A158804 -id:A158804 - cp's OEIS frontend

A089352 Numbers that are divisible by the sum of their distinct prime factors (A008472).

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 70, 71, 73, 79, 81, 83, 84, 89, 90, 97, 101, 103, 105, 107, 109, 113, 120, 121, 125, 127, 128, 131, 137, 139, 140, 149, 150, 151, 157, 163, 167, 168, 169, 173

Offset: 1

Ramin Naimi (rnaimi(AT)oxy.edu), Dec 26 2003

The Koninck & Luca bound of x / exp(c(1 + o(1))sqrt(log x log log x)) on A158804 applies equally to this sequence. - Charles R Greathouse IV, Sep 08 2012

			84=2*2*3*7 is divisible by 2+3+7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jean-Marie de Koninck, Florian Luca, Integers divisible by the sum of their prime factors, Mathematika 52:1-2 (2005), pp. 69-77.

Cf. A008472 (sopf).

Different from A071139.

primeDivisors[n_] := Select[Divisors[n], PrimeQ]; primeSumDivQ[n_] := 0 == Mod[n, Apply[Plus, primeDivisors[n]]]; Select[Range[2, 300], primeSumDivQ]
Select[Range[2, 175], Divisible[#, Plus @@ First /@ FactorInteger[#]] &] (* Jayanta Basu, Aug 13 2013 *)

is(n)=my(f=factor(n)[,1]);n%sum(i=1,#f,f[i])==0 \\ Charles R Greathouse IV, Feb 01 2013

Name edited by Michel Marcus, Jul 15 2020

A336099 Number of solutions of the equation k = n*sopf(k) in positive integers where sopf(k) is the sum of distinct prime factors of k.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 4, 1, 2, 2, 2, 1, 3, 2, 1, 1, 2, 1, 4, 1, 1, 0, 3, 1, 3, 1, 1, 2, 2, 1, 0, 1, 2, 2, 4, 1, 1, 2, 2, 1, 1, 1, 4, 2, 1, 1, 5, 1, 2, 2, 1, 2, 1, 1, 2, 1

Offset: 2

Vladimir Letsko, Jul 08 2020

nonn

Offset is 2 because a(1) cannot be defined since there are infinitely many solutions for n = 1, the primes.
If n = p^s then p^(s+1) is solution of k = n*sopf(k). Hence a(p^s) > 0. On the other hand there are infinitely many 0's in the sequence. For example a(5^s*11^t) = 0 for all positive integers s, t.
Records appear to occur only at prime n. These are seen in A336296, although note that A336296 is not monotonic, so it includes other terms. - Bill McEachen, Dec 02 2023

			a(3) = 2 because there are exactly 2 solutions of the equation k = 3*sopf(k) in positive integers (9 and 30).

Vladimir Letsko, Mathematical Marathon, Problem 227 (in Russian).

Cf. A008472, A336098, A336296.

Cf. A158804 (all possible k's).

cp's OEIS Frontend

A089352 Numbers that are divisible by the sum of their distinct prime factors (A008472).

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