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.

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

Views

Author

Antti Karttunen, Nov 21 2019

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • PARI
    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));

Formula

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.

A329890 a(1) = 1; for n > 1, a(n) = sigma((2^n)-1) - sigma((2^(n-1))-1), where sigma is A000203, the sum of divisors.

Original entry on oeis.org

1, 3, 4, 16, 8, 72, 24, 304, 160, 944, 624, 6576, -544, 14336, 16384, 72544, 19616, 342528, 50688, 1475584, 466176, 3443712, 2657376, 29486880, -3340288, 54808448, 65971360, 306781024, 77647680, 1475408064, 132153344, 5157119680, 3054411072, 12548176896, 13343981568, 130039259136, -28235160128, 228451400256, 269821673472
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2019

Keywords

Links

Crossrefs

From second term onwards, the first differences of A075708.
Cf. A000203, A329637, A329891, A329892.

Programs

  • Mathematica
    Join[{1},Table[DivisorSigma[1,2^n-1]-DivisorSigma[1,2^(n-1)-1],{n,2,40}]] (* Harvey P. Dale, Aug 18 2024 *)
  • PARI
    A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));

Formula

a(1) = 1; and for n > 1, a(n) = A075708(n) - A075708(n-1).

A329891 a(0) = 0, a(1) = 1, for n > 1, a(n) = 2^n - (sigma((2^n)-1) - sigma((2^(n-1))-1)).

Original entry on oeis.org

0, 1, 1, 4, 0, 24, -8, 104, -48, 352, 80, 1424, -2480, 8736, 2048, 16384, -7008, 111456, -80384, 473600, -427008, 1630976, 750592, 5731232, -12709664, 36894720, 12300416, 68246368, -38345568, 459223232, -401666240, 2015330304, -862152384, 5535523520, 4631692288, 21015756800, -61319782400, 165674113600, 46426506688, 279934140416, -484569911296
Offset: 0

Views

Author

Antti Karttunen, Nov 23 2019

Keywords

Links

Crossrefs

Row 1 of A329637.
Cf. A000203, A000225, A075708, A156552, A323243, A323244, A329644, A329890, A329892.

Programs

  • PARI
    A329891(n) = if(n<=1,n,(2^n - sigma((2^n)-1)) + sigma((2^(n-1))-1));

Formula

a(0) = 0; for n >= 1, a(n) = A323244(2^n) - A323244(2^(n-1)) = 2^n - A329890(n).
a(n) = A329644(2^n).

A329892 a(0) = 0, a(1) = 1, for n > 1, a(n) = 2^(n+1) - 3*(sigma((2^n)-1) - sigma((2^(n-1))-1)).

Original entry on oeis.org

0, 1, -1, 4, -16, 40, -88, 184, -400, 544, -784, 2224, -11536, 18016, -10240, 16384, -86560, 203296, -503296, 896512, -2329600, 2795776, -1942528, 8805088, -54906208, 77129728, -30207616, 70521376, -383472160, 840798784, -2278740544, 3898507264, -6881424448, 8016635968, -3284792320, 28687532032, -252678823936, 359583387328, -135598386880
Offset: 0

Views

Author

Antti Karttunen, Nov 23 2019

Keywords

Links

Crossrefs

Row 2 of A329637.
Cf. A000203, A000225, A075708, A156552, A323243, A329644, A329890, A329891.

Programs

  • PARI
    A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));
A329892(n) = if(!n,n,2^(n+1) - 3*A329890(n));
  • PARI
    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
A323243(n) = if(1==n, 0, sigma(A156552(n)));
A329644(n) = sumdiv(n, d, moebius(n/d)*((2*A156552(d))-A323243(d)));
A329892(n) = A329644(3^n);

Formula

a(n) = A329644(3^n).
a(0) = 0; for n >= 1, a(n) = 2^(n+1) - 3*A329890(n).
Showing 1-4 of 4 results.

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