A329638 -id:A329638 - cp's OEIS frontend

A329641 a(n) = gcd(A329638(n), A329639(n)).

0, 1, 1, 2, 1, 4, 1, 6, 1, 5, 1, 10, 1, 16, 2, 6, 1, 1, 1, 18, 1, 18, 1, 22, 1, 46, 1, 22, 1, 10, 1, 30, 14, 82, 2, 1, 1, 256, 2, 22, 1, 1, 1, 66, 1, 226, 1, 46, 1, 1, 8, 130, 1, 1, 1, 70, 2, 748, 1, 42, 1, 1362, 2, 2, 10, 42, 1, 214, 254, 4, 1, 1, 1, 3838, 5, 406, 2, 2, 1, 78, 1, 5458, 1, 26, 2, 12250, 2, 10, 1, 2, 1, 934

Offset: 1

Antti Karttunen, Nov 22 2019

nonn

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. A156552, A323243, A324201, A329610, A329638, A329639, A329640, A329644.

A323243(n) = if(1==n,0,sigma(A156552(n)));
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A329644(n) = sumdiv(n,d,moebius(n/d)*((2*A156552(d))-A323243(d)));
A329641(n) = { my(t=0,u=0); fordiv(n, d, if((d=A329644(d))>0, t +=d, u -= d)); gcd(u,t); };

a(n) = gcd(A329638(n), A329639(n)).

a(A324201(n)) = A329610(n).

A329610 a(n) = A329638(A324201(n)).

Original entry on oeis.org

1, 5, 269, 6181, 42467605, 6698339765, 137657144245, 2482519341068581109, 4650094754920132605500516092412538533, 172211278811469777927840379819793091108558586157350885, 6181597550035473098411130103418930148893343460909707255358224869, 15429082227467244202811178054212231454083499413948013030392138990797549495221

Offset: 1

Antti Karttunen, Nov 22 2019

nonn

The first 12 terms are all squarefree (in A005117). Does this hold in general?

Cf. A000043, A005117, A324201, A329638, A329639, A329641, A329644.

a(n) = A329638(A324201(n)) = A329639(A324201(n)) = A329641(A324201(n)).

A329640 Numbers n for which A329639(n) is equal to gcd(A329638(n), A329639(n)).

Original entry on oeis.org

1, 9, 18, 27, 45, 54, 70, 75, 84, 125, 135, 144, 153, 198, 279, 366, 390, 392, 423, 459, 747, 837, 855, 858, 891, 927, 1269, 1341, 1494, 1503, 1690, 1899, 2097, 2241, 2493, 2604, 2679, 2763, 2781, 2888, 2979, 2988, 3177, 3411, 3507, 3879, 4023, 4041, 4050, 4482, 4491, 4509, 4707, 5067, 5283, 5463, 5679, 5697, 5817, 5877, 5982, 6093

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

After the initial 1, numbers n such that A329638(n) is a multiple of A329639(n).

Antti Karttunen, Table of n, a(n) for n = 1..85 (based on Hans Havermann's factorization of A156552)

Cf. A156552, A329638, A329639, A329641.
Cf. A324201 (a subsequence).
Cf. also A326141.

isA329640(n) = { my(u=A329639(n)); (u==gcd(A329638(n), u)); };

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

Original entry on oeis.org

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)

A329644 Möbius transform of A323244, the deficiency of A156552(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 4, -1, 3, 1, 5, 1, 14, 0, 0, 1, 9, 1, 12, -5, 16, 1, 8, -5, 44, 4, 5, 1, 2, 1, 24, 12, 80, -4, -4, 1, 254, -14, 0, 1, 22, 1, 47, 7, 224, 1, 24, -13, 19, 6, 83, 1, 12, -21, 44, -14, 746, 1, 14, 1, 1360, 20, -8, 8, 9, 1, 131, 252, 24, 1, 12, 1, 3836, 13, 149, -12, 71, 1, 56, -16, 5456, 1, -21, -74, 12248, -350, -40, 1

Offset: 1

Antti Karttunen, Nov 21 2019

sign

The first eleven zeros occur at n = 1, 15, 16, 40, 96, 119, 120, 160, 893, 2464, 6731. There are 3091 negative terms among the first 10000 terms.
Applying this function to the divisors of the first four terms of A324201 reveals the following pattern:
------------------------------------------------------------------------------------
A324201(n) divisors                             a(n) applied       Sum of positive
                                                 to each:          terms, A329610
        9: [1, 3, 9]                         -> [0, 1, -1]                     1
      125: [1, 5, 25, 125]                   -> [0, 1, -5, 4]                  5
   161051: [1, 11, 121, 1331, 14641, 161051] -> [0, 1, -29, 4, -240, 264]    269
410338673: [1, 17, 289, 4913, 83521, 1419857, 24137569, 410338673]
                             -> [0, 1, -125, 4, -1008, 1032, -5048, 5144]   6181
The positive and negative terms seem to alternate, and the fourth term (from case n=125 onward) is always 4. See also array A329637.

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. A000203, A008683, A036563, A156552, A297112, A297113, A323243, A323244, A324543, A329610, A329637, A329638, A329639, A329645, A329646.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));
A329644(n) = sumdiv(n,d,moebius(n/d)*A323244(d));

a(n) = Sum_{d|n} A008683(n/d) * A323244(d).
a(n) = Sum_{d|n} A008683(n/d) * (2*A156552(d) - A323243(d)).
a(1) = 0; for n > 1, a(n) = 2*A297112(n) - A324543(n) = 2^A297113(n) - A324543(n).
a(n) = A329642(n) - A329643(n).
For all n >= 1, a(A000040(n)^2) = A323244(A000040(n)^2)-1 = -A036563(n).
For all primes p, a(p^3) = A323244(p^3) - A323244(p^2) = 4.

A329639 Sum of -A329644(d) for all such divisors d of n for which A329644(d) < 0. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 5, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 5, 0, 0, 14, 0, 0, 5, 0, 0, 1, 0, 0, 0, 13, 5, 0, 0, 0, 1, 21, 0, 14, 0, 0, 0, 0, 0, 6, 8, 0, 0, 0, 0, 0, 4, 0, 5, 0, 0, 5, 0, 12, 14, 0, 0, 17, 0, 0, 26, 74, 0, 350, 40, 0, 14, 53, 0, 0, 0, 70, 0, 0, 13, 18, 7, 0, 0, 0, 0, 15

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

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. A323244, A329638, A329640, A329641, A329644.

Cf. also A318879.

A329639(n) = sumdiv(n,d,if((d=A329644(d))<0,-d,0));

a(n) = -Sum_{d|n} [A329644(d) < 0] * A329644(d), where [ ] is Iverson bracket.

a(n) = A329638(n) - A323244(n).

cp's OEIS Frontend

A329641 a(n) = gcd(A329638(n), A329639(n)).

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A329610 a(n) = A329638(A324201(n)).

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A329640 Numbers n for which A329639(n) is equal to gcd(A329638(n), A329639(n)).

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

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A329644 Möbius transform of A323244, the deficiency of A156552(n).

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A329639 Sum of -A329644(d) for all such divisors d of n for which A329644(d) < 0. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

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