A064611 -id:A064611 - cp's OEIS frontend

A064605 Numbers k such that A064602(k) is divisible by k.

1, 2, 8, 74, 146, 150, 158, 307, 526, 541, 16157, 20289, 271343, 953614, 1002122, 2233204, 3015123, 15988923, 48033767, 85110518238

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056650, A064605, A064606, A064607, A064610, A064611, A048290, A062982, A045345.
a(20) > 3*10^10. - Donovan Johnson, Aug 31 2012
a(21) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

			Summing divisor-square sums for j = 1..8 gives 1+5+10+21+26+50+50+85 = 248, which is divisible by 8, so 8 is a term and the integer quotient is 31.

Cf. A001157, A064602, A050226, A056650, A064606, A064607, A064610, A064611, A064612, A048290, A062982, A045345.

k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[2, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G. Wilson v, Apr 25 2011 *)

(Sum_{j=1..k} sigma_2(j)) mod k = A064602(k) mod k = 0.

a(15)-a(19) from Donovan Johnson, Jun 21 2010

a(20) from Amiram Eldar, Jan 18 2024

A064609 Partial sums of A034448: sum of unitary divisors from 1 to n.

Original entry on oeis.org

1, 4, 8, 13, 19, 31, 39, 48, 58, 76, 88, 108, 122, 146, 170, 187, 205, 235, 255, 285, 317, 353, 377, 413, 439, 481, 509, 549, 579, 651, 683, 716, 764, 818, 866, 916, 954, 1014, 1070, 1124, 1166, 1262, 1306, 1366, 1426, 1498, 1546, 1614, 1664, 1742, 1814, 1884

Offset: 1

Labos Elemer, Sep 24 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A034448, A064611.

Accumulate@ Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 52}] (* Michael De Vlieger, Mar 18 2017 *)

usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ a=0; for (n=1, 1000, a+=usigma(n); write("b064609.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 20 2009

from sympy.ntheory.factor_ import udivisor_sigma
def a(n): return sum(udivisor_sigma(j,1) for j in range(1,n + 1))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 18 2017

a(n) = a(n-1) + A034448(n) = Sum_{j=1..n} usigma(j) where usigma(j) = A034448(j).

a(n) ~ Pi^2 * n^2 / (12*Zeta(3)). - Vaclav Kotesovec, Jan 11 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

A307043 Numbers n such that A307042(n) = Sum_{k=1..n} esigma(k) is divisible by n, where esigma(k) is sum of exponential divisors of k (A051377).

Original entry on oeis.org

1, 3, 4, 8, 13, 78, 94, 481, 511, 4819, 13557, 23083, 84245, 204744, 562243, 591105, 614339, 617675, 656263, 1545716, 6370802, 34882737, 534034248, 601990019, 1153304776, 2064184733, 3570196547, 10572032882

Offset: 1

Amiram Eldar, Mar 21 2019

The exponential version of A056550.

The corresponding quotients are 1, 2, 3, 5, 8, 45, ... (see the link for more values).

			3 is in the sequence since esigma(1) + esigma(2) + esigma(3) = 1 + 2 + 3 = 6 is divisible by 3.

Amiram Eldar, Table of n, a(n), A307042(a(n))/a(n) for n=1..28.

Cf. A051377, A056550, A064611, A307042.

esigma[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; seq={};s = 0; Do[s = s + esigma [n]; If[Divisible[s,n], AppendTo[seq,n]], {n, 1, 10^6}]; seq (* after Jean-François Alcover at A051377 *)

A306950 Numbers m that divide A177754(m) = Sum_{k=1..m} uphi(k), where uphi is the unitary totient function (A047994).

Original entry on oeis.org

1, 2, 32, 36, 39, 50, 62, 147, 169, 190, 203, 467, 1035, 1075, 2174, 2475, 27047, 28097, 91087, 181175, 215795, 539654, 580160, 668988, 868879, 2611450, 14359486, 118119399, 1030191204, 1109928219, 2362155122

Offset: 1

Amiram Eldar, Mar 17 2019

The unitary version of A048290.

			32 is in the sequence since A177754(32) = 384 = 32 * 12 is divisible by 32.

Amiram Eldar, Table of n, a(n), A177754(a(n))/a(n) for n = 1..31.

Cf. A047994, A048290, A064611, A177754.

uphi[1] = 1; uphi[n_] := Product[{p, e} = pe; p^e-1, {pe, FactorInteger[n]}]; seq={}; s = 0; Do[s = s + uphi[n]; If[Divisible[s,n], AppendTo[seq, n]], {n, 1, 10^6}]; seq (* after Jean-François Alcover at A047994 *)

A307161 Numbers n such that A307159(n) = Sum_{k=1..n} bsigma(k) is divisible by n, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

Original entry on oeis.org

1, 2, 17, 37, 50, 56, 391, 919, 1399, 2829, 6249, 13664, 28829, 62272, 67195, 585391, 5504271, 6798541, 10763933, 866660818, 3830393407, 11044287758, 23058607363, 83159875881, 206501883259, 297734985607, 1087473543732, 1184060078117, 2789730557061, 2821551579466, 3529184155643

Offset: 1

Amiram Eldar, Mar 27 2019

nonn

The bi-unitary version of A056550.
The corresponding quotients are 1, 2, 13, 28, 38, 43, ... (see the link for more values).
a(32) > 10^13. - Giovanni Resta, May 28 2019

Amiram Eldar and Giovanni Resta, Table of n, a(n), A307159(a(n))/a(n) for n=1..31

Cf. A056550, A064611, A188999, A307043, A307159.

fun[p_,e_] := If[OddQ[e],(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); seq={};s = 0; Do[s = s + bsigma[n]; If[Divisible[s,n], AppendTo[seq,n]], {n, 1, 10^6}]; seq

a(23)-a(31) from Giovanni Resta, Apr 20 2019

A326488 Numbers m such that A327566(m) = Sum_{k=1..m} isigma(k) is divisible by m, where isigma(k) is the sum of infinitary divisors of k (A049417).

Original entry on oeis.org

1, 2, 160, 285, 2340, 2614, 8903, 81231, 171710, 182712, 434887, 2651907, 56517068, 143714354, 922484770, 5162883263, 39421525873

Offset: 1

Amiram Eldar, Sep 20 2019

The infinitary version of A056550.

The corresponding quotients, A327566(a(n))/a(n), are 1, 2, 118, 209, 1711, 1910, 6506, 59357, 125473, 133513, 317781, 1937798, 41298052, 105014703, 674076450, 3772612983, 28806028088, ...

			2 is in the sequence since isigma(1) + isigma(2) = 1 + 3 = 4 is divisible by 2.

Cf. A049417 (isigma), A327566 (sums of isigma).

Cf. A056550 (corresponding with sigma), A064611 (unitary), A307043 (exponential), A307161 (bi-unitary).

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); seq = {}; s = 0; Do[s = s + isigma [n]; If[Divisible[s, n], AppendTo[seq, n]], {n, 1, 10^6}]; seq

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

A064605 Numbers k such that A064602(k) is divisible by k.

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A064609 Partial sums of A034448: sum of unitary divisors from 1 to 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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A307043 Numbers n such that A307042(n) = Sum_{k=1..n} esigma(k) is divisible by n, where esigma(k) is sum of exponential divisors of k (A051377).

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A306950 Numbers m that divide A177754(m) = Sum_{k=1..m} uphi(k), where uphi is the unitary totient function (A047994).

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A307161 Numbers n such that A307159(n) = Sum_{k=1..n} bsigma(k) is divisible by n, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

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