A212664 Least k with precisely n partitions k = x + y satisfying x > 0 and k’ = x’ + y’, where k’, x’, y’ are the arithmetic derivatives of k, x, y.

3, 39, 213, 903, 2379, 2343, 6545, 12325, 15015, 16107, 45045, 134225, 80535, 142545, 205205, 255255, 346035, 533715, 615615, 645645, 997815, 1601145, 1369095, 1936935

Offset

1,1

Links

Table of n, a(n) for n=1..24.

Example

n=2343, x=162, y=2181 and 2343=162+2181; n’=1027, x’=297, y’=730 and 1027=297+730.
n=2343, x=308, y=2035 and 2343=308+2035; n’=1027, x’=380, y’=647 and 1027=+380+647.
n=2343, x=377, y=1966 and 2343=377+1966; n’=1027, x’=42, y’=985 and 1027=42+985.
n=2343, x=484, y=1859 and 2343=484+1859; n’=1027, x’=572, y’=455 and 1027=572+455.
n=2343, x=505, y=1838 and 2343=505+1838; n’=1027, x’=106, y’=921 and 1027=106+921.
n=2343, x=781, y=1562 and 2343=781+1562; n’=1027, x’=82, y’=945 and 1027=82+945.

Maple

with(numtheory);
A212664:=proc(q)
local a, b, c, d, f, i, j, n, p, pfs, v;
v:=array(1..100); for n from 1 to 100 do v[n]:=0; od;
a:=0;
for n from 1 to q do
  pfs:=ifactors(n)[2]; c:=n*add(op(2, p)/op(1, p), p=pfs); b:=0;
  for i from 1 to trunc(n/2) do
    pfs:=ifactors(i)[2]; d:=i*add(op(2, p)/op(1, p), p=pfs);
    pfs:=ifactors(n-i)[2]; f:=(n-i)*add(op(2, p)/op(1, p), p=pfs);
    if c=d+f then b:=b+1; fi; od;
    if b=a+1 then a:=b; print(b, n); j:=1;
       while v[b+j]>0 do a:=b+j; print(b, v[b+j]); j:=j+1; od;
    else if b>a+1 then if v[b]=0 then v[b]:=n; fi; fi; fi;
od; end:
A212664(100000);

Crossrefs

Cf. A003415, A211223-A211225, A212662, A212663.

Sequence in context: A082954 A209366 A292542 * A342969 A050392 A292294

Adjacent sequences: A212661 A212662 A212663 * A212665 A212666 A212667

Keyword

nonn,more,hard

Author

Paolo P. Lava, May 23 2012

Extensions

a(12)-a(24) from Donovan Johnson, May 25 2012

Status

approved