A050226 -id:A050226 - cp's OEIS frontend

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.

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

A063971 Values of n for which A013939(n)/n is an integer.

Original entry on oeis.org

1, 6, 7, 8, 9, 455, 457, 458, 459, 461, 8167302, 8167314, 8167315, 8167316, 8167328, 8167330, 8167335, 8167336, 8167346, 8167347, 8167348, 8167350, 8167351, 8167352, 8167410, 8167413

Offset: 1

Labos Elemer, Sep 05 2001

nonn

For 455, 457, 458, 459, 461 the quotient is 2. The cause of "step-like" appearance of terms is that the next integer is reached slowly with the summatory function A013939. Next "island" is expected above 500000. Similar phenomenon is observable in the analogous A050226 sequence too.
The quotients for "3rd island" after 8160000 equal 3. (Sep 21 2001)
a(27) > 10^13. - Giovanni Resta, Apr 24 2017

Cf. A013939, A050226, A001221, A064612.

s = 0; Do[s = s + Length[FactorInteger[n]]; If[IntegerQ[s/n], Print[n]], {n, 1, 10000000}]

More terms from Robert G. Wilson v, Sep 19 2001

A379922 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^(k+1) * sigma_2(k).

Original entry on oeis.org

1, 2, 3, 42, 329, 633, 1039, 5689, 26621, 39245, 1101875, 1216075, 40088584, 67244920, 104332211, 549673265, 777631064, 19879301756

Offset: 1

Amiram Eldar, Jan 06 2025

Numbers m such that m | A379921(m).
The corresponding quotients, A379921(m)/m, are -1, 2, -2, 120, 5228, ... (see the link for more values).
a(19) > 5*10^10, if it exists.

Amiram Eldar, Table of n, a(n), A379921(a(n))/a(n) for n = 1..18.

Cf. A001157 (sigma_2), A379921.

Similar sequences: A050226, A048290, A056550, A064605, A067929, A067931, A379923, A379924.

With[{m = 40000}, Position[Accumulate[Table[(-1)^n * DivisorSigma[2, n], {n, 1, m}]]/Range[m], _?IntegerQ] // Flatten]

list(lim) = my(s = 0); for(k = 1, lim, s += (-1)^k * sigma(k, 2); if(!(s % k), print1(k, ", ")));

A379923 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^k * A000005(k).

Original entry on oeis.org

1, 5, 18, 22, 25, 29, 197, 1350, 1360, 1362, 1368, 1381, 1391, 1395, 10200, 75486, 75490, 557768, 557843, 557853, 557898, 4121846, 4122064, 4122112, 4122222, 30457732, 30457773, 30457835, 30458040, 30458133, 30458138, 30458140, 30458335, 225056911, 225056919, 225056925, 225056989

Offset: 1

Amiram Eldar, Jan 06 2025

nonn

Numbers m such that m | A307704(m).
The corresponding quotients, A307704(m)/m, are -1, 0, 1, 1, 1, 1, 2, 3, 3, 3, ... (see the link for more values).
a(38) > 2*10^10, if it exists.

Amiram Eldar, Table of n, a(n), A307704(a(n))/a(n) for n = 1..37.

Cf. A000005, A307704.

Similar sequences: A050226, A048290, A056550, A064605, A067929, A067931, A379922, A379924.

With[{m = 10000}, Position[Accumulate[Table[(-1)^n * DivisorSigma[0, n], {n, 1, m}]]/Range[m], _?IntegerQ] // Flatten]

list(lim) = my(s = 0); for(k = 1, lim, s += (-1)^k * numdiv(k); if(!(s % k), print1(k, ", ")));

A379924 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^(k+1) * usigma(k).

Original entry on oeis.org

1, 2, 9, 54, 101, 178, 189, 2071, 3070, 9171, 11450, 12794, 21405, 27553, 35285, 251974, 2069863, 2395894, 155931488, 387586437, 758519827, 1202435693, 9859113494, 42703260442

Offset: 1

Amiram Eldar, Jan 06 2025

Numbers m such that m | A370898(m).
The corresponding quotients, A370898(m)/m, are -1, 1, 0, 6, 9, ... (see the link for more values).
a(25) > 5*10^10, if it exists.

Amiram Eldar, Table of n, a(n), A370898(a(n))/a(n) for n = 1..24.

Cf. A034448 (usigma), A370898.

Similar sequences: A050226, A048290, A056550, A064605, A067929, A067931, A379922, A379923.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; With[{m = 260000}, Position[Accumulate[Table[(-1)^n * usigma[n], {n, 1, m}]]/Range[m], _?IntegerQ] // Flatten]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]); }
list(lim) = my(s = 0); for(k = 1, lim, s += (-1)^k * usigma(k); if(!(s % k), print1(k, ", ")));

A140221 A number n is included if n is coprime to Sum_{k=1..n} d(k), where d(k) is the number of divisors of k.

Original entry on oeis.org

1, 2, 3, 7, 9, 10, 11, 12, 13, 14, 17, 19, 23, 25, 27, 28, 29, 31, 32, 34, 35, 37, 41, 43, 45, 49, 50, 51, 52, 53, 54, 56, 58, 59, 61, 62, 65, 67, 69, 71, 73, 75, 77, 79, 81, 82, 83, 84, 86, 87, 88, 89, 92, 93, 94, 95, 97, 98, 101, 103, 107, 109, 111, 113, 115, 117, 119, 122

Offset: 1

Leroy Quet, May 12 2008

nonn

Sum_{k=1..n} d(k) = Sum_{k=1..n} floor(n/k) = A006218(n).

