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

Author

Bence Bernáth, Nov 17 2019

Keywords

• nonn
• easy

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).

Examples

			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.
		

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.

Crossrefs

Cf. A001414, A005132.

Programs

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);

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 *)