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 A286708 Powerful numbers (A001694) that are not prime powers (A000961). 48
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS If a prime p divides a(n) then p^2 must also divide a(n) and number of distinct primes dividing a(n) > 1. Intersection of A001694 and A024619. LINKS Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 5997 terms from Robert Israel) Eric Weisstein's World of Mathematics, Prime Power. Eric Weisstein's World of Mathematics, Powerful Number. Index entries for sequences related to powerful numbers FORMULA 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 EXAMPLE ------------------------------- | 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}, ...}. MAPLE 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: sort(convert(S minus P, list)); # Robert Israel, May 14 2017 MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A000961, A001221, A001694, A024619, A082695, A136141, A131605. Sequence in context: A036785 A338539 A347960 * A355462 A363216 A363169 Adjacent sequences: A286705 A286706 A286707 * A286709 A286710 A286711 KEYWORD nonn AUTHOR Ilya Gutkovskiy, May 13 2017 STATUS approved

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Last modified June 17 11:50 EDT 2024. Contains 373445 sequences. (Running on oeis4.)