););
}

{k >1: A055642(A020639(k)) <> A055642(A006530(k)) }. - R. J. Mathar, Sep 16 2009

Offset set to 1, definition shortened - R. J. Mathar, Sep 16 2009

A376740 Numbers that have at least one two-digit prime factor.

Original entry on oeis.org

11, 13, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 102, 104, 106, 110, 111, 114, 115, 116, 117

Offset: 1

Kishin Ikemoto, Oct 03 2024

Subsequence of A068191, first differing at A068191(55) = 101 which is not a term here.
Numbers k such that gcd(k,10978895066407230594062391177770267) > 1. - Chai Wah Wu, Nov 18 2024 [The big number is A109819(10) - Alois P. Heinz, Nov 18 2024]
The asymptotic density of this sequence is A051953(A109819(10))/A109819(10) = 1329644281346285477858013527/2807455661493975149742813527 = 0.473611... . - Amiram Eldar, Nov 19 2024

			201 = 3*67 is in this sequence because it has one two-digit prime factor.
202 = 2*101 is not, because neither of them is two-digit.

Kishin Ikemoto, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, order 10978895066407230594062391177770267.

Cf. A051953, A068191, A084891, A109819.

q:= convert(select(isprime, [seq(i,i=11 .. 99, 2)]),`*`):
filter:= n -> igcd(n,q) > 1:
select(filter, [$1..200]); # Robert Israel, Nov 18 2024

A376740Q[k_] := GCD[k, 10978895066407230594062391177770267] > 1;
Select[Range[200], A376740Q] (* Paolo Xausa, Jun 24 2025 *)

is(k) = {forprime(p = 11, 97, if(!(k % p), return(1))); 0;} \\ Amiram Eldar, Nov 19 2024

def ok(n): return any(n%p == 0 for p in [11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97])
print([k for k in range(1, 118) if ok(k)]) # Michael S. Branicky, Oct 15 2024

a(n + A051953(A109819(10))) = a(n) + A109819(10). - Amiram Eldar, Nov 19 2024

cp's OEIS Frontend

A162422 Numbers with at least 2 different numbers of digits among their prime factors.

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