A324055 Deficiency of Doudna-sequence: a(n) = A033879(A005940(1+n)).

1, 1, 2, 1, 4, 0, 5, 1, 6, 2, 6, -4, 19, -3, 14, 1, 10, 4, 10, -2, 22, -12, 12, -12, 41, 7, 26, -19, 94, -12, 41, 1, 12, 8, 18, 0, 38, -12, 22, -10, 58, -4, 18, -48, 102, -54, 30, -28, 109, 25, 66, -17, 148, -72, 47, -51, 286, 32, 126, -64, 469, -39, 122, 1, 16, 10, 22, 4, 46, -12, 42, -8, 70, 4, 42, -56, 178, -60, 58, -26, 118, 20

Offset: 0

Author

Antti Karttunen, Feb 14 2019

Keywords

• sign
• look

Comments

Both here and in the mirror image sequence A324185, the lowermost (asinh) scatter plot shows on the y = 0 line the numbers that correspond to the perfect numbers. Compare also to the scatter plot of A243492.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192
• Antti Karttunen, Data supplement: n, a(n) computed for n = 0..70327
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000120, A033879, A054429.

See A106737, A290077, A323915, A324052, A324054, A324056, A324057, A324058, A324114, A324335, A324340, A324348, A324349, A324394, A324395 for other sequences as permuted by A005940, and compare their scatter plots.

Cf. also A243492, A323174, A323244, A324185, A324346, A324347, A324654.

Programs

Mathematica

Array[Block[{p = Partition[Split[Join[IntegerDigits[#, 2], {2}]], 2]}, 2 # - DivisorSigma[1, #] &[Times @@ Flatten@ Table[Prime[Count[Flatten@ #, 0] + 1]^#[[1, 1]] &@ Take[p, -i], {i, Length[p]}]]] &, 82, 0] (* Michael De Vlieger, Mar 11 2019, after Robert G. Wilson v at A005940 *)

PARI

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A033879(n) = (2*n-sigma(n));
A324055(n) = A033879(A005940(1+n));

PARI

A324055(n) = { my(m1=2,m2=1,p=2,mp=p*p); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, m1 *= p; if(3==(n%4),mp *= p,m2 *= (mp-1)/(p-1))); n>>=1); (m1-m2); };

Formula

a(n) = A033879(A005940(1+n)).

a(n) = 2*A005940(1+n) - A324054(n).

For n > 0, a(n) = A324185(A054429(n)).

a(n) = A324348(n) + A000120(A005940(1+n)).