A248821
Number of cubes that divide 1!*2!*3!*...*n!.
Original entry on oeis.org
1, 1, 1, 2, 6, 10, 36, 64, 220, 468, 1024, 2052, 7590, 16224, 50400, 142800, 246240, 510300, 2261952, 3545856, 14152320, 40986000, 68428800, 178293960, 784274400, 1526805504, 2782080000, 9307872000, 15858633600, 28225260000, 143730892800, 225167040000
Offset: 1
a(5) counts these cubes that divide 34560: 1^3, 2^3, 3^3, 4^3, 6^3, 12^3.
-
b:= proc(n) option remember; add(i[2]*x^numtheory[pi](i[1]),
i=ifactors(n)[2])+`if`(n=1, 0, b(n-1))
end:
c:= proc(n) option remember; b(n)+`if`(n=1, 0, c(n-1)) end:
a:= n->(p->mul(iquo(coeff(p, x, i), 3)+1, i=1..degree(p)))(c(n)):
seq(a(n), n=1..30); # Alois P. Heinz, Oct 16 2014
-
z = 40; p[n_] := Product[k!, {k, 1, n}];
f[n_] := f[n] = FactorInteger[p[n]];
r[m_, x_] := r[m, x] = m*Floor[x/m]
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
t[m_, n_] := Apply[Times, 1 + r[m, v[n]]/m]
m = 3; Table[t[m, n], {n, 1, z}] (* A248821 *)
A248822
Number of integers k^4 that divide 1!*2!*3!*...*n!.
Original entry on oeis.org
1, 1, 1, 2, 3, 8, 10, 36, 64, 200, 432, 630, 1088, 4800, 7590, 32448, 47040, 114240, 164160, 835920, 1302840, 4804800, 7091712, 25243920, 39168000, 171555840, 320973840, 667447200, 1113944832, 3338108928, 5181926400, 19372953600, 31804416000, 132562944000
Offset: 1
a(6) counts these integers k^4 that divide 24883200: 1^4, 2^4, 4^4, 8^4, 6^4, 12^4, 24^4.
-
b:= proc(n) option remember; add(i[2]*x^numtheory[pi](i[1]),
i=ifactors(n)[2])+`if`(n=1, 0, b(n-1))
end:
c:= proc(n) option remember; b(n)+`if`(n=1, 0, c(n-1)) end:
a:= n->(p->mul(iquo(coeff(p, x, i), 4)+1, i=1..degree(p)))(c(n)):
seq(a(n), n=1..30); # Alois P. Heinz, Oct 16 2014
-
z = 40; p[n_] := Product[k!, {k, 1, n}];
f[n_] := f[n] = FactorInteger[p[n]];
r[m_, x_] := r[m, x] = m*Floor[x/m]
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
t[m_, n_] := Apply[Times, 1 + r[m, v[n]]/m]
m = 4; Table[t[m, n], {n, 1, z}] (* A248822 *)
A248823
Number of integers k^5 that divide 1!*2!*3!*...*n!.
Original entry on oeis.org
1, 1, 1, 2, 2, 6, 8, 10, 42, 64, 200, 432, 588, 1024, 3888, 6300, 21120, 33696, 52080, 114240, 328320, 816480, 3326400, 4435200, 6469632, 20616960, 57153600, 145411200, 258003900, 320973840, 791513856, 1634592960, 6403719168, 9967104000, 34939296000
Offset: 1
a(6) counts these integers k^5 that divide 24883200: 1, 32, 1024, 7776, 32768, 248832, these being k^5 for k = 1, 2, 3, 4, 6, 12.
-
b:= proc(n) option remember; add(i[2]*x^numtheory[pi](i[1]),
i=ifactors(n)[2])+`if`(n=1, 0, b(n-1))
end:
c:= proc(n) option remember; b(n)+`if`(n=1, 0, c(n-1)) end:
a:= n->(p->mul(iquo(coeff(p, x, i), 5)+1, i=1..degree(p)))(c(n)):
seq(a(n), n=1..30); # Alois P. Heinz, Oct 16 2014
-
z = 40; p[n_] := Product[k!, {k, 1, n}];
f[n_] := f[n] = FactorInteger[p[n]];
r[m_, x_] := r[m, x] = m*Floor[x/m]
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
t[m_, n_] := Apply[Times, 1 + r[m, v[n]]/m]
m = 5; Table[t[m, n], {n, 1, z}] (* A248823 *)
A248824
Number of integers k^6 that divide 1!*2!*3!*...*n!.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 6, 8, 18, 42, 64, 200, 432, 546, 960, 3888, 6000, 15180, 29952, 38976, 67200, 285600, 393984, 1632960, 3175200, 4165392, 6105600, 38413440, 55339200, 114048000, 205632000, 280219500, 448156800, 2621445120, 3777725952, 12940849152
Offset: 1
a(7) counts these integers k^6 that divide 125411328000 = A000178(6): 1, 64, 729, 4096, 46656, 2985984, these being k^6 for k = 1, 2, 3, 4, 6, 12.
-
b:= proc(n) option remember; add(i[2]*x^numtheory[pi](i[1]),
i=ifactors(n)[2])+`if`(n=1, 0, b(n-1))
end:
c:= proc(n) option remember; b(n)+`if`(n=1, 0, c(n-1)) end:
a:= n->(p->mul(iquo(coeff(p, x, i), 6)+1, i=1..degree(p)))(c(n)):
seq(a(n), n=1..30); # Alois P. Heinz, Oct 16 2014
-
z = 40; p[n_] := Product[k!, {k, 1, n}];
f[n_] := f[n] = FactorInteger[p[n]];
r[m_, x_] := r[m, x] = m*Floor[x/m]
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
t[m_, n_] := Apply[Times, 1 + r[m, v[n]]/m]
m = 6; Table[t[m, n], {n, 1, z}] (* A248824 *)
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