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 1-6 of 6 results.

A302571 Bi-unitary barely abundant numbers: bi-unitary abundant numbers k such that bsigma(k)/k < bsigma(m)/m for all bi-unitary abundant numbers m < k, where bsigma(k) is the sum of the bi-unitary divisors of k (A188999).

Original entry on oeis.org

24, 30, 40, 54, 56, 70, 80, 104, 642, 654, 678, 726, 762, 786, 822, 832, 1888, 1952, 4030, 5830, 7424, 32128, 62464, 374802, 374838, 374862, 374898, 374982, 375006, 375042, 375198, 375234, 375294, 375378, 375486, 375546, 375582, 375618, 375702, 375762, 375798
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2018

Keywords

Examples

			The values of bsigma(k)/k are: 3, 2.5, 2.4, 2.25, 2.222..., 2.142...

Links

Crossrefs

The bi-unitary version of A071927.
Cf. A188999, A292982, A302570.

Programs

  • Mathematica
    f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=  DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; r = 3; seq={}; Do[
s = bsigma[n]/n; If[s > 2 && s < r, AppendTo[seq,n]; r = s], {n, 1, 10000}]; seq
  • PARI
    babindex(n) = {my(f = factor(n), p, e); prod(k = 1, #f~, p = f[k, 1]; e = f[k, 2]; (p^(e+1)-1)/(p^(e+1)-p^e) - if(e%2, 0, 1/p^(e/2)));}
lista(kmax) = {my(bab, babm = 3); for(k = 1, kmax, bab = babindex(k); if(bab > 2 && bab < babm, babm = bab; print1(k, ", "))); }

A302572 Unitary barely deficient numbers: unitary deficient numbers k such that usigma(k)/k > usigma(m)/m for all unitary deficient numbers m < k, where usigma(k) is the sum of the unitary divisors of k (A034448).

Original entry on oeis.org

1, 2, 10, 84, 110, 1155, 6490, 34320, 55335, 80652, 163212, 449295, 676390, 1360810, 1503370, 1788490, 3214090, 22627605, 32062485, 35604492, 103712410, 365690892, 615206030, 815634435
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2018

Keywords

Examples

			The values of usigma(k)/k are 1, 1.5, 1.8, 1.904..., 1.963..., 1.994...

Crossrefs

Cf. A034448, A071927, A302570.

Programs

  • Mathematica
    usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; seq = {}; r = 0; Do[s = usigma[n]/n; If[s < 2 && s > r, AppendTo[seq, n]; r = s], {n, 1, 1000000}]; seq

A335054 Infinitary barely abundant numbers: infinitary abundant numbers whose infinitary abundancy is closer to 2 than that of any smaller infinitary abundant number.

Original entry on oeis.org

24, 30, 40, 54, 56, 70, 88, 104, 642, 654, 678, 726, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1014, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398, 1434, 1446, 1506, 1536, 1542, 1578, 1596, 2406, 2454, 2514, 2526, 2586, 2598, 2634
Offset: 1

Views

Author

Amiram Eldar, May 21 2020

Keywords

Comments

The infinitary abundancy of a number k is isigma(k)/k, where isigma(k) is the sum of infinitary divisors of k (A049417).

Examples

			The infinitary abundancies of the first terms are 2.5, 2.4, 2.25, 2.222..., 2.142..., 2.057..., ...

Links

Crossrefs

The infinitary version of A071927.
Cf. A049417, A129656, A302570, A302571.

Programs

  • Mathematica
    fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; seq = {}; r = 3; Do[s = isigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 3000}]; seq

A335052 Odd unitary abundant numbers whose unitary abundancy is closer to 2 than that of any smaller odd unitary abundant number.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 31395, 33915, 39585, 41055, 45885, 51765, 80535, 83265, 354585, 359205, 361515, 366135, 382305, 389235, 400785, 403095, 407715, 414645, 416955, 423885, 430815, 437745, 442365, 77967015, 132335385, 617102535, 724239285, 1756753845
Offset: 1

Views

Author

Amiram Eldar, May 21 2020

Keywords

Comments

The unitary abundancy of a number k is usigma(k)/k, where usigma(k) is the sum of unitary divisors of k (A034448).

Examples

			The unitary abundancies of the first terms are 2.148..., 2.112..., 2.099..., 2.085..., 2.072..., ...

Crossrefs

Cf. A034448, A129485, A188263, A302570, A335053, A335055.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); seq = {}; r = 3; Do[s = usigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 10^6, 2}]; seq

A336254 Exponential barely abundant numbers: exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller exponential abundant number.

Original entry on oeis.org

900, 1764, 3600, 4356, 4500, 4900, 12348, 47916, 79092, 112500, 605052, 2812500, 13366548, 29647548, 89139564, 231708348, 701538156, 1757812500, 14772192228, 32179382604, 43945312500, 71183762748, 620995547124, 990454107996, 3417547576788, 3488004374652, 10271220141996
Offset: 1

Views

Author

Amiram Eldar, Jul 14 2020

Keywords

Comments

The exponential abundancy of a number k is esigma(k)/k, where esigma is the sum of exponential divisors of k (A051377).
All the terms are powerful numbers (A001694) because esigma(k)/k depends only on the powerful part of k (A057521). - Amiram Eldar, May 06 2025

Examples

			The first 6 exponential abundant numbers, 900, 1764, 3600, 4356, 4500 and 4900, have decreasing values of exponential abundancy: 2.4, 2.285..., 2.2, 2.181..., 2.08, 2.057... and therefore they are in this sequence. The next exponential abundant number with a lower exponential abundancy is 12348 with eisgma(12348)/12348 = 2.040...

Links

Crossrefs

The exponential version of A071927.
Subsequence of A001694 and A328136.
Cf. A051377, A057521, A129575, A336253.
Similar sequences: A188263, A302570, A302571, A335054.

Programs

  • Mathematica
    fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; rm = 3; s={}; Do[r = esigma[n]/n; If[r <= 2, Continue[]]; If[r < rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s

Extensions

a(23)-a(27) from Amiram Eldar, May 06 2025

A336671 Unitary barely 3-abundant: numbers m such that 3 < usigma(m)/m < usigma(k)/k for all numbers k < m, where usigma is the sum of unitary divisors function (A034448).

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 62790, 67830, 79170, 82110, 91770, 103530, 161070, 166530, 709170, 718410, 723030, 732270, 764610, 778470, 801570, 806190, 815430, 829290, 833910, 847770, 861630, 875490, 884730, 155934030, 264670770, 1234205070, 1448478570
Offset: 1

Views

Author

Amiram Eldar, Jul 29 2020

Keywords

Comments

The corresponding values of usigma(m)/m are 3.222..., 3.168...., 3.149..., 3.127..., 3.109..., ...

Links

Crossrefs

The unitary version of A259312.
Subsequence of A285615.
Cf. A034448, A034683, A302570, A336672.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); s = {}; rm = 4; Do[r = usigma[n]/n; If[r > 3 && r < rm, rm = r; AppendTo[s, n]], {n, 1, 10^5}]; s
Showing 1-6 of 6 results.

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