A002093 -id:A002093 - cp's OEIS frontend

A161468 Fibonacci numbers that are more abundant than any smaller Fibonacci number.

1, 2, 3, 5, 8, 21, 34, 55, 89, 144, 377, 610, 987, 1597, 2584, 6765, 10946, 17711, 28657, 46368, 196418, 317811, 514229, 832040, 2178309, 3524578, 5702887, 9227465, 14930352, 63245986, 102334155, 267914296, 701408733, 1134903170, 2971215073

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 10 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1016

Cf. A002093

lst={};k=0;Do[p=Plus@@Divisors[f=Fibonacci[n]];If[p>k,AppendTo[lst,f];k=p],{n,5!}];lst

Name corrected by T. D. Noe, Jul 06 2010

A172516 Least number k such that sigma(k) >= 2^n.

Original entry on oeis.org

2, 3, 6, 10, 18, 30, 60, 108, 180, 360, 720, 1260, 2520, 5040, 9240, 17640, 35280, 65520, 131040, 257040, 498960, 982800, 1884960, 3603600, 7207200, 14414400, 28274400, 56548800, 110270160, 220540320, 428828400, 845404560, 1690809120

Offset: 1

T. D. Noe, Feb 05 2010

nonn

For n-bit arithmetic, m=a(n)-1 is the largest number for which sigma(m) can be computed without overflow. This is a subsequence of the highly abundant numbers, A002093, which is very useful for computing this sequence. a(63) is 1454751268447276800.

T. D. Noe, Table of n, a(n) for n=1..64

A141847 (least number k such that sigma2(k) >= 2^n)

k=1; Table[While[DivisorSigma[1,k]<2^n, k++ ]; k, {n,20}]

a(n) <= 2 * a(n-1)

A192929 Least highly abundant number with n distinct prime factors.

Original entry on oeis.org

1, 2, 6, 30, 210, 4620, 120120, 4084080, 116396280, 2677114440, 310545275040, 14440355289360, 1068586291412640, 114873026326858800, 9419588158802421600, 442720643463713815200, 54011918502573085454400, 8587895041909120587249600

Offset: 0

Enrique Pérez Herrero, Jul 12 2011

nonn

a(n) is the least number in sequence A002093, with A001221(a(n)) = n.

S. S. Pillai, Highly abundant numbers, Bull. Calcutta Math. Soc., 35, (1943), 141-156.

L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Wikipedia, Highly abundant number

Cf. A001222, A002093, A225194.

A225194 Largest highly abundant number with n distinct prime factors.

Original entry on oeis.org

1, 16, 288, 7200, 604800, 46569600, 1816214400, 92626934400, 5573053886400, 445257673660800, 11738611396512000, 1734575477357923200, 98737373326527936000, 8491414106081402496000, 1697221394453020323888000, 372257225850029124372768000

Offset: 1

Enrique Pérez Herrero, May 01 2013

nonn

a(n) is the largest number in sequence A002093, with A001221(a(n)) = n.

Pillai, S. S., Highly abundant numbers, Bull. Calcutta Math. Soc., 35, (1943), 141-156

L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Wikipedia, Highly abundant number

Cf. A002093, A001222.

Cf. A192929.

A240073 Deficient numbers k for which sigma(k), the sum of divisors of k, reaches a new maximum.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 14, 16, 21, 22, 26, 32, 44, 50, 52, 63, 64, 76, 92, 98, 105, 110, 124, 128, 136, 152, 170, 182, 184, 212, 225, 230, 232, 248, 256, 290, 296, 310, 315, 328, 344, 370, 376, 405, 410, 424, 470, 472, 484, 495, 512, 568, 584, 592, 632, 656

Offset: 1

T. D. Noe, Apr 08 2014

nonn

Every power of 2 appears. The deficient number k has sigma(k) < 2*k. In relation to the highly abundant numbers, these numbers might be termed highly deficient numbers.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A002093 (highly abundant numbers), A005100 (deficient numbers).

