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 1-7 of 7 results.

A380933 Numbers k such that k and k+1 are both in A380929.

Original entry on oeis.org

121643775, 157390064, 161019495, 275734304, 584899875, 1493214975, 1614323655, 2043708975, 3081783375, 3118599224, 3426851295, 3902652495, 3947893424, 5849043375, 11731509855, 12138531615, 13008843224, 14598032624, 17588484584, 19782621495, 20191564575, 20759209064
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

Numbers k such that A380845(k) > 2*k and A380845(k+1) > 2*(k+1).

Examples

			121643775 is a term since A380845(121643775) = 244722015 > 2 * 121643775 = 243287550, and A380845(121643776) = 256456081 > 2 * 121643776 = 243287552.

Links

Crossrefs

Subsequence of A096399 and A380929.
Cf. A380845, A380932.
Similar sequences: A283418, A318167, A327635, A327942, A331412, A333951, A357608, A364727, A364861.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k];
seq[lim_] := Module[{s = {}}, Do[If[q[k], If[q[k-1], AppendTo[s, k-1]]; If[q[k+1], AppendTo[s, k]]], {k, 3, lim, 2}]; s];
seq[3*10^8]
  • PARI
    isab(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}
list(lim) = forstep(k = 3, lim, 2, if(isab(k), if(isab(k-1), print1(k-1, ", ")); if(isab(k+1), print1(k, ", "))));

A380931 Numbers k such that A380845(k) > 4*k.

Original entry on oeis.org

5155920, 7733880, 10311840, 15467760, 20623680, 30935520, 41247360, 46403280, 61871040, 61901280, 75546240, 82494720, 87693480, 92806560, 103168800, 103194000, 113513400, 123742080, 123802560, 134152200, 140540400, 151092480, 151351200, 162162000, 164989440, 175386960
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

Analogous to 4-abundant numbers (A068404) with A380845 instead of A000203.

Examples

			5155920 is a term since A380845(5155920) = 21067042 > 4 * 5155920 = 20623680.

Links

Crossrefs

Cf. A000203, A380845, A380846, A380847, A380848.
Subsequence of A068404, A380929 and A380931.
Similar sequences: A307114, A340110.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 4*k]; Select[Range[10^8], q]
  • PARI
    isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 4*k;}

A380930 Numbers k such that A380845(k) > 3*k.

Original entry on oeis.org

1080, 2160, 3600, 4320, 7200, 7440, 8640, 11340, 13608, 14400, 14880, 15120, 17280, 18600, 22680, 22860, 27216, 28800, 29760, 30240, 30480, 31752, 33264, 34020, 34560, 37200, 41664, 45360, 45720, 45900, 51408, 53340, 54432, 57600, 59520, 60480, 60960, 61200, 63504
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

Analogous to 3-abundant numbers (A068403) with A380845 instead of A000203.

Links

Crossrefs

Cf. A000203, A380845, A380846, A380847.
Subsequence of A068403 and A380929.
Subsequences: A380848, A380931.
Similar sequences: A285615, A293187, A300664, A328135, A340109.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 3*k]; Select[Range[64000], q]
  • PARI
    isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 3*k;}

Formula

1080 is a term since A380845(1080) = 3330 > 3 * 1080 = 3240.

A380932 Odd numbers k such that A380845(k) > 2*k.

Original entry on oeis.org

322245, 590205, 874665, 966735, 1934415, 2900205, 3224025, 3378375, 3869775, 4729725, 6081075, 6449625, 6818175, 7740495, 8783775, 8906625, 9029475, 9889425, 10135125, 10961685, 11609325, 11821425, 12900825, 13378365, 14189175, 15049125, 15481935, 15909075, 16253055
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

The odd terms in A380929.
Analogous to odd abundant numbers (A005231) with A380845 instead of A000203.

Examples

			322245 is a term since it is odd, and A380845(322245) = 679582 > 2 * 322245 = 644490.

Links

Crossrefs

Cf. A000203, A380845.
Intersection of A005408 and A380929.
Subsequence of A005231.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A339938, A348275, A360526, A379031.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1,10^6,2], q]
  • PARI
    isok(k) = if(!(k % 2), 0, my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k);

A381070 Numbers k such that A380845(k)/k > A380845(m)/m for all m < k.

