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A286708 Powerful numbers (A001694) that are not prime powers (A000961). 2
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

Robert Israel, Table of n, a(n) for n = 1..5997

Eric Weisstein's World of Mathematics, Prime Power

Eric Weisstein's World of Mathematics, Powerful Number

Index entries for sequences related to powerful numbers

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[#] &]

CROSSREFS

Cf. A000961, A001221, A001694, A024619, A131605.

Sequence in context: A249726 A192026 A036785 * A114127 A322658 A224830

Adjacent sequences:  A286705 A286706 A286707 * A286709 A286710 A286711

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, May 13 2017

STATUS

approved

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Last modified May 31 16:29 EDT 2020. Contains 334748 sequences. (Running on oeis4.)