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A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes. 7
16, 64, 81, 96, 144, 160, 224, 256, 324, 352, 384, 400, 416, 486, 544, 576, 608, 625, 640, 729, 736, 784, 864, 896, 928, 960, 992, 1024, 1184, 1215, 1296, 1312, 1344, 1376, 1408, 1440, 1504, 1536, 1600, 1664, 1696, 1701, 1888, 1936, 1944, 1952, 2016, 2025 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A semiprime (A001358) is a product of any two not necessarily distinct primes.

If A025487(k) is in the sequence then so is every number with the same prime signature. - David A. Corneth, Oct 23 2018

Numbers for which A001222(n) is even and A322353(n) is zero. - Antti Karttunen, Dec 06 2018

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

EXAMPLE

A complete list of all factorizations of 1296 into semiprimes is:

  1296 = (4*4*9*9)

  1296 = (4*6*6*9)

  1296 = (6*6*6*6)

None of these is strict, so 1296 belongs to the sequence.

MATHEMATICA

strsemfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strsemfacs[n/d], Min@@#>d&]], {d, Select[Rest[Divisors[n]], PrimeOmega[#]==2&]}]];

Select[Range[1000], And[EvenQ[PrimeOmega[#]], strsemfacs[#]=={}]&]

PROG

(PARI)

A322353(n, m=n, facs=List([])) = if(1==n, my(u=apply(bigomega, Vec(facs))); (0==length(u)||(2==vecmin(u)&&2==vecmax(u))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs, d); s += A322353(n/d, d-1, newfacs))); (s));

isA300892(n) = if(bigomega(n)%2, 0, (0==A322353(n))); \\ Antti Karttunen, Dec 06 2018

CROSSREFS

Cf. A001055, A001358, A005117, A006881, A007717, A025487, A028260, A045778, A318871, A318953, A320462, A320655, A320656, A320891, A320893, A320894, A322353.

Sequence in context: A303797 A118902 A092210 * A062320 A233330 A322449

Adjacent sequences:  A320889 A320890 A320891 * A320893 A320894 A320895

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 23 2018

STATUS

approved

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Last modified October 18 02:23 EDT 2019. Contains 328135 sequences. (Running on oeis4.)