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A338093
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Composite numbers which are multiples of the sum of the squares of their prime factors (taken with multiplicity).
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1
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16, 27, 256, 540, 756, 1200, 1890, 2940, 3060, 3125, 4050, 4200, 4320, 5460, 6000, 6048, 7920, 8232, 10080, 10164, 10368, 10530, 11232, 11286, 12960, 13104, 13524, 13800, 14000, 14157, 14175, 15708, 15960, 17280, 18200, 18480, 19278, 19683, 19992, 20295, 23814
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OFFSET
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1,1
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COMMENTS
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If a(n)=p1*p2*..*pk where p1,p2,..pk primes, then a(n)=m(p1^2+p2^2+..+pk^2) with m a positive integer.
For the special case of m=1, a(n) is equal to the sum of the squares of its prime factors.
There are only 5 known numbers to have this property:
16, 27 and three more numbers with 123, 163 and 179 digits found by Giorgos Kalogeropoulos (see Rivera links).
It is not known if any smaller numbers than those three exist for the case of m=1.
Suppose n is in the sequence with n = k*A067666(n). Then n^m is in the sequence if m divides k^m (in particular for m=k).
For any prime p, p^(p^j) is in the sequence if j >= 1 (except j>=2 if p=2). (End)
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LINKS
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EXAMPLE
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16 = 2*2*2*2 = 1*(2^2 + 2^2 + 2^2 + 2^2).
7920 = 2*2*2*2*3*3*5*11 = 44*(2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 5^2 + 11^2).
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MAPLE
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filter:= proc(n) local t;
if isprime(n) then return false fi;
n mod add(t[1]^2*t[2], t=ifactors(n)[2]) = 0
end proc:
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MATHEMATICA
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Select[Range@20000, Mod[#, Total[Flatten[Table@@@FactorInteger@#]^2]]==0&]
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PROG
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(PARI) isok(m) = if (!isprime(m) && (m>1), my(f=factor(m)); (m % sum(k=1, #f~, f[k, 1]^2*f[k, 2])) == 0); \\ Michel Marcus, Oct 11 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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