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A321182 Composite numbers k such that sigma(k)/k' is an integer, where k' is the arithmetic derivative of k. 1
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
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
Sequence in context: A358219 A338486 A143202 * A268463 A108668 A201018
KEYWORD
nonn,easy
AUTHOR
Paolo P. Lava, Oct 29 2018
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)