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 1-3 of 3 results.

A357763 Numbers m such that A357761(m) > A357761(k) for all k < m.

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 56, 112, 224, 448, 728, 1456, 2912, 5824, 10192, 11648, 20384, 27664, 40768, 55328, 110656, 221312, 442624, 885248, 1263808, 1770496, 2527616, 3430336, 5055232, 6860672, 10110464, 13721344, 16155776, 20220928, 24012352, 32311552, 48024704
Offset: 1

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Author

Amiram Eldar, Oct 12 2022

Keywords

Comments

First differs from A330289 at n = 28.
Since A357761(2^n) = n + 1, A357761 is unbounded and this sequence is infinite.
The corresponding record values are 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 20, 24, 28, 30, 32, ... .

Links

Crossrefs

Cf. A330289, A355969, A356020, A357761, A357764.

Programs

  • Mathematica
    f[n_] := -DivisorSum[n, (-1)^DigitCount[#, 2, 1] &]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^5}]; s
  • PARI
    f(n) = -sumdiv(n, d, (-1)^hammingweight(d));
lista(nmax) = {my(fm = 0); for(n = 1, nmax, f1 = f(n); if(f1 > fm, fm = f1; print1(n, ", ")))};

A357764 Numbers m such that A357761(m) < A357761(k) for all k < m.

Original entry on oeis.org

1, 3, 9, 15, 30, 60, 90, 180, 360, 540, 720, 1080, 2160, 4320, 6120, 8640, 12240, 18360, 24480, 36720, 73440, 146880, 257040, 293760, 514080, 587520, 807840, 1028160, 1615680, 1884960, 2056320, 2827440, 3231360, 3769920, 5654880, 7539840, 9424800, 11309760, 18849600
Offset: 1

Views

Author

Amiram Eldar, Oct 12 2022

Keywords

Comments

Since A357761(15*2^n) = -2*(n+1), A357761 is unbounded from below and this sequence is infinite.
The corresponding records of low value are 1, 0, -1, -2, -4, -6, -8, -12, -16, -18, -20, -24, -30, -36, ... .

Crossrefs

Cf. A355969, A356020, A357761, A357763.

Programs

  • Mathematica
    f[n_] := -DivisorSum[n, (-1)^DigitCount[#, 2, 1] &]; fm = 2; s = {}; Do[f1 = f[n]; If[f1 < fm, fm = f1; AppendTo[s, n]], {n, 1, 10^5}]; s
  • PARI
    f(n) = -sumdiv(n, d, (-1)^hammingweight(d));
lista(nmax) = {my(fm = 2); for(n = 1, nmax, f1 = f(n); if(f1 < fm, fm = f1; print1(n, ", ")))};

A357762 Decimal expansion of -Sum_{k>=1} A106400(k)/k.

Original entry on oeis.org

1, 1, 9, 6, 2, 8, 3, 2, 6, 4, 3, 2, 5, 2, 5, 6, 4, 3, 7, 2, 2, 2, 2, 9, 1, 6, 3, 3, 2, 0, 0, 8, 1, 9, 1, 8, 1, 0, 1, 0, 4, 2, 6, 7, 4, 6, 4, 0, 1, 5, 9, 4, 3, 8, 1, 8, 9, 8, 7, 2, 3, 3, 3, 7, 3, 0, 7, 8, 3, 7, 5, 1, 6, 1, 0, 9, 1, 5, 8, 0, 8, 7, 7, 7, 9, 1, 1, 9, 6, 4, 5, 4, 6, 2, 1, 1, 0, 7, 4, 8, 9, 6, 3, 3, 3
Offset: 1

Views

Author

Amiram Eldar, Oct 12 2022

Keywords

Comments

The asymptotic mean of the excess of the number of odious divisors over the number of evil divisors (A357761, see formula).
The convergence of the partial sums S(m) = -Sum_{k=1..2^m-1} A106400(k)/k is fast: e.g., S(28) is already correct to 100 decimal digits (see also Jon E. Schoenfield's comment in A351404).

Examples

			1.19628326432525643722229163320081918101042674640159...

Crossrefs

Cf. A106400, A357761.
Similar constants: A215016, A351404

Programs

  • Mathematica
    sum = 0; m = 1; pow = 2; Do[sum -= (-1)^DigitCount[k, 2, 1]/k; If[k == pow - 1, Print[m, " ", N[sum, 120]]; m++; pow *= 2], {k, 1, 2^30}]
  • PARI
    default(realprecision, 150);
sm = 0.; m = 1; pow = 2; for(k = 1, 2^30, sm -= (-1)^hammingweight(k)/k; if(k == pow - 1, print(m," ",sm); m++; pow *= 2))

Formula

Equals -2 * Sum_{k>=1} A106400(2*k-1)/(2*k-1).
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A357761(k).
Showing 1-3 of 3 results.

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