A297114 -id:A297114 - cp's OEIS frontend

A317844 Difference between A294898 and its Möbius transform (A297114).

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 3, 2, 0, 0, 1, 0, 0, 3, 7, 0, -6, 2, 9, 3, 1, 0, -2, 0, 0, 7, 14, 5, -9, 0, 15, 9, -4, 0, 3, 0, 5, 5, 18, 0, -14, 3, 14, 14, 7, 0, 2, 9, -3, 15, 24, 0, -24, 0, 25, 10, 0, 11, 12, 0, 12, 18, 15, 0, -29, 0, 33, 16, 13, 10, 14, 0, -12, 10, 37, 0, -23, 16, 38, 24, 1, 0, -16, 12, 16, 25

Offset: 1

Antti Karttunen, Aug 09 2018

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Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A001065, A005187, A008683, A294898, A297114, A300244.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A317844(n) = -sumdiv(n,d,(dA005187(d)-sigma(d)));

a(n) = -Sum_{d|n, dA008683(n/d)*A294898(d).
a(n) = A294898(n) - A297114(n).
a(n) = A300244(n) - A001065(n).

A324397 a(1) = 0; for n > 1, a(n) = A297114(A156552(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 3, -2, 3, 0, 7, 0, 14, -2, 0, 0, 9, 0, 15, -6, 9, 0, 18, -4, 33, -2, 14, 0, 4, 0, 25, -2, 42, -4, 7, 0, 254, -26, 9, 0, 33, 0, 63, -2, 140, 0, 41, -8, 14, -34, 127, 0, 24, -12, 66, -90, 579, 0, 38, 0, 684, -2, 6, -4, 21, 0, 175, -2, 37, 0, 24, 0, 3587, -2, 304, -8, 85, 0, 73, -14, 2733, 0, 6, -52, 8707, -378, 11, 0, 3

Offset: 1

Antti Karttunen, Mar 05 2019

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Antti Karttunen, Table of n, a(n) for n = 1..4137

Cf. A000203, A005187, A156552, A297114, A323243, A323244, A324103, A323247, A323248, A324396, A324398, A324542.

Array[If[# == 1, 0, Function[n, DivisorSum[n, MoebiusMu[n/#] (2 # - DigitCount[2 #, 2, 1] - DivisorSigma[1, #]) &]]@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]]] &, 90] (* Michael De Vlieger, Mar 11 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))));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
\\ Slow: A297114(n) = sumdiv(n,d,moebius(n/d)*(A005187(d)-sigma(d)));
A297111(n) = sumdiv(n,d,moebius(n/d)*A005187(d));
A297114(n) = (A297111(n) - n);
A324397(n) = if(1==n,0,A297114(A156552(n)));

a(1) = 0; for n > 1, a(n) = A297114(A156552(n)).

For all n >= 1, a(2n-1) = A324103(2n-1).

A083254 a(n) = 2*phi(n) - n.

Original entry on oeis.org

1, 0, 1, 0, 3, -2, 5, 0, 3, -2, 9, -4, 11, -2, 1, 0, 15, -6, 17, -4, 3, -2, 21, -8, 15, -2, 9, -4, 27, -14, 29, 0, 7, -2, 13, -12, 35, -2, 9, -8, 39, -18, 41, -4, 3, -2, 45, -16, 35, -10, 13, -4, 51, -18, 25, -8, 15, -2, 57, -28, 59, -2, 9, 0, 31, -26, 65, -4, 19, -22, 69, -24, 71, -2, 5, -4, 43, -30, 77, -16, 27, -2, 81, -36, 43, -2, 25

Offset: 1

Labos Elemer, May 08 2003

Möbius transform of A033879, deficiency of n. - Antti Karttunen, Dec 26 2017

			Case 1# - totient(x)-cototient[x] = 0 if x is a power of 2;
Case 2# - totient(x)>cototient[x] gives odd primes and also A067800, (= A014076 except probably A036798); e.g. n = 33: a(33) = 2.20-33 = 7; n = p prime: a(p) = p-2;
Case 3# - totient(x)<cototient[x] gives even numbers without powers of 2 and most probably A036798; e.g. n = 20: a(20) = -4; n = 105: a(105) = 2.48-105 = 96-105 = -9.

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A000010, A051953, A000079, A014076, A033879, A067800, A036798, A083255, A115405, A293516, A297114, A297115, A297153, A297154.

