cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Antti Karttunen, Dec 25 2017

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    Table[DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 80}] (* Michael De Vlieger, Mar 10 2018 *)
  • PARI
    A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297111(n) = sumdiv(n,d,moebius(n/d)*A005187(d));

Formula

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)

A297114 Möbius transform of A294898, where A294898 is deficiency minus binary weight.

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, -2, 7, -4, 9, -2, 0, 0, 14, -6, 15, -4, 4, -2, 18, -8, 14, -2, 7, -4, 24, -14, 25, 0, 9, -2, 14, -12, 33, -2, 9, -8, 37, -18, 38, -4, 3, -2, 41, -16, 35, -10, 12, -4, 48, -18, 24, -8, 15, -2, 53, -28, 55, -2, 6, 0, 33, -26, 63, -4, 21, -22, 66, -24, 69, -2, 6, -4, 44, -30, 73, -16, 28, -2
Offset: 1

Views

Author

Antti Karttunen, Dec 26 2017

Keywords

Links

Crossrefs

Cf. A000203, A005187, A051953, A083254, A294898, A297111, A297115, A297117, A317844, A324397.
Cf. A378986 (even bisection, -2*A176095).

Programs

  • Mathematica
    Table[DivisorSum[n, MoebiusMu[n/#] (2 # - DigitCount[2 #, 2, 1] - DivisorSigma[1, #]) &], {n, 82}] (* Michael De Vlieger, Mar 11 2019 *)
  • PARI
    A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297114(n) = sumdiv(n,d,moebius(n/d)*(A005187(d)-sigma(d)));

Formula

a(n) = Sum_{d|n} A008683(n/d)*A294898(d).
a(n) = A297111(n) - n.
a(n) = A297117(n) - A051953(n).
a(n) = A083254(n) - A297115(n).
a(2n) = A083254(2n) = A378986(n) = -2*A176095(n).
a(n) = A294898(n) - A317844(n).

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

Views

Author

Antti Karttunen, Dec 26 2017

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Antti Karttunen, Mar 10 2018

Keywords

Links

Crossrefs

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

Programs

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

Formula

a(n) = A005187(n) - A297111(n).
a(n) = -Sum_{d|n, dA008683(n/d)*A005187(d).
Showing 1-4 of 4 results.

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