A339245 -id:A339245 - cp's OEIS frontend

A036785 Numbers divisible by the squares of two distinct primes.

36, 72, 100, 108, 144, 180, 196, 200, 216, 225, 252, 288, 300, 324, 360, 392, 396, 400, 432, 441, 450, 468, 484, 500, 504, 540, 576, 588, 600, 612, 648, 675, 676, 684, 700, 720, 756, 784, 792, 800, 828, 864, 882, 900, 936, 968, 972, 980, 1000, 1008, 1044

Offset: 1

Alford Arnold

Not squarefree, not a nontrivial prime power and not in {squarefree} times {nontrivial prime powers}.

Numbers k such that A056170(k) > 1. The asymptotic density of this sequence is 1 - (6/Pi^2) * (1 + A154945) = 0.05668359058... - Amiram Eldar, Nov 01 2020

CRC Standard Mathematical Tables and Formulae, 30th ed., (1996) page 102-105.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005117, A025475, A056170, A154945.
Equivalent sequence for 3 distinct primes: A318720.
Cf. A085986, A338539, A339245 (subsequences).
Subsequence of A038838.

Select[Range@ 1050, And[Length@ # > 1, Total@ Boole@ Map[# > 1 &, #[[All, -1]]] > 1] &@ FactorInteger@ # &] (* Michael De Vlieger, Apr 25 2017 *)
dstdpQ[n_]:=Length[Select[Sqrt[#]&/@Divisors[n],PrimeQ]]>1; Select[ Range[ 1100],dstdpQ] (* Harvey P. Dale, Jan 15 2020 *)

is(n)=my(f=vecsort(factor(n)[,2],,4));#f>1&&f[2]>1 \\ Charles R Greathouse IV, Nov 15 2012

More terms from Larry Reeves (larryr(AT)acm.org), Apr 03 2000

New name from Charles R Greathouse IV, Nov 15 2012

A017643 a(n) = (12n+10)^3.

Original entry on oeis.org

1000, 10648, 39304, 97336, 195112, 343000, 551368, 830584, 1191016, 1643032, 2197000, 2863288, 3652264, 4574296, 5639752, 6859000, 8242408, 9800344, 11543176, 13481272, 15625000, 17984728, 20570824, 23393656, 26463592, 29791000, 33386248, 37259704

Offset: 0

N. J. A. Sloane

6n + 5 = (12n + 10) / 2 is never a square, as 5 is not a quadratic residue modulo 6. Using this, we can show that each term has an even square part and an even squarefree part, neither part being a power of 2. (Less than 2% of integers have this property - see A339245.) - Peter Munn, Dec 14 2020

Eric Weisstein's World of Mathematics, Quadratic Residue.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A000578, A017641 are used in a formula defining this sequence.
Subsequence of A339245.
Cf. A017642, A017644.

A017643:=(12*n+10)^3; seq(A017643(n), n=0..100); # Wesley Ivan Hurt, Nov 25 2013

(12Range[0,30]+10)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{1000,10648,39304,97336},30] (* Harvey P. Dale, Sep 30 2011 *)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with a(0)=1000, a(1)=10648, a(2)=39304, a(3)=97336. [Harvey P. Dale, Sep 30 2011]

a(n) = A017641(n)^3 = A000578(A017641(n)). - Michel Marcus, Nov 25 2013

A017139 a(n) = (8*n + 6)^3.

Original entry on oeis.org

216, 2744, 10648, 27000, 54872, 97336, 157464, 238328, 343000, 474552, 636056, 830584, 1061208, 1331000, 1643032, 2000376, 2406104, 2863288, 3375000, 3944312, 4574296, 5268024, 6028568, 6859000, 7762392, 8741816, 9800344, 10941048, 12167000, 13481272, 14886936

Offset: 0

N. J. A. Sloane

4*n + 3 = (8*n + 6) / 2 is never a square, as 3 is not a quadratic residue modulo 4. Using this, we can show that each term has an even square part and an even squarefree part, neither part being a power of 2. (Less than 2% of integers have this property - see A339245.) - Peter Munn, Dec 14 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Quadratic Residue.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A000578, A016839, A017137 are used in a formula defining this sequence.
Subsequence of A339245.
Cf. A002117, A017138, A017140.

[(8*n+6)^3: n in [0..35]]; // Vincenzo Librandi, Jul 22 2011

Table[(8*n+6)^3,{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2010 *)
LinearRecurrence[{4,-6,4,-1},{216,2744,10648,27000},30] (* Harvey P. Dale, Dec 11 2012 *)

From R. J. Mathar, Mar 22 2010: (Start)
G.f.: 8*(27 + 235*x + 121*x^2 + x^3)/(x-1)^4.
a(n) = 8*A016839(n). (End)
a(0)=216, a(1)=2744, a(2)=10648, a(3)=27000, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Dec 11 2012
a(n) = A017137(n)^3 = A000578(A017137(n)). - Peter Munn, Dec 20 2020
Sum_{n>=0} 1/a(n) = 7*zeta(3)/128 - Pi^2/512. - Amiram Eldar, Apr 26 2023

More terms from Vladimir Joseph Stephan Orlovsky, Mar 17 2010

A190892 Numbers that can be written as ab = cd*e, where a, b, c, d, and e are distinct composite numbers.

Original entry on oeis.org

192, 216, 240, 288, 320, 336, 360, 384, 432, 448, 480, 504, 528, 540, 560, 576, 600, 624, 640, 648, 672, 704, 720, 756, 768, 792, 800, 810, 816, 832, 840, 864, 880, 896, 900, 912, 936, 960, 972, 1000, 1008, 1024, 1040, 1056, 1080, 1088, 1104, 1120, 1134

Offset: 1

T. D. Noe, May 23 2011

nonn

Similar to A175340, but without the requirement that the composite numbers be consecutive. In this case, the k in A175340 can be taken to be 2.

Almost all numbers are in this sequence. Its complement has density O(n (log log n)^4/log n). - Charles R Greathouse IV, May 23 2011

			192 = 12*16 = 4*6*8.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A162247 (all factorizations of numbers), A175340.

A339245 is a subsequence.

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A036785 Numbers divisible by the squares of two distinct primes.

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A017643 a(n) = (12n+10)^3.

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A017139 a(n) = (8*n + 6)^3.

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A190892 Numbers that can be written as a*b = c*d*e, where a, b, c, d, and e are distinct composite numbers.

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A190892 Numbers that can be written as ab = cd*e, where a, b, c, d, and e are distinct composite numbers.