A083254 := proc(n)
    2*numtheory[phi](n)-n ;
end proc: # R. J. Mathar, Jan 13 2014

Mathematica
```
Table[2*EulerPhi[w]-w, {w, 1, 1000}]
```

a(n)=2*eulerphi(n)-n \\ Charles R Greathouse IV, Feb 21 2013

a(n) = totient(n) - cototient(n) = A000010(n) - A051953(n).
From Antti Karttunen, Dec 26 2017: (Start)
a(n) = A065620(A297153(n)) = A117966(A297154(n)).
a(n) = A297114(n) + A297115(n).
a(2n) = A297114(2n).
For all n >= 1, -a(A000010(n)) = A293516(n).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where  c = 6/Pi^2 - 1/2 = 0.107927... . - Amiram Eldar, Sep 07 2023

A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 0, 7, -6, 9, 1, 2, 0, 14, -5, 15, -4, 7, 5, 18, -14, 16, 7, 10, -3, 24, -16, 25, 0, 16, 12, 19, -21, 33, 13, 18, -12, 37, -15, 38, 1, 8, 16, 41, -30, 38, 4, 26, 3, 48, -16, 33, -11, 30, 22, 53, -52, 55, 23, 16, 0, 44, -14, 63, 8, 39, -7, 66, -53, 69, 31, 22, 9, 54, -16, 73, -28, 38, 35, 78, -59, 58

Offset: 1

Antti Karttunen, Nov 25 2017

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"Least deficient numbers" or "almost perfect numbers" are those k for which A033879(k) = 1, or equally, for which a(k) = -A048881(k-1). The only known solutions are powers of 2 (A000079), all present also in A295296. See also A235796 and A378988. - Antti Karttunen, Dec 16 2024

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n).

Cf. A000120, A000203, A001065, A005187, A011371, A013661, A033879, A048881, A235796, A294896, A294899, A297114 (Möbius transform), A317844 (difference from a(n)), A326133, A326138, A324348 (a(n) applied to Doudna sequence), A379008 (a(n) applied to prime shift array), A378988.

Cf. A295296 (positions of zeros), A295297 (parity of a(n)).

Array[IntegerExponent[(2 #)!, 2] - DivisorSigma[1, #] &, 85] (* Michael De Vlieger, Nov 26 2017 *)

A294898(n) = ((2*n-sigma(n))-hammingweight(n)); \\ Antti Karttunen, Dec 16 2024

(define (A294898 n) (- (A005187 n) (A000203 n)))

a(n) = A005187(n) - A000203(n).
a(n) = A011371(n) - A001065(n).
a(n) = A033879(n) - A000120(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.177532... . - Amiram Eldar, Feb 22 2024

Name edited by Antti Karttunen, Dec 16 2024

A297111 Möbius transform of A005187, where A005187(n) = 2n - (number of 1's in binary representation of n).

Original entry on oeis.org

1, 2, 3, 4, 7, 4, 10, 8, 12, 8, 18, 8, 22, 12, 15, 16, 31, 12, 34, 16, 25, 20, 41, 16, 39, 24, 34, 24, 53, 16, 56, 32, 42, 32, 49, 24, 70, 36, 48, 32, 78, 24, 81, 40, 48, 44, 88, 32, 84, 40, 63, 48, 101, 36, 79, 48, 72, 56, 112, 32, 116, 60, 69, 64, 98, 40, 130, 64, 90, 48, 137, 48, 142, 72, 81, 72, 121, 48, 152, 64

Offset: 1

Antti Karttunen, Dec 25 2017

nonn

Sequence differs from A035532 for the first time at n = 15, 21, 25, 27, 33, 35, 51, etc., i.e., at those composite n where A297115 has a nonzero value. - Antti Karttunen & M. F. Hasler, Mar 10 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A005187, A008683, A297108, A297110, A297114, A297115, A297117, A300244, A300723, A300724, A300725.

Table[DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 80}] (* Michael De Vlieger, Mar 10 2018 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297111(n) = sumdiv(n,d,moebius(n/d)*A005187(d));

a(n) = Sum_{d|n} A005187(d)*A008683(n/d).
a(n) = n + A297114(n).
From Antti Karttunen, Mar 11 2018: (Start)
Sum A005187(n) x^n = Sum a(n)*x^n/(1-x^n). [Another way of saying that this is the Möbius transform of A005187. This was originally included in A035532 by mistake.]
a(n) = 2*phi(n) - A297115(n) = phi(n) + A297117(n).
a(n) = A005187(n) - A300244(n).
a(1) = 1; for n > 1, a(n) = A300723(n) + 2*A300724(n).
(End)

A297115 Möbius transform of A000120, binary weight of n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, -1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 4, 0, -2, 0, -1, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 4, 0, 0, 0, 1, 0, 3, 0, 1, 0, 0, 0, 4, 0, 4, 0, 3, 0, -2, 0, 2, 0, -2, 0, 3, 0, 2, 0, -1, 0, -1, 0, 4, 0, -1, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, -1, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 0, 0

Offset: 1

Antti Karttunen, Dec 26 2017

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Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A000120, A008683, A083254, A297114, A297116 (odd bisection), A297117.

A297115(n) = sumdiv(n,d,moebius(n/d)*hammingweight(d));

a(n) = Sum_{d|n} A008683(n/d)*A000120(d).
a(n) = A083254(n) - A297114(n).
a(2n) = 0.

A300244 Difference between A005187 and its Möbius transform (A297111).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 10, 1, 14, 1, 13, 11, 15, 1, 22, 1, 22, 14, 21, 1, 30, 8, 25, 16, 29, 1, 40, 1, 31, 22, 34, 18, 46, 1, 37, 26, 46, 1, 57, 1, 45, 38, 44, 1, 62, 11, 57, 35, 53, 1, 68, 26, 61, 38, 56, 1, 84, 1, 59, 51, 63, 30, 90, 1, 70, 45, 89, 1, 94, 1, 73, 65, 77, 29, 104, 1, 94, 50, 81, 1, 117, 39, 84, 57, 93, 1, 128, 33, 92, 60, 91, 42

Offset: 1

Antti Karttunen, Mar 10 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005187, A008683, A297111, A297114, A297117.

