A324115 -id:A324115 - cp's OEIS frontend

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

Offset: 1

Antti Karttunen, Jan 10 2019

sign

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.

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)

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.

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 *)

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));

from sympy import divisor_sigma, primepi, factorint
def A323244(n): return (lambda n: (n<<1)-divisor_sigma(n))(sum((1< 1 else 0 # Chai Wah Wu, Mar 10 2023

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)

A324117 Number of odd divisors in A156552(n): a(1) = 0, for n > 1, a(n) = A001227(A156552(n)) = A000005(A322993(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 3, 4, 1, 2, 2, 4, 2, 4, 1, 4, 1, 2, 2, 4, 2, 4, 1, 2, 4, 4, 1, 2, 1, 2, 2, 8, 1, 2, 2, 3, 4, 2, 1, 2, 3, 2, 4, 6, 1, 2, 1, 4, 2, 6, 2, 4, 1, 4, 2, 2, 1, 4, 1, 4, 2, 4, 2, 4, 1, 2, 4, 4, 1, 6, 4, 8, 8, 8, 1, 6, 3, 4, 6, 12, 4, 4, 1, 3, 4, 4, 1, 6, 1, 2, 4

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4473
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A001227, A156552, A246277, A322813, A322993, A324105, A324115, A324118.

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))));
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
A322993(n) = A156552(2*A246277(n));
A324117(n) = if(1==n, 0, numdiv(A322993(n)));

a(1) = 0; for n > 1, a(n) = A000005(A322993(n)) = A000005(A156552(2*A246277(n))) = A324105(2*A246277(n)).

A324116 a(n) = A002487(1+A323247(n)) = A324288(A156552(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 2, 5, 5, 1, 3, 1, 6, 7, 8, 1, 2, 1, 4, 9, 7, 1, 13, 7, 8, 5, 5, 1, 3, 1, 7, 11, 9, 10, 12, 1, 10, 13, 18, 1, 4, 1, 6, 8, 11, 1, 12, 9, 2, 15, 7, 1, 11, 13, 23, 17, 12, 1, 19, 1, 13, 11, 13, 16, 5, 1, 8, 19, 3, 1, 13, 1, 14, 7, 9, 13, 6, 1, 17, 10, 15, 1, 26, 19, 16, 21, 28, 1, 18, 17, 10, 23, 17, 22, 23, 1, 2

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

Like A323902, this also has quite a moderate growth rate, even though a certain subset of terms of A156552 soon grow quite big.

Cf. A002487, A005187, A156552, A323247, A323902, A324115, A324288.

A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ So we use this one, modified from the one given in A002487
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
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))));
A324288(n) = A002487(1+A005187(n));
A324116(n) = A324288(A156552(n));

a(n) = A002487(1+A323247(n)) = A324288(A156552(n)).

a(p) = 1 for all primes p.

A324049 a(n) = A002487(A324051(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 2, 0, 1, 4, 3, 2, 1, 1, 2, 11, 2, 1, 4, 4, 4, 7, 5, 1, 1, 1, 5, 3, 3, 26, 4, 3, 11, 4, 3, 2, 1, 2, 5, 21, 4, 3, 6, 1, 5, 37, 7, 2, 5, 1, 7, 1, 3, 12, 9, 4, 40, 14, 8, 4, 76, 1, 6, 4, 8, 63, 5, 7, 3, 3, 1, 53, 27, 3, 9, 7, 5, 5, 10, 2, 199, 98, 10, 4, 37, 6, 4, 2, 4, 8, 25, 34, 104, 12, 3, 43, 6, 1, 4, 5, 2, 117, 13, 2, 1, 3, 7, 76, 4

Offset: 2

Antti Karttunen, Feb 22 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 2..4473

Cf. A002487, A324051.

Cf. also A323902, A324115, A324116.

PARI
```
A324049(n) = A002487(A324051(n));
```

a(n) = A002487(A324051(n)).

A324285 a(n) = A002487(A297168(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 3, 0, 3, 0, 4, 2, 3, 0, 4, 0, 5, 3, 5, 0, 4, 1, 6, 2, 7, 0, 5, 0, 4, 4, 7, 2, 7, 0, 8, 5, 7, 0, 7, 0, 9, 3, 9, 0, 5, 1, 5, 6, 11, 0, 8, 3, 10, 7, 10, 0, 9, 0, 11, 5, 5, 4, 13, 0, 13, 8, 6, 0, 10, 0, 12, 4, 15, 2, 19, 0, 9, 3, 13, 0, 11, 5, 14, 9, 13, 0, 11, 3, 17, 10, 15, 6, 6, 0, 6, 7, 9, 0, 25, 0, 16, 5

Offset: 1

Antti Karttunen, Feb 22 2019

nonn

Cf. A002487, A156552, A297168.

Cf. also A323902, A324115, A324116, A324049.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n,0,2^A297167(n));
A297168(n) = sumdiv(n,d,(dA297112(d)); \\ Antti Karttunen, Feb 22 2019
A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
A324285(n) = A002487(A297168(n));

a(n) = A002487(A297168(n)).

cp's OEIS Frontend

A323244 a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).

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A324117 Number of odd divisors in A156552(n): a(1) = 0, for n > 1, a(n) = A001227(A156552(n)) = A000005(A322993(n)).

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A324116 a(n) = A002487(1+A323247(n)) = A324288(A156552(n)).

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A324049 a(n) = A002487(A324051(n)).

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A324285 a(n) = A002487(A297168(n)).

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