A085829 -id:A085829 - cp's OEIS frontend

A050226 Numbers m such that m divides Sum_{k = 1..m} A000005(k).

1, 4, 5, 15, 42, 44, 47, 121, 336, 340, 347, 930, 2548, 6937, 6947, 51322, 379097, 379131, 379133, 2801205, 20698345, 56264090, 56264197, 152941920, 152942012, 8350344420, 61701166395, 455913379395, 455913379831, 1239301050694, 3368769533660, 3368769533812

Offset: 1

Labos Elemer, Dec 20 1999

			For k = 15 the sum is 1 + 2 + 2 + 3 + 2 + 4 + 2 + 4 + 3 + 4 + 2 + 6 + 2 + 4 + 4 = 45 which is divisible by 15.

Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 112-113, 2003.

Donovan Johnson, Table of n, a(n) for n = 1..39 (quotients <= 40)

Cf. A000005, A006218, A057494, A085567, A085829.

s = 0; Do[ s = s + DivisorSigma[ 0, n ]; If[ Mod[ s, n ] == 0, Print[ n ] ], {n, 1, 2*10^9} ]
k=10^6; a[1]=1;a[n_]:=a[n]=DivisorSigma[0,n]+a[n-1]; nd=a/@Range@k; Select[Range@k,Divisible[nd[[#]],#]&] (* Ivan N. Ianakiev, Apr 30 2016 *)
Module[{nn=400000},Select[Thread[{Range[nn],Accumulate[DivisorSigma[0,Range[nn]]]}],Divisible[#[[2]],#[[1]]]&]][[All,1]] (* The program generates the first 19 terms of the sequence. To generate more, increase the nn constant. *) (* Harvey P. Dale, Jul 03 2022 *)

lista(nn) = {my(s = 0); for (n=1, nn, s += numdiv(n); if (!(s % n), print1(n, ", ")););} \\ Michel Marcus, Dec 14 2015

def A050226_list(len):
    a, L = 0, []
    for n in (1..len):
        a += sigma(n,0)
        if n.divides(a): L.append(n)
    return L
A050226_list(10000) # Peter Luschny, Dec 18 2015

m is in the sequence if Sum_{i = 1..m} d(i) = m*k, k an integer, where d(i) = number of divisors of i.

More terms from Robert G. Wilson v, Sep 21 2000
Further terms from Naohiro Nomoto, Aug 03 2001
a(26)-a(30) from Donovan Johnson, Dec 21 2008

A057494 a(n) = Sum_{k = 1..10^n} d(k) where d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 27, 482, 7069, 93668, 1166750, 13970034, 162725364, 1857511568, 20877697634, 231802823220, 2548286736297, 27785452449086, 300880375389757, 3239062263181054, 34693207724724246, 369957928177109416, 3929837791070240368, 41600963003695964400, 439035480966899467508

Offset: 0

Robert G. Wilson v, Sep 21 2000

nonn

The Polymath project describes an algorithm for computing a(n) in time O(2.154...^n), see Tao, Croot, and Helfgott link. - Charles R Greathouse IV, Apr 16 2012

Henri Lifchitz, Table of n, a(n) for n = 0..36
Terence Tao, Ernest Croot III, and Harald Helfgott, Deterministic methods to find primes, Mathematics of Computation, 81 (2012), 1233-1246. arXiv:1009.3956, [math.NT], 2010-2012.

Cf. A006218, A050226, A085567, A085829, A085831.

k = s = 0; Do[ While[ k < 10^n, k++; s = s + DivisorSigma[ 0, k ] ]; Print[s], {n, 0, 8} ]

a(n) = sum(k=1, 10^n, numdiv(k)); \\ Michel Marcus, Feb 19 2017

from math import isqrt
def A057494(n): return -(s:=isqrt(m:=10**n))**2+(sum(m//k for k in range(1,s+1))<<1) # Chai Wah Wu, Oct 23 2023

a(n) = A006218(10^n). - Max Alekseyev, May 10 2009

a(10)-a(16) from Max Alekseyev, Jan 25 2010
a(17)-a(19) from Donovan Johnson, Dec 26 2012
a(20)-a(27) from Hiroaki Yamanouchi, Sep 22 2015
a(28)-a(36) from Henri Lifchitz, Feb 19 2017

A085831 a(n) = Sum_{k=1..2^n} d(k) where d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 3, 8, 20, 50, 119, 280, 645, 1466, 3280, 7262, 15937, 34720, 75108, 161552, 345785, 736974, 1564762, 3311206, 6985780, 14698342, 30850276, 64607782, 135030018, 281689074, 586636098, 1219788256, 2532608855, 5251282902, 10874696106, 22493653324, 46475828418