Cf. A228450 (deficient numbers with increasing abundancy).

t = {}; mn = 0; n = 0; While[Length[t] < 100, n++; d = DivisorSigma[1, n]; If[mn < d < 2*n, AppendTo[t, n]; mn = d]]; t

lista(kmax) = {my(sigmax = 0, sig); for(k = 1, kmax, sig = sigma(k); if(sig < 2*k && sig > sigmax, sigmax = sig; print1(k, ", ")));} \\ Amiram Eldar, Apr 06 2024

A243915 a(n) = sigma(omega(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 4, 1, 1, 3, 3, 3, 3, 1, 3, 3, 3, 1, 4, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 4, 1, 3, 3, 1, 3, 4, 1, 3, 3, 4, 1, 3, 1, 3, 3, 3, 3, 4, 1, 3, 1, 3, 1, 4, 3, 3, 3

Offset: 2

Wesley Ivan Hurt, Jun 14 2014

If n is the product of k distinct primes, then a(n) = sigma(k).
Records occur at n = 2, 6, 30, 210, 30030, ... . - R. J. Mathar, Jun 18 2014 [The position of the n-th record is A002110(A002093(n)). - Amiram Eldar, Dec 29 2024]
If n = p^k where p is prime and k is a positive integer, a(p^k) = sigma(omega(p^k)) = sigma(1) = 1. - Wesley Ivan Hurt, May 21 2021

G. C. Greubel, Table of n, a(n) for n = 2..5000

Cf. A000203 (sigma), A001221 (omega), A002110, A002093.

with(numtheory):
A243915 := proc(n)
    sigma(nops(factorset(n))) ;
end proc:
seq(A243915(n), n=2..100); # R. J. Mathar, Jun 18 2014

Table[DivisorSigma[1, PrimeNu[n]], {n, 2, 100}]

for(n=2,50, print1(sigma(omega(n)), ", ")) \\ G. C. Greubel, May 17 2017

a(n) = A000203(A001221(n)).

A244043 Numbers n for new peaks of floor(sigma(n)/primepi(n)).

Original entry on oeis.org

2, 6, 12, 24, 30, 36, 60, 96, 120, 180, 240, 360, 600, 720, 840, 1080, 1260, 1680, 2520, 5040, 7560, 10080, 12600, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 138600, 166320, 196560, 221760, 277200

Offset: 1

Bill McEachen, Jun 17 2014

sigma(n) = A000203(n), primepi(n) = A000720(n).
The sequence entries frequently are members of A002182 (highly composite numbers). Similar sequences can be generated by varying the "k" seen in the PARI code, for example to k=2.
Subsequence of A002093 (highly abundant numbers). - Jens Kruse Andersen, Jul 15 2014

			Example at n=2 (start), sigma(2)=3, primepi(2)=1 so the initial peak is 3.
We see a new peak (4) at n=6 from floor(12/3), a(2)=6.
We see new peak (5) at n=12 from floor(28/5), a(3)=12. No entry is defined for n<2.

Jens Kruse Andersen, Table of n, a(n) for n = 1..82

Cf. A000203, A000720, A002093, A002182.

Reap[For[peak = 0; n = 2, n < 10^5, n++, f = Floor[DivisorSigma[1, n] / PrimePi[n]]; If[f > peak, peak = f; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 12 2018 *)
DeleteDuplicates[Table[{n,Floor[DivisorSigma[1,n]/PrimePi[n]]},{n,2,85000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Mar 13 2025 *)

genit={my(maxx=100000);peak=3;k=1;n=3;optr=2;sptr=1;
write("A244043.csv",sptr," , ",2);while(npeak,sptr++;peak=c;
write("A244043.csv",sptr," , ",optr););n++);}

Define A(n) = floor(A000203(n)/A000720(n)) for n >= 2. Then a(1) = 2 and for n >= 2 a(n) is the least k > a(n-1) such that A(k) > A(a(n-1)). - Wolfdieter Lang, Jul 03 2014

Edited. Crossrefs for sigma and primepi added. - Wolfdieter Lang, Jul 03 2014

More terms from Harvey P. Dale, Mar 13 2025

A253248 Number of k <= n with A000203(k) <= A000203(n).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 6, 8, 8, 10, 8, 12, 10, 13, 14, 16, 13, 18, 14, 20, 19, 20, 17, 24, 20, 25, 24, 27, 19, 30, 23, 31, 29, 30, 30, 36, 25, 35, 34, 39, 30, 42, 31, 41, 41, 41, 34, 48, 38, 48, 44, 51, 36, 53, 46, 55, 48, 51, 42, 60, 43, 57, 59, 63, 52

Offset: 1

Robert Israel, Jun 04 2015

			A000203(7) = 8 >= A000203(k) for k = 1,2,3,4,5,7, so a(7) = 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A002093, A007609.

