A329596 A variation of Recamán's sequence (A005132): a(0) = 0; a(1) = 1; a(2) = 3; for n > 2, a(n) = sopfr(a(n-1))- sopfr(n) if positive and not already in the sequence, otherwise a(n) = sopfr(a(n-1)) + sopfr(n).

0, 1, 3, 6, 9, 11, 16, 15, 2, 8, 13, 24, 16, 21, 19, 27, 17, 34, 27, 28, 20, 19, 32, 33, 5, 15, 23, 14, 20, 38, 31, 62, 43, 29, 10, 19, 29, 66, 37, 53, 42, 53, 41, 84, 29, 18, 33, 61, 50, 26, 27, 29, 12, 60, 23, 7, 20, 31, 62, 92, 39, 77, 51, 33, 26, 33, 30, 77, 39, 42

Offset

0,3

Comments

On the graph it seems that there are lines where the density of points is higher than elsewhere. These lines correspond to those which are easily observable on A001414. Up to n=100000 there are 13413 prime numbers in this sequence while at A001414 there are 21877; surely the primes are distributed differently in these sequences.

Regarding the MATLAB code: factor(0)=sum(factor(0))=0 and factor(1)=sum(factor(1))=1, this can be very misleading, attention needed during using sum(factor(n)) as sopfr(n).

Links

Bence Bernáth, Table of n, a(n) for n = 0..10000

Bence Bernáth, Table of n, a(n) for n = 0..200000

Michael De Vlieger, First 65 terms drawn as a spiral, akin to the Harriss drawing at A005132.

Michael De Vlieger, video of first 128 steps of this sequence, with audio accompaniment generated by aspects of the sequence. Nov 19, 2019.

Example

a(3)=6, factor(6)=[2 3], sum of factor(6) is 5. Then n=4, sum of factor(4) is 2+2=4. 5-4 = 1 but 1 is already in the sequence so a(4)=5+4=9.

Mathematica

Block[{f}, f[n_] := Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[n]]; Nest[Append[#1, If[And[#3 >= 0, FreeQ[#1, #3]], #3, f@ #1[[-1]] + f@ #2]] & @@ {#1, #2, f@ #1[[-1]] - f@#2} & @@ {#, Length@ #} &, {0}, 69] ] (* Michael De Vlieger, Nov 19 2019 *)

Prog

(MATLAB)
length_seq=100000;
sequence(1)=0; %sum(factor(0))=0
sequence(2)=1; %sum(factor(1))=1
for i1=2:1:length_seq
if  (sum(factor(sequence(i1)))-sum(factor((i1))))>0 && (ismember((sum(factor(sequence(i1)))-sum(factor((i1)))), sequence)==0)
      sequence(i1+1)=(sum(factor(sequence(i1)))-sum(factor((i1))));
   else
       sequence(i1+1)=(sum(factor(sequence(i1)))+sum(factor((i1))));
end
end
result=transpose(sequence);

Crossrefs

Cf. A001414, A005132.

Sequence in context: A310149 A310150 A169924 * A344956 A332369 A101723

Adjacent sequences: A329593 A329594 A329595 * A329597 A329598 A329599

Keyword

nonn,easy

Author

Bence Bernáth, Nov 17 2019

Status

approved