A064605 -id:A064605 - cp's OEIS frontend

A064602 Partial sums of A001157: Sum_{j=1..n} sigma_2(j).

1, 6, 16, 37, 63, 113, 163, 248, 339, 469, 591, 801, 971, 1221, 1481, 1822, 2112, 2567, 2929, 3475, 3975, 4585, 5115, 5965, 6616, 7466, 8286, 9336, 10178, 11478, 12440, 13805, 15025, 16475, 17775, 19686, 21056, 22866, 24566, 26776, 28458, 30958

Offset: 1

Labos Elemer, Sep 24 2001

nonn

In general, for m >= 0 and j >= 0, Sum_{k=1..n} k^m * sigma_j(k) = Sum_{k=1..s} (k^m * F_{m+j}(floor(n/k)) + k^(m+j) * F_m(floor(n/k))) - F_{m+j}(s) * F_m(s), where s = floor(sqrt(n)) and F_m(x) are the Faulhaber polynomials defined as F_m(x) = (Bernoulli(m+1, x+1) - Bernoulli(m+1, 1)) / (m+1). - Daniel Suteu, Nov 27 2020

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Project Euler, Problem 401: Sum of squares of divisors, 2012.

Cf. A001157, A064605.

Cf. A064603, A064604, A248076.

Accumulate@ Array[DivisorSigma[2, #] &, 42] (* Michael De Vlieger, Jan 02 2017 *)

a(n) = sum(j=1, n, sigma(j, 2)); \\ Michel Marcus, Dec 15 2013

f(n) = n*(n+1)*(2*n+1)/6; \\ A000330
a(n) = my(s=sqrtint(n)); sum(k=1, s, f(n\k) + k^2*(n\k)) - s*f(s); \\ Daniel Suteu, Nov 26 2020

from math import isqrt
def f(n): return n*(n+1)*(2*n+1)//6
def a(n):
    s = isqrt(n)
    return sum(f(n//k) + k*k*(n//k) for k in range(1, s+1)) - s*f(s)
print([a(k) for k in range(1, 43)]) # Michael S. Branicky, Oct 01 2022 after Daniel Suteu

a(n) = a(n-1) + A001157(n) = Sum_{j=1..n} sigma_2(j) where sigma_2(j) = A001157(j).
a(n) = Sum_{i=1..n} i^2 * floor(n/i). - Enrique Pérez Herrero, Sep 15 2012
G.f.: (1/(1 - x))*Sum_{k>=1} k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 02 2017
a(n) ~ zeta(3) * n^3 / 3. - Vaclav Kotesovec, Sep 02 2018
a(n) = Sum_{k=1..s} (A000330(floor(n/k)) + k^2*floor(n/k)) - s*A000330(s), where s = floor(sqrt(n)). - Daniel Suteu, Nov 26 2020

A064610 Places k where A064608(k) (partial sums of unitary tau) is divisible by k.

Original entry on oeis.org

1, 35, 37, 1015, 27417, 27421, 27449, 27453, 19774739, 530743781, 530743799, 530743807, 530743813

Offset: 1

Labos Elemer, Sep 24 2001

The corresponding quotients are 1, 3, 3, 5, 7, 7, 7, 7, 11, 13, 13, 13, 13, ...

a(14) > 7.5*10^10, if it exists. - Amiram Eldar, Jun 04 2021

			For n = 37, the sum A064608(37) = 1+2+2+2+2+4+2+...+4+4+4+2 = 111 = 3*37, so 37 is in the sequence.

Cf. A064608.

Analogous "integer-mean" sequences for various arithmetical functions are A050226, A056650, A064605, A064606, A064607, A048290, A063986, A063971, A064911, A062982, A045345.

s[1] = 1; s[n_] := s[n] = s[n - 1] + 2^PrimeNu[n]; Select[Range[30000], Divisible[s[#], #] &] (* Amiram Eldar, Jun 04 2021 *)

{n: A064608(n) == 0 (mod n)}.

a(10)-a(13) from Donovan Johnson, Jul 20 2012

A064611 Partial sum of usigma is divisible by n, where usigma(n) = A034448(n) and summatory-usigma(n) = A064609(n).

Original entry on oeis.org

1, 2, 8, 11, 12, 174, 212, 524, 1567, 14096, 19795, 38466, 42114, 55575, 338809, 498001, 1175281, 2424880, 3994532, 7908519, 48453784, 696840720, 5497869355, 7479239685

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056550, A064605-A064607, A064610, A064612, A048290, A062982, A045345.

			udivisor sums[=usigma(j) values] from 1 to 8 are added: 1+3+4+5+6+12+8+9=48; it is divisible by 8, thus 8 is here.

Amiram Eldar, Table of n, a(n), A064609(a(n))/a(n) for n=1..24.

Cf. A034448, A064609, A050226, A056550, A064605, A064606, A064607, A064610, A064612, A048290, A062982, A045345.

s = Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 10^6}]; Module[{a = First@ s, b = {First@ s}}, Do[a += s[[i]]; If[Divisible[a, i], AppendTo[b, i]], {i, 2, Length@ s}]; b] (* Michael De Vlieger, Mar 18 2017 *)

A064609(n) mod n = 0.

a(17)-a(22) from Donovan Johnson, Jul 20 2012

a(23)-a(24) from Amiram Eldar, Mar 17 2019

A064607 Numbers k such that A064604(k) is divisible by k.

