A002093 -id:A002093 - cp's OEIS frontend

A340111 Coreful highly abundant numbers: numbers m such that csigma(m) > csigma(k) for all k < m, where csigma is the sum of the coreful divisors function (A057723).

1, 2, 3, 4, 7, 8, 12, 16, 24, 32, 36, 48, 56, 64, 72, 96, 108, 128, 144, 192, 200, 216, 288, 360, 400, 432, 504, 576, 648, 720, 792, 800, 864, 1008, 1080, 1152, 1296, 1440, 1512, 1584, 1728, 1800, 1944, 2016, 2160, 2304, 2592, 2880, 3024, 3240, 3456, 3600

Offset: 1

Amiram Eldar, Dec 28 2020

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

Analogous to highly abundant numbers (A002093) with the sum of the coreful divisors function (A057723) instead of the sum of divisors function (A000203).

			The first 10 values of A057723(n) for n=1..10 are: 1, 2, 3, 6, 5, 6, 7, 14, 12, 10. The record values, 1, 2, 3, 6, 7 and 14 occur at 1, 2, 3, 4, 7 and 8, the first 6 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..533 (terms below 10^10)

Cf. A007947, A057723, A283052, A308053.

Similar sequences: A002093, A285614, A292983, A327634, A328134, A329883.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; 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, 3600}]; seq

A342401 Numbers k such that the number of distinct sums of distinct unitary divisors of k sets a new record.

Original entry on oeis.org

1, 2, 6, 10, 30, 42, 60, 66, 78, 90, 110, 130, 170, 190, 210, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1050, 1110, 1218, 1230, 1290, 1410, 1470, 1554, 1590, 1722, 1770, 1830, 1974, 2010, 2130, 2190, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006

Offset: 1

Amiram Eldar, Mar 10 2021

nonn

The corresponding record values are 1, 3, 12, 15, 72, 96, 108, 144, 168, 172, 183, 207, 231, 255, 576, 864, ... (see the link for more values).

The analogous sequence of records with all the divisors (A119347) is conjecturally the highly abundant numbers (A002093), excluding 3 and 10.

			The first 6 terms of A342400 are 1, 3, 3, 3, 3 and 12. The record values, 1, 3 and 12, occur are 1, 2 and 6, the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..99
Amiram Eldar, Table of n, a(n), A342400(a(n)) for n = 1..99

Cf. A002093, A119347, A342400.

f[n_] := Module[{d = Select[Divisors[n], CoprimeQ[#, n/#] &], x, s, m, c}, m = Length[d]; s = Plus @@ d; c = Rest @ CoefficientList[Series[Product[1 + x^d[[i]], {i, 1, m}], {x, 0, s}], x]; Count[c, _?(# > 0 &)]]; fmax = -1; seq = {}; Do[f1 = f[n]; If[f1 > fmax, fmax = f1; AppendTo[seq, n]], {n, 1, 10^3}]; seq

A350299 Numbers k > 1 with sigma(k)/(k * log(log(k))) > sigma(m)/(m * log(log(m))) for all m > k, sigma(k) being A000203(k), the sum of the divisors of k.

Original entry on oeis.org

3, 4, 6, 12, 24, 60, 120, 180, 360, 2520, 5040

Offset: 1

Thomas Strohmann, Dec 23 2021

nonn

Gronwall's theorem says that lim sup_{k -> infinity} sigma(k)/(k*log(log(k))) = exp(gamma). Moreover if the Riemann hypothesis is true, we have sigma(k)/(k*log(log(k))) < exp(gamma) when k > 5040 (gamma = Euler-Mascheroni constant).

The terms in the sequence listed above are provably correct since their ratios: sigma(k)/(k * log(log(k))) are greater than exp(gamma).

Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.

Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), pp. 251-256.
S. Nazardonyavi and S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
Thomas Strohmann, C++ code

Cf. A000203, A067698, A004394, A002093, A004490.

Cf. also A001620, A073004.

