A293252 Numbers k such that k = x + y, k' = x' + y' and k'' = x'' + y'', where k' and k'' are the first and second arithmetic derivatives of k.

3, 778, 1331, 1575, 1589, 3111, 5368, 14060, 17649, 17714, 23232, 33813, 34353, 36234, 52936, 53391, 66375, 74544, 80938, 88945, 93475, 94905, 97470, 98434, 156816, 180804, 207754, 229502, 238830, 267120, 274065, 357318, 367921, 400500, 406700, 411872, 418037

Offset: 1

Author

Paolo P. Lava, Oct 04 2017

Keywords

• nonn

Comments

A226779(n) + 1 are terms of the sequence: for these numbers the relation stands for any following derivative because n = 1 + (n-1), n' = 0 + (n-1)' and n' = (n-1)' by definition. Apart 3, no other prime p can be in the sequence because p = x + y implies p' = 1 = x' + y' that is impossible (for 3 we have 3 = 1 + 2 and 3' = 1 = 1' + 2' = 0 + 1). Similarly, x and y cannot be both primes.

Is there any number that admits two or more different partitions?

Examples

			1331 = 198 + 1133, 1331' = 363 = 198' + 1133' = 249 + 114, 1331'' = 187 = 198'' + 1133'' = 86 + 101.
		

Links

• Paolo P. Lava, List of k, x, and y

Crossrefs

Cf. A003415, A226779.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,k,n,p; for n from 1 to q do
for k from 1 to trunc(n/2) do a:=k*add(op(2,p)/op(1,p),p=ifactors(k)[2]);
b:=(n-k)*add(op(2,p)/op(1,p),p=ifactors(n-k)[2]); c:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); if c=a+b then a:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]); b:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]); c:=c*add(op(2,p)/op(1,p),p=ifactors(c)[2]);
if c=a+b then print(n); break; fi; fi; od; od; end: P(10^5);

Mathematica

f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]; Select[Range[2000], Function[k, Count[IntegerPartitions[k, {2}], ?(And[f@ k == f@ #1 + f@ #2, Nest[f, k, 2] == Nest[f, #1, 2] + Nest[f, #2, 2]] & @@ # &)] > 0]] (* _Michael De Vlieger, Oct 08 2017 *)

Extensions

a(25)-a(37) from Giovanni Resta, Oct 05 2017