A000422 -id:A000422 - cp's OEIS frontend

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

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

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

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

Cf. A000422, A001222, A007908, A048288.

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;

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

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

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

A386985 Smallest k > 0 such that the base-n number formed by concatenating k, k - 1, ..., 2, 1 (each written in base n) is prime, or -1 if no such k exists for the given n.

Original entry on oeis.org

2, 2, 4, 2, 2, 373, 2, 2, 82, 2, 3

Offset: 2

Marco Ripà, Aug 11 2025

If a(13) != -1, then the corresponding prime must have more than 4178 decimal digits.
Sequence continues n=13..36: ?, 2, 2, 28, 362, 2, ?, 2, 2, 4, 2, 3, ?, 2, 5, 4, 2, 2, 37, 3, 2, 4, 2, 2.
The increasing-order analog begins 15, 2, ?. See A048436.

			a(3) = 2 since 21_(base-3) = 7_(base-10), which is prime.

Mathematics Stack Exchange, Concatenation of consecutive positive integers: Does a prime also exist in base 4 (as in bases 2, 3, 5, 6, 7, 8, 9,...)?.
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992.

Cf. A000422, A048436, A281571.

f(n, b) = my(p=1, L=1); sum(k=1, n, k*p*=L+(k==L&&!L*=b)); \\ adapted from A000422
a(n) = my(k=1); while (!ispseudoprime(f(k, n)), k++); k; \\ Michel Marcus, Aug 16 2025

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Python

Extensions