A377929 Quasi-practical numbers: positive integers m such that every k <= sigma(m)-m is a sum of distinct proper divisors of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 47, 48, 53, 54, 56, 59, 60, 61, 64, 66, 67, 71, 72, 73, 78, 79, 80, 83, 84, 88, 89, 90, 96, 97, 100, 101, 103, 104, 107, 108, 109, 112, 113, 120, 126, 127, 128

Offset: 1

Andrzej Kukla, Nov 11 2024

Equivalently, positive integers m such that every number k <= d is a sum of distinct proper divisors of m, where d is the largest proper divisor of m (follows from Corollary 2.11 in the Kukla and Miska paper).

Rao and Peng (2013) proved that a number is quasi practical if and only if it is prime or practical (also Theorem 2.9 in Kukla/Miska paper).

Andrzej Kukla, Table of n, a(n) for n = 1..10000
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Union of A000040 and A005153.

Cf. A002093, A103288, A033630, A119348, A174533, A174973, A083207.

QuasiPracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]] || PrimeQ[n]]; Select[Range[200], QuasiPracticalQ] (* Created based on code by T. D. Noe, Apr 02 2010 *)

A195864 Numbers k such that sigma(k) + tau(k) is a perfect number (A000396).

Original entry on oeis.org

3, 14, 15, 168, 326, 5414, 33357341, 4324809536

Offset: 1

Jeff Kruse, Oct 26 2011

A002093(848) < a(9) <= 1155321467378283328. - Donovan Johnson, Nov 03 2011

Cf. A000396, A002093, A007503 (sigma(n) + tau(n)).

perfect = {6, 28, 496, 8128, 33550336, 8589869056, 137438691328}; Select[Range[10000], MemberQ[perfect, DivisorSigma[0, #] + DivisorSigma[1, #]] &] (* T. D. Noe, Nov 03 2011 *)

a(8) from Donovan Johnson, Nov 03 2011

A247334 Highly abundant numbers which are not abundant.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 16

Offset: 1

Andrew Rodland, Sep 13 2014

A number n is called "abundant" if sigma(n) > 2n, and "highly abundant" if sigma(n) > sigma(m) for all m < n. With these definitions, it's possible for a number to be highly abundant but not abundant. (A similar situation occurs with 2 being prime and highly composite.)

Fischer shows that all highly abundant numbers greater than 20 are multiples of 6. Since 6 is perfect and multiples of perfect numbers are abundant, this list is finite and complete.

			10 is in the sequence because sigma(10) > sigma(m) for m = 1 to 9, yet sigma(10) = 17 < 20.

Daniel Fischer, Prove that if Fn is highly abundant, then so is n, Mathematics Stack Exchange, Aug 13 2013

Members of A002093 not in A005101. Members of A002093 in (A000396 union A005100).

for(n=1, 1000, if((sum(i=1, n-1, sign(sigma(n)-sigma(i))) == n-1) && (sigma(n) <= 2*n), print1(n, ", "))) \\ Michel Marcus, Sep 21 2014

is_A247334(n)={!for(i=2,n-1, sigma(n)>sigma(i)||return) && sigma(n)<=2*n} \\ M. F. Hasler, Oct 15 2014

A306621 Highly abundant numbers that are powerful.

Original entry on oeis.org

1, 4, 8, 16, 36, 72, 108, 144, 216, 288, 1800, 3600, 7200

Offset: 1

Amiram Eldar, Mar 01 2019

Alaoglu and Erdős proved that 7200 is the last term.

Intersection of A001694 and A002093.

Leonidas Alaoglu and Paul Erdős, On highly composite and similar numbers, Transactions of the American Mathematical Society, Vol. 56, No. 3 (1944), pp. 448-469.

