A262996
Ten-digit semiprimes with exactly one 1, two 2's, three 3's and four 4's.
Original entry on oeis.org
1223444433, 1224343443, 1233444243, 1233444423, 1234424343, 1234442343, 1243344243, 1243442433, 1244332443, 1244343423, 1244344323, 1244442333, 1323442443, 1324244433, 1324344423, 1324443243, 1324443423, 1324444323, 1332244443, 1334244243, 1334244423, 1334424243, 1342443243, 1343242443
Offset: 1
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Select[(FromDigits[#] & /@ Permutations[{1, 2, 2, 3, 3, 3, 4, 4, 4, 4}]), PrimeQ[#/3] &]
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forprime(p=407814811, 1481443741, d=digits(3*p); if(vecsort(d)==[1, 2, 2, 3, 3, 3, 4, 4, 4, 4], print1(3*p", "))) \\ Charles R Greathouse IV, Sep 24 2015
A064382
Number of ways to put numbers 1, 2, ..., n*(n+1)/2 in a triangular array of n rows in such a way that each row is increasing or decreasing.
Original entry on oeis.org
1, 6, 240, 100800, 605404800, 65703372134400, 155590841484029952000, 9416463528385701835407360000, 16689045681854870055279680279347200000, 976264092939656812770160178186221444104192000000, 2097176933095579995533106263090939623313258344938995712000000
Offset: 1
Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Sep 27 2001
A172528
Triangular array T(n,k) n,k>=0 is the number of k letter words formed using at most 1a,2b's,3c's,...,n#'s.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 3, 1, 3, 8, 19, 38, 60, 60, 1, 4, 15, 53, 175, 535, 1490, 3675, 7700, 12600, 12600, 1, 5, 24, 111, 494, 2111, 8634, 33635, 123998, 428820, 1373820, 4003230, 10325700, 22522500, 37837800, 37837800
Offset: 0
T(3,2) = 8 because there are 8 two letter words that can be formed using the letters a,b,b,c,c,c: {a, b}, {a, c}, {b, a}, {b, b}, {b, c}, {c, a}, {c, b}, {c, c}.
Triangle Begins
1;
1,1;
1,2,3,3;
1,3,8,19,38,60,60;
1,4,15,53,175,535,1490,3675,7700,12600,12600;
The last entry in row n is
A022915(n).
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Table[CoefficientList[Series[Product[Sum[x^i/i!, {i, 0, n}], {n, 0, m}], {x,0, (m^2 + m)/2}], x]*Table[n!, {n, 0, (m^2 + m)/2}], {m, 0,5}] // Grid
A249675
Multinomial coefficient (2+3+...+prime(n); 2,3,5,7,...,prime(n)) = A007504(n)! / A111180(n).
Original entry on oeis.org
1, 10, 2520, 49008960, 1052427228652800, 18543848132205515927808000, 3663312848979081767807855264321333760000, 1860046674511129497317809046448809961990598435124736000000, 46250563343969315569958487234182581699417368835772148395100189090556723200000000
Offset: 1
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a := n -> add(ithprime(k),k=1..n)!/mul(ithprime(k)!,k=1..n);
seq(a(n),n=0..9); # Peter Luschny, Nov 04 2014
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a(n) = my(d=1,t=0); forprime(p=2,prime(n),d*=p!;t+=p);t!\d
A269516
Number of T(n) X T(n) matrices containing the multiset {1, 2, 2, 3, 3, 3, ..., n, n, ..., n} in each row and column, where T(n) = A000217(n).
Original entry on oeis.org
The a(3) = 5450400 (= 6!*7570) 6 X 6 matrices include
1 2 2 3 3 3
2 1 2 3 3 3
2 2 1 3 3 3
3 3 3 1 2 2
3 3 3 2 1 2
3 3 3 2 2 1.
Using terminology such as that suggested by Rebecca J. Stones in a "Chess Set Latin Square" comment, this matrix is a (3, 2, 1)-frequency square corresponding to a decomposition of K_{6,6} into one 3-regular spanning subgraph, one 2-regular spanning subgraph, and one 1-regular spanning subgraph.
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