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

Cf. A006218, A050226, A140222.

N:= 500: # for terms <= N
S:= ListTools:-PartialSums(map(numtheory:-tau, [$1..N])):
select(t -> igcd(t,S[t])=1, [$1..N]); # Robert Israel, Feb 20 2024

With[{r = Range[200]}, PositionIndex[CoprimeQ[r, Accumulate[DivisorSigma[0, r]]]][True]] (* Paolo Xausa, Feb 21 2024 *)

from math import gcd, isqrt
def A140221_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: gcd(n,-(s:=isqrt(n))**2+(sum(n//k for k in range(1,s+1))<<1))==1,count(max(startvalue,1)))
A140221_list = list(islice(A140221_gen(),30)) # Chai Wah Wu, Oct 23 2023

More terms from Max Alekseyev, May 10 2009

A344731 Numbers k such that k divides A306069(k).

Original entry on oeis.org

1, 275, 277, 3337, 3353, 3359, 39675, 39689, 472467, 797806459, 9501109507

Offset: 1

Amiram Eldar, May 27 2021

The corresponding quotients A306069(k)/k are 1, 5, 5, 7, 7, 7, 9, 9, 11, 17, 19, ...

a(12) > 7.5*10^10, if it exists.

			a(1) = 1 since A306069(1) = 1 is divisible by 1.
a(2) = 275 since A306069(275) = 1375 = 5 * 275 is divisible by 275.

Cf. A286324, A306069.
The bi-unitary version of A050226.
Similar sequences: A064610, A344732, A344733.

f[p_, e_] := If[Mod[e, 2] == 1, (e + 1), e]; s[1] = 1; s[n_] := s[n] = s[n - 1] + Times @@ f @@@ FactorInteger[n]; Select[Range[40000], Divisible[s[#], #] &]

A344732 Numbers k such that k divides Sum_{j=1..k} A048105(j).

Original entry on oeis.org

1, 2, 3, 54, 58, 62, 71, 10535, 10541, 10579, 135242, 135243, 1733777, 1733781, 1733895, 1733905, 1733999, 22216757, 22216765, 22216790, 22216808, 22216814, 46745561148, 46745561156

Offset: 1

Amiram Eldar, May 27 2021

The corresponding quotients Sum_{j=1..k} A048105(j)/k are 0, 0, 0, 1, 1, 1, 1, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 9, 9, ...

a(25) > 7.5*10^10, if it exists.

			a(1) = 1 since A048105(1) = 0 is divisible by 1.
a(4) = 54 since Sum_{j=1..54} A048105(j) = 54 is divisible by 54.

Cf. A048105.
The non-unitary version of A050226.
Similar sequences: A064610, A344731, A344733.

s[1] = 0; s[n_] := s[n] = s[n - 1] + DivisorSigma[0, n] - 2^PrimeNu[n]; Select[Range[140000], Divisible[s[#], #] &]

A344733 Numbers k such that k divides A327573(k).

Original entry on oeis.org

1, 25, 387, 6063, 1416379, 1416403, 1416411, 331362359, 5068450527

Offset: 1

Amiram Eldar, May 27 2021

The corresponding quotients A327573(k)/k are 1, 3, 5, 7, 11, 11, 11, 15, 17, ...

a(10) > 7.5*10^10, if it exists.

			a(1) = 1 since A327573(1) = 1 is divisible by 1.
a(2) = 25 since A327573(25) = 75 = 3 * 25 is divisible by 25.

Cf. A037445, A327573.
The infinitary version of A050226.
Similar sequences: A064610, A344731, A344732.

f[p_, e_] := 2^DigitCount[e, 2, 1]; s[1] = 1; s[n_] := s[n] = s[n - 1] + Times @@ f @@@ FactorInteger[n]; Select[Range[1.5*10^6], Divisible[s[#], #] &]

A073128 Integer quotients arising in A063986.

Original entry on oeis.org

0, 1, 1, 5, 5, 49, 118, 121, 2406, 2698, 4182, 4946, 153627, 3087192, 8203485, 38394376, 487844934, 2822741576, 4140154385, 4397137572, 8583966231

Offset: 1

Labos Elemer, Jul 16 2002

			n=15, A063986(15)=41846733, sum of first 41846733 cototients is s=343289046464505, a(15)=s/41846733. Only in knowledge of either the quotients or of partial sums of cototients is it possible to continue A063986 without recomputing previous terms!

Cf. A051953, A000010, A002088, A048920, A000217, A050226, A063986.

a(n)=Sum[cototient[j], j=1..A063986(n)]/A063986(n)

Changed A063896 to A063986 and a(16)-a(21) from Donovan Johnson, May 11 2010

cp's OEIS Frontend

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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A063971 Values of n for which A013939(n)/n is an integer.

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A379922 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^(k+1) * sigma_2(k).

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A379923 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^k * A000005(k).

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A379924 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^(k+1) * usigma(k).

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A140221 A number n is included if n is coprime to Sum_{k=1..n} d(k), where d(k) is the number of divisors of k.

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A344731 Numbers k such that k divides A306069(k).

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A344732 Numbers k such that k divides Sum_{j=1..k} A048105(j).

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