Original entry on oeis.org

1, 2, 7, 151, 257, 1823, 3048, 5588, 6875, 7201, 8973, 24099, 5249801, 9177919, 18926164, 70079434, 78647747, 705686794, 2530414370, 3557744074, 25364328389, 32487653727, 66843959963

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
a(19) > 2*10^9. - Donovan Johnson, Jun 21 2010
a(24) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

			Adding 4th-power divisor-sums for j = 1..7 gives 1+17+82+273+626+1394+2402 = 4795 which is divisible by 7, so 7 is a term and the integer quotient is 655.

Cf. A001159, A064604, A050226, A056650, A064605, A064606, A064610-A064612, A048290, A062982, A045345.

k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[4, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G.Wilson v, Aug 25 2011 *)

(Sum_{j=1..k} sigma_4(j)) mod k = A064604(k) mod k = 0.

a(13)-a(18) from Donovan Johnson, Jun 21 2010

a(19)-a(23) from Amiram Eldar, Jan 18 2024

A064612 Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n).

Original entry on oeis.org

1, 4, 5, 2178, 416417176, 416417184, 416417185, 416417186, 416417194, 416417204, 416417206, 416417208, 416417213, 416417214, 416417231, 416417271, 416417318, 416417319, 416417326, 416417335, 416417336, 416417338, 416417339, 416417374

Offset: 1

Labos Elemer, Sep 24 2001

nonn

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
Partial sums of A001222, similarly to summatory A001221 increases like loglog(n), explaining small quotients.
a(25) > 10^13. - Giovanni Resta, Apr 25 2017

			Sum of bigomega values from 1 to 5 is: 0+0+1+1+2+1=5, which is divisible by n=5, so 5 is here, with quotient=1. For the last value,2178,below 1000000 the quotient is only 3.

A001222, A022559, A050226, A056650, A064602-A064611, A048290, A062982, A045345.

Mod[A022559(n), n]=0

a(5)-a(24) from Donovan Johnson, Nov 15 2009

A064606 Numbers k such that A064603(k) is divisible by k.

Original entry on oeis.org

1, 2, 7, 45, 184, 210, 267, 732, 1282, 3487, 98374, 137620, 159597, 645174, 3949726, 7867343, 13215333, 14153570, 14262845, 317186286, 337222295, 2788845412, 10937683400, 72836157215, 95250594634

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
a(22) > 2*10^9. - Donovan Johnson, Jun 21 2010
a(26) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

			Adding divisor-cube sums for j = 1..7 gives 1+9+28+73+126+252+344 = 833 = 7*119, which is divisible by 7, so 7 is a term and the integer quotient is 119.

Cf. A001158, A064603, A050226, A056650, A064605, A064607, A064610-A064612, A048290, A062982, A045345.

A064603 = Accumulate[Table[DivisorSigma[3, k], {k, 1, 1000000}]]; Select[Range[Length[A064603]], Divisible[A064603[[#]], #] &] (* Vaclav Kotesovec, Jul 11 2021 *)

(Sum_{j=1..k} sigma_3(j)) mod k = A064603(k) mod k = 0.

a(15)-a(21) from Donovan Johnson, Jun 21 2010

a(22)-a(25) from Amiram Eldar, Jan 18 2024

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, ", ")));

A355544 Numbers k such that the arithmetic mean of the first k squarefree numbers is an integer.

Original entry on oeis.org

1, 3, 6, 37, 75, 668, 1075, 37732, 742767, 1811865, 3140083, 8937770, 108268896, 282951249, 633932500, 1275584757, 60455590365

Offset: 1

Amiram Eldar, Jul 06 2022

Numbers k such that A173143(k) is divisible by k.

The corresponding quotients A173143(k)/k are 1, 2, 4, 29, ..., and the corresponding values of A005117(k) are 1, 3, 7, 59, ... (see the link for more values).

			3 is a term since the arithmetic mean of the first 3 squarefree numbers, (1+2+3)/3 = 2, is an integer.

Amiram Eldar, Table of n, a(n), A173143(a(n))/a(n), A005117(a(n)) for n = 1..16.

Cf. A005117, A173143.

Similar sequences: A045345, A050226, A056550, A064605, A064606, A064607, A064610, A064611, A064612, A048290.

s = Select[Range[10^6], SquareFreeQ]; r = Accumulate[s]/Range[Length[s]]; ind = Position[r, _?IntegerQ] // Flatten

upto(n) = my(s=0,k=0); forsquarefree(m=1, n, s+=m[1]; k+=1; if(s%k == 0, print1(k, ", "))); \\ Daniel Suteu, Jul 06 2022

a(17) from Daniel Suteu, Jul 06 2022

cp's OEIS Frontend

A064602 Partial sums of A001157: Sum_{j=1..n} sigma_2(j).

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A064610 Places k where A064608(k) (partial sums of unitary tau) is divisible by k.

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A064611 Partial sum of usigma is divisible by n, where usigma(n) = A034448(n) and summatory-usigma(n) = A064609(n).

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A064607 Numbers k such that A064604(k) is divisible by k.

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A064612 Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n).

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A064606 Numbers k such that A064603(k) is divisible by k.

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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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