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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.
4
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
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
KEYWORD
nonn,more,hard
AUTHOR
Paolo P. Lava, May 23 2012
EXTENSIONS
a(12)-a(24) from Donovan Johnson, May 25 2012
STATUS
approved