Table[IntegerExponent[(2 n)!, 2] - DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 95}] (* or *)
Fold[Function[{a, n}, Append[a, {Abs@ Total@ Map[MoebiusMu[n/#] a[[#, -1]] &, Most@ Divisors@ n], IntegerExponent[(2 n)!, 2]}]], {{0, 1}}, Range[2, 95]][[All, 1]] (* Michael De Vlieger, Mar 10 2018 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A300244(n) = -sumdiv(n,d,(dA005187(d));

a(n) = A005187(n) - A297111(n).

a(n) = -Sum_{d|n, dA008683(n/d)*A005187(d).

A297117 Möbius transform of A011371, n minus (number of 1's in binary expansion of n).

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 4, 4, 6, 4, 8, 4, 10, 6, 7, 8, 15, 6, 16, 8, 13, 10, 19, 8, 19, 12, 16, 12, 25, 8, 26, 16, 22, 16, 25, 12, 34, 18, 24, 16, 38, 12, 39, 20, 24, 22, 42, 16, 42, 20, 31, 24, 49, 18, 39, 24, 36, 28, 54, 16, 56, 30, 33, 32, 50, 20, 64, 32, 46, 24, 67, 24, 70, 36, 41, 36, 61, 24, 74, 32, 55, 40, 79, 24

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A000120, A008683, A011371, A051953, A297111, A297114, A297115.

A297117(n) = sumdiv(n,d,moebius(n/d)*(d-hammingweight(d)));

a(n) = Sum_{d|n} A008683(n/d) * (d - A000120(d)).
a(n) = A297111(n) - A000010(n).
a(n) = A297114(n) + A051953(n).

A378986 a(n) = 2phi(2n) - 2*n, where phi is Euler totient function.

Original entry on oeis.org

0, 0, -2, 0, -2, -4, -2, 0, -6, -4, -2, -8, -2, -4, -14, 0, -2, -12, -2, -8, -18, -4, -2, -16, -10, -4, -18, -8, -2, -28, -2, 0, -26, -4, -22, -24, -2, -4, -30, -16, -2, -36, -2, -8, -42, -4, -2, -32, -14, -20, -38, -8, -2, -36, -30, -16, -42, -4, -2, -56, -2, -4, -54, 0, -34, -52, -2, -8, -50, -44, -2, -48, -2, -4

Offset: 1

Antti Karttunen, Dec 14 2024

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Antti Karttunen, Table of n, a(n) for n = 1..32769

Even bisection of A083254, and of A297114.
First row of A379011.
Cf. A000010, A008683, A033879, A062570, A176095, A378978.
Cf. also A378987.

PARI
```
A378986(n) = (2*eulerphi(2*n)-(2*n));
```

a(n) = 2*A000010(2*n) - 2*n.
a(n) = A083254(2*n) = A297114(2*n).
a(n) = -2*A176095(n).
a(n) = Sum_{d|2n} A008683(d)*A033879(2*n/d).

A318448 a(n) = Sum_{d|n} A294898(d), where A294898(d) = A005187(d) - sigma(d).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 2, 7, -8, 9, 4, 4, 0, 14, -4, 15, -2, 10, 12, 18, -22, 18, 16, 13, 1, 24, -14, 25, 0, 23, 26, 24, -31, 33, 28, 27, -14, 37, -6, 38, 13, 15, 34, 41, -52, 41, 22, 40, 19, 48, -10, 42, -10, 45, 46, 53, -76, 55, 48, 29, 0, 55, 12, 63, 34, 57, 18, 66, -98, 69, 64, 42, 37, 64, 16, 73, -42, 51, 72, 78, -74, 74, 74, 73, 6

Offset: 1

Antti Karttunen, Aug 27 2018

sign

Inverse Möbius transform of A294898.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A005187, A007429, A093653, A294898, A296075, A318446, A318447.

Cf. also A297114, A317844.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n) - sigma(n));
A318448(n) = sumdiv(n,d,A294898(d));

a(n) = Sum_{d|n} A294898(d).
a(n) = A318447(n) + A294898(n).
a(n) = A318446(n) - A007429(n).
a(n) = A296075(n) - A093653(n).

cp's OEIS Frontend

A317844 Difference between A294898 and its Möbius transform (A297114).

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A324397 a(1) = 0; for n > 1, a(n) = A297114(A156552(n)).

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A083254 a(n) = 2*phi(n) - n.

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A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

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A297111 Möbius transform of A005187, where A005187(n) = 2n - (number of 1's in binary representation of n).

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A297115 Möbius transform of A000120, binary weight of n.

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A300244 Difference between A005187 and its Möbius transform (A297111).

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A297117 Möbius transform of A011371, n minus (number of 1's in binary expansion of n).

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A378986 a(n) = 2phi(2n) - 2*n, where phi is Euler totient function.