A342020 Row sums of A342000.

0, 2, 0, 1, 2, 6, 13, 32, 63, 124, 244, 453, 862, 1568, 2835, 5150, 9251, 16093, 27830, 48605, 84765, 145300, 245730, 417251, 721100, 1267411, 2247106, 3997263

Offset

0,2

Comments

Question: Will a(n)/(2^n) converge towards zero? After local maximum a(7)/(2^7) = 32/128 = 1/4, the ratio seems to be decreasing, and is already only 0.02978... (< 1/32) at a(27) = 3997263.

Links

Table of n, a(n) for n=0..27.

Mathematica

Block[{a, b, c, f, nn = 12}, b[0] = c[1] = 1; f[n_] := DivisorSigma[1, n]/(n Log[Log[n]]); Do[b[i] = Prime[1 + BitLength[i] - DigitCount[i, 2, 1]]*b[i - 2^Floor@ Log2@ i]; c[i + 1] = Apply[Times, Flatten@ MapIndexed[ConstantArray[Prime[First[#2]], #1] &, Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &@ Sort[Flatten[ConstantArray[PrimePi@ #1, #2] & @@@ FactorInteger[b[i]]], Greater]]]; a[i - 1] = Boole[f[c[i]] >= f[c[Floor[(i + 1)/2]]]], {i, 2^nn}]; Total /@ Table[a[2^n + k], {n, 0, nn - 2}, {k, 0, 2^n - 1}]] (* Michael De Vlieger, Mar 07 2021 *)

Prog

(PARI)
default(parisizemax, 2^31);
default(realprecision, 121);
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A329886(n) = if(n<2, 1+n, if(!(n%2), A283980(A329886(n/2)), 2*A329886(n\2)));
Gi(n) = (log(n)^(n/sigma(n)));
A342000(n) = if(n<=3, !!(n-1), my(p=A329886(n\2)); if(n%2, Gi(2*p)<=Gi(p), Gi(A283980(p))<=Gi(p))); \\ Program less vulnerable to the loss of precision.
s=0; k=0; for(n=2, 2^31, if(!bitand(n, n-1), write("b342020.txt", k, " ", s); print(s, "/2^", k, " = ", (1.0)*(s/(2^k))); k++; s = 0); s += A342000(n));

Crossrefs

Cf. A342000.

Sequence in context: A355297 A260663 A241857 * A300485 A014511 A333687

Adjacent sequences: A342017 A342018 A342019 * A342021 A342022 A342023

Keyword

nonn,more

Author

Antti Karttunen and Michael De Vlieger, Mar 07 2021

Status

approved