A336499
Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 0, 1, 3, 2, 5, 3, 3, 2, 1, 1, 4, 2, 7, 4, 4, 3, 2, 0, 1, 4, 2, 7, 4, 5, 7, 7, 6, 3, 2, 0, 1, 4, 2, 8, 8, 9, 10, 11, 11, 7, 8, 5, 2, 0, 1, 4, 3, 11, 8, 11, 16, 16, 15, 15, 15, 13, 9, 6, 3, 1, 1, 5, 3, 14, 10, 13, 21, 21, 20, 19, 21, 18, 13, 9, 5, 2, 0
Offset: 0
Triangle begins:
1
1
1 1
1 2 0
1 2 1 2 1
1 3 1 3 2 0
1 3 2 5 3 3 2 1
1 4 2 7 4 4 3 2 0
1 4 2 7 4 5 7 7 6 3 2 0
1 4 2 8 8 9 10 11 11 7 8 5 2 0
1 4 3 11 8 11 16 16 15 15 15 13 9 6 3 1
1 5 3 14 10 13 21 21 20 19 21 18 13 9 5 2 0
1 5 3 14 10 14 25 23 27 24 30 28 28 25 20 16 11 5 2 0
Row n = 7 counts the following divisors:
1 2 4 8 16 48 144 720 {}
3 9 12 24 72 360 1008
5 18 40 80 504
7 20 56 112
28
45
63
A022559 gives row lengths minus one.
A056172 appears to be column k = 2.
A336420 is the version for superprimorials.
A336498 is the version counting all divisors.
A336865 is the generalization to non-factorials.
A336866 lists indices of rows with a final 1.
A336867 lists indices of rows with a final 0.
A336868 gives the final terms in each row.
A000110 counts divisors of superprimorials with distinct prime exponents.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327498 gives the maximum divisor of n with distinct prime exponents.
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Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,6},{k,0,PrimeOmega[n!]}]
A336865
Irregular triangle read by rows where T(n,k) is the number of divisors of n with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 0, 0
Offset: 1
The triangle begins as follows. The n-th row is shown to the right of "n:".
1: (1) 16: (1,1,1,1,1) 31: (1,1)
2: (1,1) 17: (1,1) 32: (1,1,1,1,1,1)
3: (1,1) 18: (1,2,1,1) 33: (1,2,0)
4: (1,1,1) 19: (1,1) 34: (1,2,0)
5: (1,1) 20: (1,2,1,1) 35: (1,2,0)
6: (1,2,0) 21: (1,2,0) 36: (1,2,2,2,0)
7: (1,1) 22: (1,2,0) 37: (1,1)
8: (1,1,1,1) 23: (1,1) 38: (1,2,0)
9: (1,1,1) 24: (1,2,1,2,1) 39: (1,2,0)
10: (1,2,0) 25: (1,1,1) 40: (1,2,1,2,1)
11: (1,1) 26: (1,2,0) 41: (1,1)
12: (1,2,1,1) 27: (1,1,1,1) 42: (1,3,0,0)
13: (1,1) 28: (1,2,1,1) 43: (1,1)
14: (1,2,0) 29: (1,1) 44: (1,2,1,1)
15: (1,2,0) 30: (1,3,0,0) 45: (1,2,1,1)
Row n = 72 counts the following divisors:
1 2 4 8 24 72
3 9 12
18
Row n = 1200 counts the following divisors:
1 2 4 8 16 48 400 1200
3 25 12 24 80 600
5 20 40 200
50
75
A130092 gives positions of rows ending with 0.
A146291 is the version not requiring distinct prime multiplicities.
A336499 is the restriction to factorial numbers.
A001222 counts prime factors, counting multiplicity.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor of n with distinct prime multiplicities.
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Table[Length[Select[Divisors[n],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,20},{k,0,PrimeOmega[n]}]
A336871
Number of divisors d of A076954(n) with distinct prime multiplicities such that the numerator of A006939(n)/d also has distinct prime multiplicities.
Original entry on oeis.org
1, 2, 4, 11, 28, 96, 309, 1256, 4676, 21647
Offset: 0
The a(0) = 1 through a(3) = 11 divisors:
1 2 18 2250
1 9 1125
3 375
1 125
75
45
25
18
9
5
1
A336419 is the version for superprimorials.
A336500 is the generalization to all positive integers.
A006939 lists superprimorials or Chernoff numbers.
A007425 counts divisors of divisors.
A076954 is a sister of superprimorials.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
Cf.
A001055,
A022559,
A022915,
A027423,
A091050,
A124010,
A317829,
A327498,
A327527,
A336420,
A336421,
A336571.
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chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
cochern[n_]:=Product[Prime[i]^i,{i,n}];
Table[Length[Select[Divisors[cochern[n]],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[chern[n]/#]&]],{n,0,5}]
Comments