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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 19 07:13 EDT 2022. Contains 356216 sequences. (Running on oeis4.)