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A190882 Numbers other than prime powers divisible by the sum of the squares of their prime divisors. 4
46206, 72105, 73346, 92412, 96096, 97440, 98098, 99528, 113883, 117040, 127680, 134805, 138618, 143520, 146692, 150024, 165880, 165886, 184824, 192192, 194880, 196196, 199056, 216315, 234080, 255360, 269192, 276640, 277236, 287040, 288288, 292320, 293384, 294216, 298584, 300048, 331760 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The number of distinct prime divisors of n is > = 3, because if n = p^a * q^b where p and q are distinct primes, p^2+q^2 | n => p+q ==0 (mod p) or 0 (mod (q), but p==0 (mod q), or q==0 (mod p) is impossible.
Koninck & Luca show that this sequence is infinite. - Charles R Greathouse IV, Sep 08 2012
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jean-Marie de Koninck and Florian Luca, Integers divisible by sums of powers of their prime factors, Journal of Number Theory, Volume 128, Issue 3, March 2008, Pages 557-563.
EXAMPLE
46206 is in the sequence because the prime distinct divisors of this number are {2, 3, 17, 151} and 2^2 + 3^2 + 17^2 + 151^2 = 23103, then 46206 = 23103*2.
MAPLE
with(numtheory):for n from 1 to 200000 do:x:=factorset(n):n1:=nops(x):s:=0:for
p from 1 to n1 do: s:=s+x[p]^2:od:if n1 >= 2 and irem(n, s)=0 then printf(`%d, `, n):else fi:od:
MATHEMATICA
Select[Range[2, 332000], !PrimePowerQ[#]&&Divisible[#, Total[Select[ Divisors[#], PrimeQ]^2]]&] (* Harvey P. Dale, May 24 2022 *)
PROG
(PARI) is(n)=my(f=factor(n)[, 1]); #f>2&n%sum(i=1, #f, f[i]^2)==0 \\ Charles R Greathouse IV, May 23 2011
(PARI) is(n)=n>9 && !isprimepower(n) && n%norml2(factor(n)[, 1])==0 \\ Charles R Greathouse IV, Feb 03 2016
CROSSREFS
Cf. A066031.
Sequence in context: A115939 A168630 A361035 * A251278 A202897 A185518
KEYWORD
nonn
AUTHOR
Michel Lagneau, May 23 2011
STATUS
approved

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Last modified May 26 11:55 EDT 2024. Contains 372824 sequences. (Running on oeis4.)