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 1-10 of 12 results. Next

A188263 Odd abundant numbers whose abundancy is closer to 2 than any smaller odd abundant number.

Original entry on oeis.org

945, 2205, 7425, 8415, 8925, 31815, 32445, 351351, 442365, 14571585, 20355825, 20487159, 78524145, 159030135, 1756753845, 2586415095, 82014476355, 93128205975, 125208115065, 127595519865, 154063853475, 394247024535, 948907364895
Offset: 1

Views

Author

T. D. Noe, Mar 30 2011

Keywords

Comments

The abundancy of a number n is defined as sigma(n)/n. Abundant numbers have an abundancy greater than 2. All these numbers must be odd primitive abundant numbers, A006038.
These numbers might be considered the opposite of A119239, which has odd numbers whose abundancy increases. This sequence has terms in common with A171929. A similar sequence for deficient numbers is A188597.
These are odd numbers that are barely abundant. See A071927 for the even version.
a(24) > 10^12. - Donovan Johnson, May 05 2012

Links

Crossrefs

Cf. A171929 (odd numbers whose abundancy is closer to 2 than any smaller odd number)

Programs

  • Mathematica
    k = 1; minDiff = 1; Table[k = k + 2; While[abun = DivisorSigma[1, k]/k; abun - 2 > minDiff || abun < 2, k = k + 2]; minDiff = abun - 2; k, {10}]

Extensions

a(15)-a(16) from Donovan Johnson, Mar 31 2011
a(17)-a(22) from Donovan Johnson, Apr 02 2011
a(23) from Donovan Johnson, May 05 2012

A302570 Unitary barely abundant numbers: unitary abundant numbers k such that usigma(k)/k < usigma(m)/m for all unitary abundant numbers m < k, where usigma(k) is the sum of the unitary divisors of k (A034448).

Original entry on oeis.org

30, 42, 66, 70, 222, 246, 258, 282, 294, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 726, 750, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1014, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398, 1434
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2018

Keywords

Comments

The unitary version of A071927.

Examples

			The values of usigma(k)/k are 2.4, 2.285..., 2.181..., 2.057..., 2.054...

Links

Crossrefs

Cf. A034448, A034683, A071927, A302572.

Programs

  • Mathematica
    usigma[n_] := If[n == 1, 1, 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, 10000}]; seq

A228450 Deficient numbers with increasing abundancy without being powers of 2.

Original entry on oeis.org

3, 9, 10, 44, 110, 136, 592, 884, 2144, 8384, 18632, 32896, 116624, 391612, 527872, 1090912, 2102272, 8394752, 15370304, 73995392, 536920064, 815634435, 2147516416, 34360131584, 217898810368, 546409576448, 549759483904
Offset: 1

Views

Author

Michel Marcus, Oct 27 2013

Keywords

Comments

Without the additional condition one would have obtained A000079, see "least deficient" comment there. Subsequence of A005100.

Examples

			First term is 3 with sigma(n)/n = 4/3 ~ 1.33, then 4 with 13/9 ~ 1.44, then 10 with 9/5 = 1.80.

Crossrefs

Cf. A000079, A000203, A005100, A071927, A171929, A188597.

Programs

  • Mathematica
    abun[n_] := DivisorSigma[1, n]/n; mx = 0; t = {}; Do[m = abun[n]; If[m < 2 && m > mx && ! IntegerQ[Log[2, n]], mx = m; AppendTo[t, n]], {n, 10000}]; t (* T. D. Noe, Apr 09 2014 *)
  • PARI
    lista(nn) = {rab = 0; for (n=1, nn, if (n != 2^valuation(n, 2), ab = sigma(n)/n; if ((ab < 2) && (ab > rab), print1(n, ", "); rab = ab;);););} \\ Michel Marcus, Oct 27 2013

Extensions

a(21)-a(22) from Michel Marcus, Oct 28 2013
a(23)-a(27) from Donovan Johnson, Nov 13 2013

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

A307122 3-deficient numbers with increasing abundancy: Numbers k such that sigma(m)/m < sigma(k)/k < 3 for all numbers m < k such that sigma(m)/m < 3.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 168, 252, 300, 336, 630, 2268, 2310, 5472, 6804, 20412, 47424, 61236, 161304, 183708, 486096, 551124, 1215216, 1653372, 4081104, 4960116, 14880348, 44641044, 133923132, 401769396, 1205308188, 1631268870, 3615924564, 10847773692
Offset: 1

Views

Author

Amiram Eldar, Mar 26 2019

Keywords

Comments

Analogous to A259312 with 3-deficient numbers instead of 3-abundant numbers.
Analogous to A228450 with ratio 3 instead of 2.
The values of sigma(a(n))/a(n) are 1, 1.5, 1.75, 2, 2.333..., 2.5, 2.527..., 2.583..., 2.8, ...

Crossrefs

Cf. A071927, A023197, A228450, A259312.

Programs

  • Mathematica
    sm=0; seq={}; Do[s=DivisorSigma[1,n]/n; If[s<3 && s>sm, sm=s; AppendTo[seq, n]], {n,1,100000}]; 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

A259312 n such that 3 < sigma(n)/n < sigma(m)/m for all abundant numbers m

Original entry on oeis.org

180, 780, 1872, 2352, 16830, 17850, 20496, 51060, 59724, 64890, 430272, 884730, 3767100, 4946508, 15287976, 33806052, 34747416, 40974318, 137168580, 140492772, 157048290, 184773312, 3615557148, 16709332608, 40564903620, 40936559976, 60782804964
Offset: 1

Views

Author

Michel Marcus, Jun 24 2015

Keywords

Comments

This sequence is similar to A071927, but with ratio 3 rather than 2.
a(28) > 7*10^10. - Giovanni Resta, Jun 24 2015

Crossrefs

Cf. A068403 (sigma(n)>3n), A071927 (barely abundant).

Programs

  • PARI
    lista(nn) =  {abk = 4; for (n = 1, nn, ab = sigma(n)/n; if ((ab > 3) && (ab < abk), print1(n, ", "); abk = ab););}

Extensions

a(23)-a(27) from Giovanni Resta, Jun 24 2015

A333967 Subsequence of A071395. The extra constraint is m is not a term if m*q/p is abundant where prime p|m and q is the least prime larger than p.

Original entry on oeis.org

70, 2002, 3230, 4030, 5830, 8415, 8925, 20482, 32445, 45885, 51765, 83265, 107198, 131054, 133042, 178486, 206770, 253270, 253946, 258970, 270470, 310930, 330310, 334305, 362710, 442365, 474045, 497835, 513890, 544310, 567765, 589095, 592670, 602175, 617265, 631670, 654675
Offset: 1

Views

Author

David A. Corneth, Jul 05 2020

Keywords

Examples

			70 is in the sequence as it's abundant. Its prime factorization is 2 * 5 * 7. Each of 3 * 5 * 7, 2 * 7 * 7 and 2 * 5 * 11 are deficient and no divisor of 70 is in this sequence.

Links

Crossrefs

Cf. A071395, A071927, A335557.

Programs

  • Mathematica
    primabQ[n_] := DivisorSigma[1, n] > 2n && AllTrue[Most @ Divisors[n], DivisorSigma[1, #] < 2# &]; seqQ[n_] := Module[{f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]]; q = NextPrime[p]; AllTrue[n*(q/p), DivisorSigma[1, #] <= 2# &]]; Select[Range[10^5], primabQ[#] && seqQ[#] &] (* Amiram Eldar, Jul 05 2020 *)
Showing 1-10 of 12 results. Next

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

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