A020492 -id:A020492 - cp's OEIS frontend

A023897 a(n) = sigma_1(k) / phi(k) where k = A020492(n) is the n-th balanced number.

1, 3, 2, 6, 7, 4, 3, 9, 2, 8, 5, 6, 7, 4, 7, 10, 5, 12, 4, 9, 10, 3, 4, 14, 10, 8, 6, 13, 9, 8, 5, 15, 7, 2, 6, 8, 4, 5, 12, 6, 7, 10, 10, 11, 14, 12, 9, 4, 3, 4, 12, 9, 4, 4, 7, 5, 7, 10, 3, 5, 4, 13, 14, 12, 10, 9, 10, 8, 7, 4, 8, 6, 18, 9, 3, 8, 13, 8, 15, 15, 8, 3, 14, 9, 10, 8, 8, 10, 5, 7, 8, 11, 6, 11, 13, 6

Offset: 1

Olivier Gérard

nonn

sigma_1(n) is the sum of the divisors of n [same as sigma(n)] (A000203).

Jud McCranie, Table of n, a(n) for n = 1..10000 (first 800 terms from Vincenzo Librandi)

Cf. A000010, A000203, A020492.

[ q: n in [1..20000] | r eq 0 where q, r is Quotrem(SumOfDivisors(n), EulerPhi(n)) ]; // Klaus Brockhaus, Nov 09 2008

Select[ Array[ DivisorSigma[ 1, # ]/EulerPhi[ # ]&, 20000 ], IntegerQ ]

s(n) = {my(f = factor(n)); sigma(f)/eulerphi(f);}
list(lim) = select(x -> denominator(x) == 1, vector(lim, i, s(i))); \\ Amiram Eldar, Dec 25 2024

from math import prod
from itertools import count, islice
from sympy import factorint
def A023897_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        f = factorint(m)
        q, r = divmod(prod(p**(e+2)-p for p,e in f.items()),m*prod((p-1)**2 for p in f))
        if not r:
            yield q
A023897_list = list(islice(A023897_gen(),20)) # Chai Wah Wu, Aug 12 2024

A078539 Least non-balanced x (i.e., not in A020492) such that sigma(2n-1,x)/phi(x) is an integer.

Original entry on oeis.org

38, 46, 295, 38, 235, 749, 38, 3687, 6128, 38, 1415, 46, 38, 4254, 10451, 38, 46, 8351, 38, 334, 4511, 38, 3398, 295, 38, 1286, 46, 38, 148870, 11015, 38, 46, 35519, 38, 10239, 14072, 38, 235, 76088, 38, 5991, 46, 38, 718, 295, 38, 46, 11654, 38, 30761

Offset: 2

Labos Elemer, Dec 02 2002

nonn

			n=7: 2n-1 = 13, cases of sigma(13,x)/phi(x) is an integer listed in A015771: 1, 2, 3,6, 12, etc,; the first term which is non-balanced, i.e., not in A020492 is a(7) = 749 = A020492(31); increasing n, the trend of a(n) is roughly the same. If 2n-1 = 3s, i.e., is divisible by 3, then a(3s) = 38. Similar relationships hold for 2n - 1 = 5s, 7s, 11s, etc.

Amiram Eldar, Table of n, a(n) for n = 2..1200

Cf. A000005, A000010, A015759-A015774, A020492, A078538.

Table[fl=1; Do[s1=DivisorSigma[1, n]/EulerPhi[n]; sk=DivisorSigma[2*k-1, n]/EulerPhi[n]; If[ !IntegerQ[s1]&&IntegerQ[sk]&&Equal[fl, 1], Print[{n, 2*k-1}]; fl=0], {n, 1, 1000000}], {k, 2, 100}]

a(n) = min{x; sigma(1,x) mod phi(x) = 0 but sigma(2n-1, x) mod phi(x) is not 0}.

a(31) corrected by Amiram Eldar, Jul 21 2019

A078557 Squarefree balanced numbers (i.e., squarefree members of A020492).

