A323244 - OEIS

A323244

a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).

0, 1, 1, 2, 1, 4, 1, 6, 0, 5, 1, 10, 1, 16, 2, 6, 1, 12, 1, 18, -3, 18, 1, 22, -4, 46, 4, 22, 1, 10, 1, 30, 14, 82, -2, 14, 1, 256, -12, 22, 1, 36, 1, 66, 8, 226, 1, 46, -12, 19, 8, 130, 1, 28, -19, 70, -12, 748, 1, 42, 1, 1362, 16, 22, 10, 42, 1, 214, 254, 40, 1, 38, 1, 3838, 10, 406, -10, 106, 1, 78, -12, 5458, 1, 26, -72, 12250, -348, 30, 1, 12

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OFFSET

1,4

COMMENTS

After a(1) = 0, the other zeros occur for k >= 1, at A005940(1+A000396(k)), which, provided no odd perfect numbers exist, is equal to A324201(k) = A062457(A000043(k)): 9, 125, 161051, 410338673, ..., etc.

There are 2321 negative terms among the first 10000 terms.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)

Index entries for sequences related to binary expansion of n

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

FORMULA

a(n) = 2*A156552(n) - A323243(n).

a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).

a(n) = A323248(n) + A001222(n) = (A323247(n) - A323243(n)) + A001222(n).

From Antti Karttunen, Mar 12 2019 & Nov 23 2019: (Start)

a(n) = Sum_{d|n} (2*A297112(d) - A324543(d)) = Sum_{d|n} A329644(d).

A002487(a(n)) = A324115(n).

a(n) = A329638(n) - A329639(n).

a(n) = A329645(n) - A329646(n).

(End)

MATHEMATICA

Array[2 # - If[# == 0, 0, DivisorSigma[1, #]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 90] (* Michael De Vlieger, Apr 21 2019 *)

PROG

(PARI)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};

A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));

A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));

(Python)

from sympy import divisor_sigma, primepi, factorint

def A323244(n): return (lambda n: (n<<1)-divisor_sigma(n))(sum((1<<primepi(p)-1)<<i for i, p in enumerate(factorint(n, multiple=True)))) if n > 1 else 0 # Chai Wah Wu, Mar 10 2023

CROSSREFS

Cf. A000043, A000396, A033879, A064989, A156552, A297112, A323240, A323243, A323245, A323248, A324115, A324051, A324103, A324396, A324398, A324543, A324713.

Cf. A324201 (positions of zeros, conjectured), A324551 (of negative terms), A324720 (of nonnegative terms), A324721 (of positive terms), A324731, A324732.

Cf. A329644 (Möbius transform).

Cf. A323174, A324055, A324185, A324546 for other permutations of deficiency, and also A324574, A324575, A324654.

Sequence in context: A114326 A308175 A241423 * A329642 A214052 A276094

Adjacent sequences: A323241 A323242 A323243 * A323245 A323246 A323247

KEYWORD

sign

AUTHOR

Antti Karttunen, Jan 10 2019

STATUS

approved