Cf. A001694, A002093.

seq={1}; sm = 0; Do[s = DivisorSigma[1, n]; If[s>sm, sm = s; If[ Min @  FactorInteger[n][[;;, 2]] > 1, AppendTo[seq, n]]], {n, 2, 72000}]; seq

A381708 a(n) is the smallest nonnegative integer k such that sigma_k(n) > sigma_k(j) for all 1 <= j < n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 1, 2, 1, 3, 0, 3, 2, 2, 1, 3, 1, 3, 1, 3, 2, 3, 0, 4, 3, 3, 2, 4, 1, 4, 2, 4, 2, 4, 0, 4, 3, 4, 2, 4, 1, 4, 2, 4, 2, 4, 0, 4, 3, 4, 2, 4, 2, 4, 2, 4, 3, 4, 0, 5, 3, 4, 2, 5, 2, 5, 3, 4, 2, 5, 1, 5, 3, 4, 3, 5, 2, 5, 2, 5, 3, 5, 1, 5, 3, 5, 3, 5, 1, 5, 3, 5, 3, 5, 1, 5, 3, 5, 2, 5, 2, 5, 3, 5, 3, 5, 1, 5

Offset: 1

Matthew Conroy, Mar 04 2025

nonn

sigma_k(n) is the sum of the k-th powers of the divisors of n.

a(n) exists since one can prove that for k > n*(log 2 + 1/2 log(n-1)), sigma_k sets a record at n.

			For n = 1, k = 0 is enough so a(1) = 0.
For n = 2, k = 0 works since sigma_0(2) = 2 > 1 = sigma_0(1) so a(2) = 0.
For n = 3, sigma_0(3) = 2 = sigma_0(2), but sigma_1(3) = 1^1+3^1 = 4 > 3 = sigma_1(2) > 1 = sigma_1(1) so a(3) = 1.
For n = 4, sigma_0(4) = 1^0+2^0+4^0 = 3 > 2 = sigma_0(3) = sigma_0(2) > 1 = sigma_0(1) so a(4) = 0.
For n = 5, sigma_0(5) = 2 = sigma_0(2) and sigma_1(5) = 6 < sigma_1(4) = 7 but sigma_2(5) = 26 > sigma_2(4) > sigma_2(3) > sigma_2(2) > sigma_2(1) so a(5) = 2.

Cf. A000203, A001157, A002093, A002182, A098475, A109974, A193988.

check(n,k) =  my(m=0);for(i=1,n-1, my(s=sigma(i,k)); if(s>m,m=s)); if(sigma(n,k)>m,return(1),return(0));
a(n) = my(ii=0); while(!check(n, ii), ii++);  ii;

a(n) = 0 precisely when n is highly composite number A002182.
a(n) = 1 precisely when n is highly abundant A002093 and not highly composite.
a(n) = 2 precisely when n is in A193988 and is not highly composite and is not highly abundant.
a(n) <= m if n < A098475(m). Empirically, it appears that a(A098475(m)) = m+1. - Pontus von Brömssen, Mar 16 2025

A382930 a(n) is the smallest k such that A382506(k) + sigma(k) = A000396(n).

Original entry on oeis.org

1, 4, 16, 180, 2520, 7207200

Offset: 1

Leo Hennig, Apr 09 2025

Numbers k for which A382506 "jumps", there appears to be a remarkable relationship to A002182, the highly composites, in line with jumps for A382483 being in that series. See also the graph for A382506.

			A382506(16) = 465, 465 + sigma(16) = 496, which is perfect.

Cf. A002093, A002182, A382506.

New name and a(1) from Jinyuan Wang, Apr 10 2025

a(6) from David A. Corneth, Apr 10 2025

cp's OEIS Frontend

A340111 Coreful highly abundant numbers: numbers m such that csigma(m) > csigma(k) for all k < m, where csigma is the sum of the coreful divisors function (A057723).

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A342401 Numbers k such that the number of distinct sums of distinct unitary divisors of k sets a new record.

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A350299 Numbers k > 1 with sigma(k)/(k * log(log(k))) > sigma(m)/(m * log(log(m))) for all m > k, sigma(k) being A000203(k), the sum of the divisors of k.

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A377929 Quasi-practical numbers: positive integers m such that every k <= sigma(m)-m is a sum of distinct proper divisors of m.

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A195864 Numbers k such that sigma(k) + tau(k) is a perfect number (A000396).

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A247334 Highly abundant numbers which are not abundant.

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A306621 Highly abundant numbers that are powerful.

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A381708 a(n) is the smallest nonnegative integer k such that sigma_k(n) > sigma_k(j) for all 1 <= j < n.

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A382930 a(n) is the smallest k such that A382506(k) + sigma(k) = A000396(n).

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