Offset: 0

Robert G. Wilson v, Sep 21 2000

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..82 (terms 0..64 from Donovan Johnson)

Cf. A006218, A050226, A085567, A085829, A057494.

k = s = 0; Do[ While[ k < 2^n, k++; s = s + DivisorSigma[ 0, k ]]; Print[s], {n, 0, 29} ]

a(n) = sum(k=1, 2^n, numdiv(k)); \\ Michel Marcus, Oct 10 2021

from math import isqrt
def A085831(n): return (lambda m, r: 2*sum(r//k for k in range(1, m+1))-m*m)(isqrt(2**n),2**n) # Chai Wah Wu, Oct 08 2021

A328331 a(n) is the least k such that the average number of unitary divisors of {1..k} is >= n.

Original entry on oeis.org

1, 6, 35, 190, 1015, 5304, 27417, 142142, 736782, 3816852, 19774690, 102446730, 530743749, 2749606626, 14244797910

Offset: 1

Amiram Eldar, Oct 22 2019

The unitary version of A085829.

			a(2) = 6 because the average number of unitary divisors of {1..6} is  A064608(6)/6 = 13/6 > 2.

Cf. A013661, A034444, A064608, A085829.

seq={}; s = 0; k = 1; Do[While[s += 2^PrimeNu[k]; s < k*n, k++]; AppendTo[seq, k]; k++, {n, 1, 10}]; seq

Lim_{n->oo} a(n+1)/a(n) = exp(zeta(2)) = exp(Pi^2/6) = 5.180668... (since A064608(n) ~ n*log(n)/zeta(2)).

A085567 Least m such that the average number of divisors of all integers from 1 to m equals n, or 0 if no such number exists.

Original entry on oeis.org

1, 4, 15, 42, 121, 336, 930, 2548, 6937, 0, 51322, 0, 379097, 0, 2801205, 0, 20698345, 56264090, 152941920, 0, 0, 0, 8350344420, 0, 61701166395, 0, 455913379395, 1239301050694, 3368769533660, 0, 24892027072619, 0, 183928584450999, 0, 0, 0

Offset: 1

Jason Earls, Jul 06 2003

nonn

"In 1838 Lejeune Dirichlet (1805-1859) proved that (1/n)*sum_{r=1..n} #(divisors(r)), the average number of divisors of all integers from 1 to n, approaches ln n + 2gamma - 1 as n increases." [Havil]

a(n+1)/a(n) ~ e. - Robert G. Wilson v

			a(2) = 4 because (1/4)*(1+2+2+3) = 2.

Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 112-113, 2003.

Donovan Johnson, Table of n, a(n) for n = 1..40

Cf. A050226, A057494, A085829.

Edited and extended by Robert G. Wilson v, Jul 07 2003
Corrected by Rick L. Shepherd, Aug 28 2003
Missing terms a(16)-a(17) and a(20)-a(29) added by Donovan Johnson, Dec 21 2008
a(30)-a(36) from Donovan Johnson, Jul 20 2011

A338891 a(n) is the least number k such that the average number of odd divisors of {1..k} is >= n.

Original entry on oeis.org

1, 21, 165, 1274, 9435, 69720, 515230, 3807265, 28132035, 207869515, 1535959665, 11349295155, 83860579775

Offset: 1

Ilya Gutkovskiy, Nov 14 2020

			a(5) = 9435 because the average number of odd divisors of {1..9435} is >= 5.

Eric Weisstein's World of Mathematics, Odd Divisor Function.

Cf. A001227, A060831, A072334, A085829, A328331, A338943.

m = 1; sum = 0; s = {}; Do[sum += DivisorSigma[0, k/2^IntegerExponent[k, 2]]; If[sum >= m*k, AppendTo[s, k]; m++], {k, 1, 10^6}]; s (* Amiram Eldar, Nov 15 2020 *)

a(n) = my(s=1, k=1); while(s>valuation(k, 2))); k; \\ Michel Marcus, Nov 14 2020

a(n+1)/a(n) approaches e^2.

a(11)-a(12) from Amiram Eldar, Nov 16 2020

a(13) from Bill McEachen, Sep 01 2025

A338943 a(n) is the least number k such that the average number of distinct prime divisors of {1..k} is >= n.

