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- #!/usr/bin/ruby
- # a(n) is the smallest n-gonal number with exactly n prime factors (counted with multiplicity).
- # https://oeis.org/A358863
- # Known terms:
- # 28, 16, 176, 4950, 8910, 1408, 346500, 277992, 7542080, 326656, 544320, 120400000, 145213440, 48549888, 4733575168, 536813568, 2149576704, 3057500160, 938539560960, 1358951178240
- # PARI/GP program:
- #`(
- bigomega_polygonals(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p,ceil(A/m)), B\m, my(t=m*q); if(ispolygonal(t,k), listput(list, t))), forprime(q = p, sqrtnint(B\m, n), my(t=m*q); if(ceil(A/t) <= B\t, list=concat(list, f(t, q, n-1))))); list); vecsort(Vec(f(1, 2, n)));
- a(n, k=n) = if(k < 3, return()); my(x=2^n, y=2*x); while(1, my(v=bigomega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ ~~~~
- )
- func upper_bound(n, from = 2, upto = 2*from) {
- say "\n:: Searching an upper-bound for a(#{n})\n"
- loop {
- var count = n.almost_prime_count(from, upto)
- if (count > 0) {
- say "Sieving range: [#{from}, #{upto}]"
- say "This range contains: #{count.commify} elements\n"
- n.almost_primes_each(from, upto, {|v|
- if (v.is_polygonal(n)) {
- say "a(#{n}) = #{v}"
- return v
- }
- })
- }
- from = upto+1
- upto *= 2
- }
- }
- upper_bound(29)
- __END__
- a(3) = 28
- a(4) = 16
- a(5) = 176
- a(6) = 4950
- a(7) = 8910
- a(8) = 1408
- a(9) = 346500
- a(10) = 277992
- a(11) = 7542080
- a(12) = 326656
- a(13) = 544320
- a(14) = 120400000
- a(15) = 145213440
- a(16) = 48549888
- a(17) = 4733575168
- a(18) = 536813568
- a(19) = 2149576704
- a(20) = 3057500160
- a(21) = 938539560960
- a(22) = 1358951178240
- a(23) = 36324805836800
- a(24) = 99956555776
- a(25) = 49212503949312
- a(26) = 118747221196800
- a(27) = 59461613912064
- a(28) = 13749193801728
- a(29) = 7526849672380416
- a(30) = 98516240758210560
- a(31) = 4969489493917696
- a(32) = 78673429816934400
- a(33) = 4467570822566903808
- a(34) = 1013309912383488000
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