A380758 -id:A380758 - cp's OEIS frontend

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

Alexander M. Domashenko, Feb 15 2025

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.

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

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

Union of A000961 and A380758.

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

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

isok(k) = if (k==1, 1, my(f=factor(k)); #Set(vector(#f~, i, f[i, 1] % 10)) == 1); \\ Michel Marcus, Feb 16 2025

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

A387364 Least number which is not a prime power and whose prime factors are equal modulo n.

Original entry on oeis.org

6, 15, 10, 21, 14, 55, 46, 33, 22, 39, 26, 85, 82, 51, 34, 57, 38, 115, 118, 69, 46, 141, 142, 145, 159, 87, 58, 93, 62, 259, 314, 185, 202, 111, 74, 205, 226, 123, 82, 129, 86, 235, 262, 141, 94, 371, 291, 265, 298, 159, 106, 321, 327, 295, 334, 177, 118, 183

Offset: 1

Yaroslav Deryavko, Aug 27 2025

			For n = 2, factors of a(2) = 15 are 3 and 5, they both have a residue of 1 mod 2.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

First term of A380758 when n = 10.

a[n_]:=Module[{k=1},Until[!PrimePowerQ[k]&&Min[Mod[First/@FactorInteger[k],n]]==Max[Mod[First/@FactorInteger[k],n]],k++];k];Array[a,58] (* James C. McMahon, Sep 03 2025 *)

f(k, n) = if (!isprimepower(k), my(f=factor(k)[,1]); #Set(apply(x->Mod(x, n), f)) == 1);
a(n) = my(k=1); while (!f(k,n), k++); k; \\ Michel Marcus, Aug 27 2025

cp's OEIS Frontend

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

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