Original entry on oeis.org

1, 6, 455, 8167302

Offset: 0

Ilya Gutkovskiy, Nov 17 2020

10^18 < a(4) < 10^19. - Daniel Suteu, Nov 17 2020

			a(2) = 455 because the average number of distinct prime divisors of {1..455} is >= 2.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curio for 455
Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Cf. A001221, A013939, A085829, A328331, A338891.

A336304 a(n) is the least number k such that the average number of prime divisors of {1..k} counted with multiplicity is >= n.

Original entry on oeis.org

1, 4, 32, 2178, 416417176

Offset: 0

Ilya Gutkovskiy, Nov 18 2020

			a(1) = 4 since the average number of prime divisors of {1..4} counted with multiplicity equals (0 + 1 + 1 + 2)/4 = 1 which is >= 1 and this is the least such number.
a(3) = 2178 because the average number of prime divisors of {1..2178} counted with multiplicity is >= 3 and this is the least such number.

Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A001222, A022559, A085829, A328331, A338891, A338943.

s[n_] := Module[{m = 0, c = 0, k = 1, sum = 0, seq = {}}, While[c < n, sum += PrimeOmega[k]; If[sum >= m*k, c++; AppendTo[seq, k]; m++]; k++]; seq]; s[4] (* Amiram Eldar, Nov 18 2020 *)

a(n)=my(m=0,k=1);while(k>0, m+=bigomega(k); if(m>=k*n,break);k++);k \\ Derek Orr, Nov 18 2020

A344272 a(n) is the least k such that the average number of nonunitary divisors of {1..k} is >= n.

Original entry on oeis.org

54, 816, 10530, 135200, 1733760, 22216752, 284685408, 3647978320, 46745561100, 599002268832, 7675674748560

Offset: 1

Amiram Eldar, May 13 2021

			a(1) = 54 since the average of the number of nonunitary divisors of {1..54} is (Sum_{k=1..54} A056175(k))/54 = 1.

Cf. A013661, A056175.
The nonunitary version of A085829.
Similar sequences: A328331, A336304, A338891, A338943, A344273, A344274.

nd[n_] := DivisorSigma[0,n] - 2^PrimeNu[n]; seq={}; s = 0; k = 1; Do[While[s = s + nd[k]; s < k*n, k++]; AppendTo[seq, k]; k++, {n, 1, 5}]; seq

Lim_{n->oo} a(n+1)/a(n) = exp(1/(1-1/zeta(2))) = exp(Pi^2/(Pi^2-6)) = 12.8140996101...

a(10)-a(11) from Martin Ehrenstein, May 23 2021

A344273 a(n) is the least k such that the average number of bi-unitary divisors of {1..k} is >= n.

Original entry on oeis.org

1, 6, 24, 80, 273, 960, 3336, 11480, 39648, 136952, 472416, 1630164, 5625480, 19412736, 66992016, 231184800, 797806152, 2753187210, 9501109380, 32787848746

Offset: 1

Amiram Eldar, May 13 2021

			a(2) = 6 since the average number of bi-unitary divisors of {1..6} is A306069(6)/6 = 13/6 > 2.

Cf. A286324, A306069, A306071.
The unitary version of A085829.
Similar sequences: A328331, A336304, A338891, A338943, A344272, A344274.

f[p_, e_] := If[OddQ[e], e + 1, e]; bdivnum[1] = 1; bdivnum[n_] := Times @@ (f @@@ FactorInteger[n]); seq={}; s = 0; k = 1; Do[While[s = s + bdivnum[k]; s < k*n, k++]; AppendTo[seq, k]; k++, {n, 1, 10}]; seq

Lim_{n->oo} a(n+1)/a(n) = exp(1/A) = 3.4509501567..., where A is A306071.

cp's OEIS Frontend

A050226 Numbers m such that m divides Sum_{k = 1..m} A000005(k).

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A057494 a(n) = Sum_{k = 1..10^n} d(k) where d(n) = number of divisors of n (A000005).

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A085831 a(n) = Sum_{k=1..2^n} d(k) where d(n) = number of divisors of n (A000005).

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A328331 a(n) is the least k such that the average number of unitary divisors of {1..k} is >= n.

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A085567 Least m such that the average number of divisors of all integers from 1 to m equals n, or 0 if no such number exists.

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A338891 a(n) is the least number k such that the average number of odd divisors of {1..k} is >= n.

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A338943 a(n) is the least number k such that the average number of distinct prime divisors of {1..k} is >= n.

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A336304 a(n) is the least number k such that the average number of prime divisors of {1..k} counted with multiplicity is >= n.

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