A321182 Composite numbers k such that sigma(k)/k' is an integer, where k' is the arithmetic derivative of k.

15, 35, 45, 95, 119, 143, 209, 287, 319, 323, 377, 527, 559, 693, 779, 899, 923, 989, 1007, 1189, 1199, 1343, 1349, 1763, 1919, 2159, 2507, 2759, 2911, 3239, 3599, 3827, 4031, 4607, 5183, 5207, 5249, 5459, 5543, 6439, 6811, 6887, 7067, 7279, 7739, 8159, 8639, 9179

Offset: 1

Author

Paolo P. Lava, Oct 29 2018

Keywords

• nonn
• easy

Comments

Alternative definition: Composite numbers such that the ratio between the sum of the reciprocal of their divisors and the sum of the reciprocal of their prime factors, counted with multiplicity, is an integer.

Mainly squarefree numbers, which are a subset of A242152, apart from some sporadic terms: 45, 693, 6811, 17296, 24016, 71753, 1165669, etc.

A230164 is a subsequence (ratio equal to 1).

Examples

			Divisors of 45 are 1, 3, 5, 9, 15, 45 and prime factors 3^2, 5: (1/1 + 1/3 + 1/5 + 1/9 + 1/15 + 1 /45)/(1/3 + 1/3 + 1/5) = 2
Divisors of 119 are 1, 7, 17, 119 and prime factors 7, 17: (1/1 + 1/7 + 1/17 + 1 /119)/(1/7 + 1/17) = 6.
Divisors of 552521 are 1, 37, 109, 137, 4033, 5069, 14933, 552521 and prime factors 37, 109, 137: (1/1 + 1/37 + 1/109 + 1/137 + 1 /4033 + 1/5069 + 1/14933 + 1/552521)/(1/37 + 1/109 + 1/137) = 24.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A003415, A230164, A242152.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,k,n; for n from 2 to q do if not isprime(n) then a:=add(1/a,a=divisors(n)); b:=ifactors(n)[2]; c:=add(b[k][2]/b[k][1],k=1..nops(b)); if frac(a/c)=0 then print(n); fi; fi; od; end: P(10^7);

Mathematica

Select[Range[4, 10^4], And[CompositeQ@ #, IntegerQ[DivisorSigma[1, #]/If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]]]] &] (* Michael De Vlieger, Oct 31 2018 *)

PARI

ard(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]); \\ A003415
isok(n) = (n>1) && !isprime(n) && (frac(sigma(n)/ard(n)) == 0); \\ Michel Marcus, Oct 30 2018