N:= 1000:
B:= map(numtheory:-sigma,[$1..N]):
M:= max(B):
X:= Vector(M):
for n from 1 to N do
  b:= B[n];
  X[b..-1]:= X[b..-1] + <(1$(M-b+1))>;
  A[n]:= X[b];
od:
seq(A[n],n=1..N);

f[v_] := Count[v, ?(# <= v[[-1]] &)]; seq[lim] := Module[{v = DivisorSigma[1, Range[lim]]}, f[v[[1 ;; #]]] & /@ Range[Length[v]]]; seq[65] (* Amiram Eldar, Dec 19 2024 *)

a(n) <= n, with equality if and only if n is in A002093.

Empirically it appears that lim inf_(n -> infinity) a(n)/n = 2/3, with minimum value a(29)/29 = 19/29.

A290490 Numbers k such that (sum of proper unitary divisors of k) > (sum of proper unitary divisors of m) for all m < k.

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 22, 26, 30, 42, 60, 66, 78, 102, 114, 138, 150, 174, 186, 210, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1050, 1110, 1218, 1230, 1290, 1410, 1470, 1590, 1722, 1770, 1830, 2010, 2130, 2190, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6510, 7410, 7590, 7770

Offset: 1

Ilya Gutkovskiy, Aug 03 2017

nonn

Numbers k such that A034460(k) > A034460(m) for all m < k.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Unitary Divisor
Index entries for sequences related to sums of divisors

Cf. A002182, A002093, A034090, A034448, A034460, A063919, A280013, A281782, A285614.

mx = -1; t = {}; Do[u = DivisorSum[n, # &, GCD[#, n/#] == 1 &] - n; If[u > mx, mx = u; AppendTo[t, n]], {n, 8000}]; t

sumud(n) = sumdiv(n, d, if (gcd(d, n/d)==1, d)) - n;
lista(nn) = {lasts = -1; for (n=1, nn, if ((news = sumud(n)) > lasts, print1(n, ", "); lasts = news););} \\ Michel Marcus, Aug 04 2017

A309943 Numbers k such that k * d(k) > j * d(j) for all j < k, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1008, 1080, 1200, 1260, 1440, 1680, 1980, 2016, 2100, 2160, 2520, 3120

Offset: 1

Amiram Eldar, Aug 24 2019

nonn

Differs from A002093 for n >= 41.

Nicolas asks if there are infinitely many terms of this sequence that are not largely composite (A067128).

Amiram Eldar, Table of n, a(n) for n = 1..457
Jean-Louis Nicolas, Nombres hautement composés, Acta Arithmetica, Vol. 49 (1988), pp. 395-412, alternative link. See p. 411.

Cf. A000005, A002093, A038040, A067128.

s = {}; dm = 0; Do[d1 = n * DivisorSigma[0, n]; If[d1 > dm, dm = d1; AppendTo[s, n]], {n, 1, 10^4}]; s

cp's OEIS Frontend

A161468 Fibonacci numbers that are more abundant than any smaller Fibonacci number.

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A172516 Least number k such that sigma(k) >= 2^n.

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A192929 Least highly abundant number with n distinct prime factors.

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A225194 Largest highly abundant number with n distinct prime factors.

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A240073 Deficient numbers k for which sigma(k), the sum of divisors of k, reaches a new maximum.

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PARI

A243915 a(n) = sigma(omega(n)).

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Maple

Mathematica

PARI

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A244043 Numbers n for new peaks of floor(sigma(n)/primepi(n)).

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PARI

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A253248 Number of k <= n with A000203(k) <= A000203(n).

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Maple