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

1,1

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?

Links

Table of n, a(n) for n=1..37.

Paolo P. Lava, List of k, x, and y

Example

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

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 *)

Crossrefs

Cf. A003415, A226779.

Sequence in context: A259369 A259371 A294794 * A341567 A287695 A083250

Adjacent sequences: A293249 A293250 A293251 * A293253 A293254 A293255

Keyword

nonn

Author

Paolo P. Lava, Oct 04 2017

Extensions

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

Status

approved