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

%I A381159 #37 Feb 21 2025 07:19:45
%S A381159 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,
%T A381159 53,59,61,64,67,69,71,73,79,81,83,89,97,101,103,107,109,113,117,119,
%U A381159 121,125,127,128,129,131,137,139,149,151,157,159,163,167,169,173,179
%N A381159 Numbers whose prime divisors all end in the same digit.
%C A381159 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)
%C A381159 All powers of primes (A000961) are terms.
%e A381159 16, 69, 117 are included in the sequence because 16 = 2*2*2*2, 69 = 3*23, 117 = 3*3*13.
%p A381159 q:= n-> nops(map(p-> irem(p, 10), numtheory[factorset](n)))<2:
%p A381159 select(q, [$1..250])[];  # _Alois P. Heinz_, Feb 15 2025
%t A381159 q[n_] := SameQ @@ Mod[FactorInteger[n][[;; , 1]], 10]; Select[Range[2, 180], q] (* _Amiram Eldar_, Feb 16 2025 *)
%o A381159 (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
%o A381159 (Python)
%o A381159 from sympy import factorint, isprime
%o A381159 def ok(n): return n == 1 or isprime(n) or len(set(p%10 for p in factorint(n))) == 1
%o A381159 print([k for k in range(1, 180) if ok(k)]) # _Michael S. Branicky_, Feb 16 2025
%Y A381159 Union of A004618 (9), A004618 (3), A090652 (7), A004615 (1), A000351 (5), and A000079 (2).
%Y A381159 Union of A000961 and A380758.
%K A381159 nonn,base
%O A381159 1,2
%A A381159 _Alexander M. Domashenko_, Feb 15 2025