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 36, 72, 144, 288, 540, 1080, 2160, 4320, 8640, 17280, 34560, 45360, 68040, 90720, 106680, 136080, 213360, 272160, 320040, 640080, 1280160, 2560320, 2577960, 5155920, 10311840, 15467760, 30935520, 61871040, 123742080, 247484160, 494968320, 681080400
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2025

Keywords

Comments

Analogous to superabundant numbers (A004394) with A380845 instead of A000203.
The least number k for which A380845(k)/k >= 2 is k = a(6) = A380846(1) = 18.
The least number k for which A380845(k)/k >= 3 is k = a(12) = A380930(1) = 1080.
The least number k for which A380845(k)/k >= 4 is k = a(30) = A380931(1) = 5155920.
It seems that A380845(k)/k is unbounded (see the plot in the links section). What is the least number k for which A380845(k)/k >= 5?

Links

Crossrefs

Cf. A000203, A004394, A380845, A380846, A380929, A380930, A380931, A381069.

Programs

  • Mathematica
    r[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, # &, DigitCount[#, 2, 1] == h &]/n]; seq[lim_] := Module[{s = {}, rm = 0, r1}, Do[r1 = r[k]; If[r1 > rm, rm = r1; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^5]
  • PARI
    r(n) = {my(h = hammingweight(n)); sumdiv(n, d, d * (hammingweight(d) == h)) / n;}
list(lim) = {my(rm = 0, r1); for(k = 1, lim, r1 = r(k); if(r1 > rm, rm = r1; print1(k, ", "))); }

A381071 Numbers k such that the sum of the proper divisors of k that have the same binary weight as k is larger than k, and no subset of these divisors sums to k.

Original entry on oeis.org

1050, 3150, 4284, 4410, 5148, 6292, 6790, 7176, 8890, 10764, 17850, 18648, 19000, 19530, 32886, 33072, 33150, 35088, 35530, 35720, 35770, 38850, 41360, 43164, 45084, 49368, 49764, 50456, 50730, 52884, 54280, 54340, 58410, 58696, 59010, 59408, 63492, 66010, 68376
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2025

Keywords

Comments

Analogous to weird numbers (A006037), as A380846 is analogous to perfect numbers (A000396).

Links

Crossrefs

Cf. A000396, A380845, A380846.
Subsequence of A380929.
A381072 is a subsequence.
Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948, A339939, A348525, A348631, A349285, A364862.

Programs

  • Mathematica
    divs[n_] := Module[{hw = DigitCount[n, 2, 1]}, Select[Divisors[n], DigitCount[#, 2, 1] == hw &]];
weirdQ[n_, d_, s1_, m1_] :=  weirdQ[n, d, s1, m1] = Module[{s = s1, m = m1}, If[m == 0, False, While[m > 0 && d[[m]] > n, s -= d[[m]]; m--]; If[m == 0, True, d[[m]] < n && If[s > n, weirdQ[n - d[[m]], d, s - d[[m]], m - 1] && weirdQ[n, d, s - d[[m]], m - 1], s < n && m < Length[d] - 1]]]];
q[n_] := Module[{d = divs[n], s, m}, s = Total[d] - n; m = Length[d] - 1; weirdQ[n, d, s, m]]; Select[Range[70000], q] (* based on a Pari code by M. F. Hasler at A006037 *)
  • PARI
    divs(n) = {my(h = hammingweight(n)); select(x -> hammingweight(x)==h, divisors(n));}
is(n, d = divs(n), s = vecsum(d)-n, m = #d-1) = {if(m == 0, return(0)); while(m > 0 && d[m] > n, s -= d[m]; m--); if(m==0, return(1)); (d[m] < n &&
if(s > n, is(n-d[m], d, s-d[m], m-1) && is(n, d, s-d[m], m-1), s < n && m < #d-1));} \\ based on a code by M. F. Hasler at A006037

A381072 Odd terms in A381071.

Original entry on oeis.org

322245, 590205, 874665, 3378375, 4729725, 6081075, 6818175, 8783775, 8906625, 9889425, 10135125, 13378365, 15049125, 15909075, 16253055, 18922365, 32684085, 34754265, 36916425, 38144925, 38439765, 39471705, 44778825, 46990125, 57506085, 75200265, 84047355, 88852995
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2025

Keywords

Comments

The first 142 terms are all divisible by 3.
The least term that is not divisible by 5 is a(57) = 885593709.

Links

Crossrefs

Intersection of A005408 and A381071.
Subsequence of A380929 and A380932.

Programs

  • Mathematica
    Select[Range[1, 70000, 2], q] (* using the function q[n_] from A381071 *)
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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