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%I #26 Feb 08 2023 13:12:05
%S 36,72,100,108,144,196,200,216,225,288,324,392,400,432,441,484,500,
%T 576,648,675,676,784,800,864,900,968,972,1000,1089,1125,1152,1156,
%U 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
%N Powerful numbers (A001694) that are not prime powers (A000961).
%C If a prime p divides a(n) then p^2 must also divide a(n) and number of distinct primes dividing a(n) > 1.
%C Intersection of A001694 and A024619.
%H Michael De Vlieger, <a href="/A286708/b286708.txt">Table of n, a(n) for n = 1..10000</a> (first 5997 terms from Robert Israel)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePower.html">Prime Power</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerfulNumber.html">Powerful Number</a>.
%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>
%F 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
%e -------------------------------
%e | n | a(n) | prime |
%e | | | factorization |
%e |------------------------------
%e | 1 | 36 | {{2, 2}, {3, 2}} |
%e | 2 | 72 | {{2, 3}, {3, 2}} |
%e | 3 | 100 | {{2, 2}, {5, 2}} |
%e | 4 | 108 | {{2, 2}, {3, 3}} |
%e | 5 | 144 | {{2, 4}, {3, 2}} |
%e | 6 | 196 | {{2, 2}, {7, 2}} |
%e | 7 | 200 | {{2, 3}, {5, 2}} |
%e | 8 | 216 | {{2, 3}, {3, 3}} |
%e | 9 | 225 | {{3, 2}, {5, 2}} |
%e -------------------------------
%e a(n) = p_1^e_1*p_2^e_2*... : {{p_1, e_1}, {p_2, e_2}, ...}.
%p N:= 10000:
%p S:= {1}: P:= {1}:
%p p:= 1:
%p do
%p p:= nextprime(p);
%p if p^2 > N then break fi;
%p S:= map(s -> (s, seq(s*p^k, k = 2 .. floor(log[p](N/s)))), S);
%p P:= P union {seq(p^k, k=2..floor(log[p](N)))}:
%p od:
%p sort(convert(S minus P, list)); # _Robert Israel_, May 14 2017
%t Select[Range@2750, Min@FactorInteger[#][[All, 2]] > 1 && ! PrimePowerQ[#] &]
%t (* Second program *)
%t 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 *)
%o (Python)
%o from sympy import primefactors, factorint
%o 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
%Y Cf. A000961, A001221, A001694, A024619, A082695, A136141, A131605.
%K nonn
%O 1,1
%A _Ilya Gutkovskiy_, May 13 2017
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