A381159 Numbers whose prime divisors all end in the same digit.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 53, 59, 61, 64, 67, 69, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 117, 119, 121, 125, 127, 128, 129, 131, 137, 139, 149, 151, 157, 159, 163, 167, 169, 173, 179

Offset

1,2

Comments

51st All-Russian Mathematical Olympiad for Schoolchildren. Problem. Let us call a natural number "lopsided" if it is greater than 1 and all its prime divisors end with the same digit. Is there an increasing arithmetic progression with a difference not exceeding 2025, consisting of 150 natural numbers, each of which is "lopsided"? (A. Chironov)

All powers of primes (A000961) are terms.

Links

Table of n, a(n) for n=1..62.

Example

16, 69, 117 are included in the sequence because 16 = 2*2*2*2, 69 = 3*23, 117 = 3*3*13.

Maple

q:= n-> nops(map(p-> irem(p, 10), numtheory[factorset](n)))<2:
select(q, [$1..250])[];  # Alois P. Heinz, Feb 15 2025

Mathematica

q[n_] := SameQ @@ Mod[FactorInteger[n][[;; , 1]], 10]; Select[Range[2, 180], q] (* Amiram Eldar, Feb 16 2025 *)

Prog

(PARI) isok(k) = if (k==1, 1, my(f=factor(k)); #Set(vector(#f~, i, f[i, 1] % 10)) == 1); \\ Michel Marcus, Feb 16 2025
(Python)
from sympy import factorint, isprime
def ok(n): return n == 1 or isprime(n) or len(set(p%10 for p in factorint(n))) == 1
print([k for k in range(1, 180) if ok(k)]) # Michael S. Branicky, Feb 16 2025

Crossrefs

Union of A004618 (9), A004618 (3), A090652 (7), A004615 (1), A000351 (5), and A000079 (2).

Union of A000961 and A380758.

Sequence in context: A306013 A268082 A382351 * A247199 A087441 A326645

Adjacent sequences: A381156 A381157 A381158 * A381160 A381161 A381162

Keyword

nonn,base

Author

Alexander M. Domashenko, Feb 15 2025

Status

approved