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A321182 Composite numbers k such that sigma(k)/k' is an integer, where k' is the arithmetic derivative of k. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

LINKS

Table of n, a(n) for n=1..48.

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

Cf. A000203, A003415, A230164, A242152.

Sequence in context: A244969 A090196 A143202 * A268463 A108668 A201018

Adjacent sequences:  A321179 A321180 A321181 * A321183 A321184 A321185

KEYWORD

nonn,easy

AUTHOR

Paolo P. Lava, Oct 29 2018

STATUS

approved

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Last modified February 25 08:21 EST 2020. Contains 332221 sequences. (Running on oeis4.)