A351741 Numbers k such that the concatenation of 1,2,...,k and the concatenation of k,k-1,...,1 have the same number of prime factors, counted with multiplicity.

1, 3, 4, 5, 7, 10, 13, 16, 23, 26, 31, 32, 37, 39, 51, 54, 56

Offset: 1

Author

J. M. Bergot and Robert Israel, Feb 17 2022

Keywords

• nonn
• base
• more

Comments

Numbers k such that A001222(A007908(k)) = A001222(A000422(k)).

Examples

			a(3) = 4 is a term because 1234 = 2*617 and 4321 = 29*149 each have two prime factors.
		

Crossrefs

Cf. A000422, A001222, A007908, A048288.

Programs

Maple

dcat:= (a,b) -> a*10(1+ilog10(b))+b:
a:= 1: b:= 1: R:= 1:
for n from 2 to 40 do
a:= dcat(n,a);
b:= dcat(b,n);
if numtheory:-bigomega(a) = numtheory:-bigomega(b) then R:= R,n fi
od:
R;

Mathematica

Select[Range[32], SameQ @@ PrimeOmega@{FromDigits@ Flatten@ #, FromDigits@ Flatten@ Reverse[#]} &@ IntegerDigits@ Range[#] &] (* Michael De Vlieger, Feb 17 2022 *)

Python

from sympy import primeomega
def afind(limit, startk=1):
    k = startk
    sk = "".join(str(i) for i in range(1, k))
    skr = "".join(str(i) for i in range(k-1, 0, -1))
    for k in range(startk, limit+1):
        sk += str(k)
        skr = str(k) + skr
        if primeomega(int(sk)) == primeomega(int(skr)):
            print(k, end=", ")
afind(23) # Michael S. Branicky, Feb 17 2022

Extensions

a(17) from Michael S. Branicky, Feb 19 2022