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.

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A353899 Indices of records in A353898.

Original entry on oeis.org

1, 2, 4, 6, 12, 30, 36, 60, 180, 420, 900, 1260, 4620, 6300, 13860, 44100, 55440, 69300, 180180, 485100, 720720, 900900, 3063060, 6306300, 12252240, 15315300, 58198140, 107207100, 232792560, 290990700, 1163962800, 2036934900, 5354228880, 6692786100, 22406283900
Offset: 1

Views

Author

Amiram Eldar, May 10 2022

Keywords

Comments

First differs from A333931 at n=23.
The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 48, 54, 72, 81, 96, 108, 144, 162, ... (see the link for more values).

Links

Crossrefs

Cf. A333931, A353898.
Subsequence of A025487 and A138302.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386.

Programs

  • Mathematica
    f[p_, e_] := Floor[Log2[e]] + 2; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; sm = 0; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^6}]; seq

A358263 Numbers with a record number of noninfinitary square divisors.

Original entry on oeis.org

1, 16, 144, 256, 1296, 2304, 20736, 57600, 331776, 518400, 2822400, 8294400, 12960000, 25401600, 132710400, 207360000, 228614400, 406425600, 635040000, 2057529600, 3073593600, 6502809600, 10160640000, 27662342400, 31116960000, 51438240000, 76839840000, 248961081600
Offset: 1

Views

Author

Amiram Eldar, Nov 06 2022

Keywords

Comments

Numbers m such that A358261(m) > A358261(k) for all k < m.
The corresponding record values are 0, 1, 2, 3, 5, 6, 11, 12, 13, 22, 24, 26, 37, 44, 46, 47, 48, ... (see the link for more values).

Links

Crossrefs

Cf. A358261, A358262.
Subsequence of A025487.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386, A348632, A358253.

Programs

  • Mathematica
    f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^DigitCount[If[OddQ[e], e - 1, e], 2, 1]; f[1] = 0; f[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; s = {}; fmax = -1; Do[If[(fn = f[n]) > fmax, fmax = fn; AppendTo[s, n]], {n, 1, 6*10^5}]; s
  • PARI
    s(n) = {my(f = factor(n));  prod(i=1, #f~, 1+f[i,2]\2) - prod(i=1, #f~, 2^hammingweight(if(f[i,2]%2, f[i,2]-1, f[i,2])))};
lista(nmax) = {my(smax = -1, sn); for(n = 1, nmax, sn = s(n); if(sn > smax, smax = sn; print1(n, ", "))); }

A362853 Numbers with a record number of divisors that are both bi-unitary and exponential.

Original entry on oeis.org

1, 8, 64, 216, 1728, 27000, 46656, 110592, 216000, 2985984, 5832000, 13824000, 74088000, 373248000, 2000376000, 4741632000, 46656000000, 98611128000, 128024064000, 2662500456000, 6311112192000, 16003008000000, 93329542656000, 170400029184000, 5489031744000000
Offset: 1

Views

Author

Amiram Eldar, May 05 2023

Keywords

Comments

Indices of records in A362852.
The first 80 terms are cubes. Are there noncubes in this sequence?
The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, ... (see the link for more values).

Links

Crossrefs

Cf. A362852.
Subsequence of A025487.
Similar sequences: A293185, A318278.

Programs

  • Mathematica
    f[p_, e_] := DivisorSigma[0, e] - If[OddQ[e], 0, 1]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n];
v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]];
seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq

A361785 Indices of records in the sequence of bi-unitary harmonic means A361782(k)/A361783(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 54, 56, 60, 84, 96, 120, 168, 210, 240, 270, 280, 360, 420, 480, 672, 840, 1080, 1320, 1512, 1680, 1890, 2160, 2310, 2520, 3080, 3360, 4320, 5280, 6048, 7392, 7560, 9240, 10920, 11880, 14040, 15120, 18480, 20790
Offset: 1

Views

Author

Amiram Eldar, Mar 24 2023

Keywords

Examples

			The harmonic means of the bi-unitary divisors of the first 6 positive integers are 1 < 4/3 < 3/2 < 8/5 < 5/3 < 2. A361782(7)/A361783(7) = 9/5 < 2, and the next record, A361782(8)/A361783(8) = 32/15, occurs at 8. Therefore, the first 7 terms of this sequence are 1, 2, 3, 4, 5, 6 and 8.

Links

Crossrefs

Cf. A361782, A361783.
Similar sequences: A179971, A348654, A361319.
Other sequences related to records of bi-unitary divisors: A293185, A292983, A292984.

Programs

  • Mathematica
    f[p_, e_] := p^e * If[OddQ[e], (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))]; buhmean[1] = 1; buhmean[n_] := Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{buh, buhmax = 0, s = {}}, Do[buh = buhmean[k]; If[buh > buhmax, buhmax = buh; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[20000]
  • PARI
    buhmean(n) = {my(f = factor(n), p, e); n * prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  if(e%2, (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2)))); }
lista(kmax) = {my(buh, buhmax=0); for(k = 1, kmax, buh = buhmean(k); if(buh > buhmax, buhmax = buh; print1(k, ", "))); }

A363333 Numbers with a record number of divisors that are both coreful and bi-unitary.

Original entry on oeis.org

1, 8, 32, 128, 216, 864, 3456, 7776, 13824, 31104, 108000, 279936, 432000, 972000, 1728000, 3888000, 15552000, 34992000, 62208000, 97200000, 139968000, 248832000, 333396000, 559872000, 592704000, 874800000, 1333584000, 5334336000, 12002256000, 21337344000, 33339600000
Offset: 1

Views

Author

Amiram Eldar, May 28 2023

Keywords

Comments

Indices of records in A363332.
The corresponding record values are 1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63, 75, 81, ... (see the link for more values).

Links

Crossrefs

Cf. A363332.
Subsequence of A025487.
Similar sequences: A005934, A293185.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], e, e - 1]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 120]
v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]];
seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq

A302936 Bi-unitary highly composite deficient numbers: bi-unitary deficient numbers k whose number of bi-unitary divisors bd(k) > bd(m) for all bi-unitary deficient numbers m < k.

Original entry on oeis.org

1, 2, 8, 32, 84, 512, 972, 1155, 13365, 25740, 318087, 612612, 11223927, 14549535, 440374077, 746503065, 19013596875
Offset: 1

Views

Author

Amiram Eldar, Apr 16 2018

Keywords

Comments

The record numbers of bi-unitary divisors are 1, 2, 4, 6, 8, 10, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, ...
The bi-unitary version of A302934.

Crossrefs

Cf. A188999, A286324, A292982, A293185, A302934.

Programs

  • Mathematica
    f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdivnum[n_] := DivisorSum[n, 1 &, Last@Intersection[f@#, f[n/#]] == 1 &]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; dm = 0; Do[sig = bsigma[n]; If[sig >= 2 n, Continue[]]; d = bdivnum[n]; If[d > dm, Print[n]; dm = d], {n, 1, 1000000000}] (* after Michael De Vlieger at A188999 and A286324 *)
  • PARI
    udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
lista(nn) = {my(maxd = 0); for(n=1, nn, vbiudiv = biudivs(n); if ((vecsum(vbiudiv) < 2*n) && (#vbiudiv > maxd), print1(n, ", "); maxd = #vbiudiv;););} \\ Michel Marcus, Apr 17 2018

Extensions

a(15)-a(17) from Amiram Eldar, Jan 26 2019
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