A276655 Numbers j such that Sum_{p|j} 0.p is an integer where p ranges over the prime divisors of j.

1, 21, 30, 60, 63, 90, 120, 147, 150, 180, 189, 240, 270, 300, 360, 441, 450, 480, 540, 567, 600, 720, 750, 810, 900, 960, 979, 1029, 1080, 1200, 1323, 1350, 1411, 1440, 1463, 1500, 1547, 1620, 1701, 1742, 1800, 1920, 1947, 2059, 2090, 2160, 2210, 2250, 2318

Offset: 1

Author

Jaroslav Krizek, Sep 10 2016

Keywords

• nonn
• base

Comments

Here 0.p means the decimal fraction obtained by writing p after the decimal point, e.g., 0.11 = 11/100.

The first few values of Sum_{p|n} 0.p for n >= 1 are 0, 1/5, 3/10, 1/5, 1/2, 1/2, 7/10, 1/5, 3/10, 7/10, ...

Numbers j such that Sum_{p|j} 0.p (where p ranges over the prime divisors of j) = numbers j such that A276651(j) / A276652(j) is an integer.

See A276513 - the smallest number k such that Sum_{p|k} 0.p = n where p ranges over the prime divisors of k.

Sum_{p|a(n)} 0.p = 1 for first 133 terms of this sequence; Sum_{p|a(134)} 0.p = Sum_{p|16102} 0.p = 2. For number 16102 with set of prime divisors {2, 83, 97} holds: 0.2 + 0.83 + 0.97 = 2.

It is clear from the definition that if j is in the sequence so are all numbers m with rad(m) = rad(j). For example, since 21 is in the sequence, so are 63, 147, 189, 441, 567, 1029, 1323, 1701, etc. - Charles R Greathouse IV, Sep 10 2016

Examples

			The prime divisors of 60 are 2, 3, and 5, and 0.2 + 0.3 + 0.5 = 1, so 60 is a term.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A276513, A276651, A276652, A276653, A276654.

Programs

Magma

[n: n in [1..1000000] | Denominator(&+[d/(10^(#Intseq(d))): d in PrimeDivisors(n)]) eq 1]

Mathematica

{1}~Join~Select[Range[2400], IntegerQ@ Total[# 10^(-Floor@ Log10@ # - 1) &@ FactorInteger[#][[All, 1]]] &] (* Michael De Vlieger, Sep 12 2016 *)

PARI

is(n)=my(f=factor(n)[,1]); denominator(sum(i=1,#f, f[i]/10^#Str(f[i])))==1 \\ Charles R Greathouse IV, Sep 10 2016

Formula

A276652(a(n)) = 1.

Extensions

a(1) inserted by Charles R Greathouse IV, Sep 10 2016