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

Author

Paolo P. Lava, May 23 2012

Keywords

• nonn
• more
• hard

Examples

			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.
		

Crossrefs

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

Programs

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);

Extensions

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