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A286708
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Powerful numbers (A001694) that are not prime powers (A000961).
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45
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36, 72, 100, 108, 144, 196, 200, 216, 225, 288, 324, 392, 400, 432, 441, 484, 500, 576, 648, 675, 676, 784, 800, 864, 900, 968, 972, 1000, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2025, 2116, 2304, 2312, 2500, 2592, 2601, 2700, 2704, 2744
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OFFSET
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1,1
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COMMENTS
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If a prime p divides a(n) then p^2 must also divide a(n) and number of distinct primes dividing a(n) > 1.
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{p prime} 1/(p*(p-1)) - 1 = A082695 - A136141 - 1 = 0.17043976777096407719... - Amiram Eldar, Feb 12 2021
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EXAMPLE
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-------------------------------
| n | a(n) | prime |
| | | factorization |
|------------------------------
| 1 | 36 | {{2, 2}, {3, 2}} |
| 2 | 72 | {{2, 3}, {3, 2}} |
| 3 | 100 | {{2, 2}, {5, 2}} |
| 4 | 108 | {{2, 2}, {3, 3}} |
| 5 | 144 | {{2, 4}, {3, 2}} |
| 6 | 196 | {{2, 2}, {7, 2}} |
| 7 | 200 | {{2, 3}, {5, 2}} |
| 8 | 216 | {{2, 3}, {3, 3}} |
| 9 | 225 | {{3, 2}, {5, 2}} |
-------------------------------
a(n) = p_1^e_1*p_2^e_2*... : {{p_1, e_1}, {p_2, e_2}, ...}.
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MAPLE
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N:= 10000:
S:= {1}: P:= {1}:
p:= 1:
do
p:= nextprime(p);
if p^2 > N then break fi;
S:= map(s -> (s, seq(s*p^k, k = 2 .. floor(log[p](N/s)))), S);
P:= P union {seq(p^k, k=2..floor(log[p](N)))}:
od:
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MATHEMATICA
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Select[Range@2750, Min@FactorInteger[#][[All, 2]] > 1 && ! PrimePowerQ[#] &]
(* Second program *)
nn = 2^25; Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], ! PrimePowerQ[#] &] (* Michael De Vlieger, Jun 22 2022 *)
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PROG
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(Python)
from sympy import primefactors, factorint
print([n for n in range(4, 2745) if len(primefactors(n)) > 1 and min(list(factorint(n).values())) > 1]) # Karl-Heinz Hofmann, Feb 07 2023
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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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