Original entry on oeis.org

1, 2, 3, 6, 14, 15, 30, 35, 42, 70, 78, 105, 190, 210, 357, 418, 570, 714, 910, 1045, 1254, 2090, 2730, 3135, 4522, 4674, 5278, 6270, 10659, 12441, 13566, 14630, 15834, 16770, 20026, 21318, 23374, 24871, 24882, 24969, 25070, 25714, 27170, 29029, 33915, 35074

Offset: 1

Labos Elemer, Dec 06 2002

nonn

			210 = 2*3*5*7, sigma(210) = 576, phi(210) = 48, 576/48 = 12.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Intersection of A005117 and A020492.

Do[s=DivisorSigma[1, n]/EulerPhi[n]; If[IntegerQ[s]&&!Equal[MoebiusMu[n], 0], k=k+1; Print[n]], {n, 1, 2100000}]

isok(k) = {my(f = factor(k)); issquarefree(f) && denominator(prod(i = 1, #f~, (f[i, 1]+1)/(f[i, 1]-1))) == 1;} \\ Amiram Eldar, Feb 24 2025

A342103 Balanced numbers (A020492) that are also arithmetic numbers (A003601).

Original entry on oeis.org

1, 3, 6, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, 270, 357, 418, 420, 570, 594, 616, 630, 714, 744, 812, 840, 910, 1045, 1240, 1254, 1485, 1672, 1848, 2090, 2214, 2376, 2436, 2580, 2730, 2970, 3080, 3135, 3339, 3596, 3720, 3828, 3956, 4064, 4180

Offset: 1

Bernard Schott, Feb 28 2021

nonn

Equivalently, numbers m such that phi(m) (A000010) and tau(m) (A000005) both divide sigma(m) (A000203). In this case, the quotients sigma(m)/phi(m) = A023897(m) and sigma(m)/tau(m) = A102187(m).
Phi, tau and sigma are multiplicative functions and for this reason if k and q are coprime and included in this sequence then k*q is another term.
The only prime in the sequence is 3, because sigma(2)/tau(2) = 3/2 and when p is an odd prime, sigma(p)/phi(p) = (p+1)/(p-1) is an integer iff p=3 with sigma(3)/phi(3) = 4/2 = 2, and also sigma(3)/tau(3) = 4/2 = 2.

			phi(30) = tau(30) = 8, sigma(30) = 72 and 72/8 = 9, hence 30 is a term.
phi(12) = 4, tau(12) = 6, sigma(12) = 28, phi(12) divides sigma(12), but tau(12) does not divide sigma(12), hence 12 is a balanced number but is not an arithmetic number, and 12 is not a term.
phi(20) = 8, tau(20) = 6, sigma(20) = 42, tau(20) divides sigma(20), but phi(20) does not divide sigma(20), hence 20 is an arithmetic number but is not a balanced number, and 20 is not a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A003601 and A020492.
Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A023897 (sigma/phi), A102187 (sigma/tau).
Cf. A342104, A342105, A342106.

with(numtheory): filter:= q -> (sigma(q) mod phi(q) = 0) and (sigma(q) mod tau(q) = 0) : select(filter, [$1..5000]);

Select[Range[5000], And @@ Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] &] (* Amiram Eldar, Feb 28 2021 *)

isok(m) = my(s=sigma(m)); !(s % eulerphi(m)) && !(s % numdiv(m)); \\ Michel Marcus, Mar 01 2021

A076375 Numbers k such that both k and 2*k are balanced numbers (A020492).

Original entry on oeis.org

1, 3, 6, 15, 35, 70, 105, 210, 357, 420, 1045, 1485, 2090, 2970, 3135, 3339, 5049, 5940, 6270, 10659, 12441, 12540, 16065, 24871, 24969, 29029, 33915, 35343, 39105, 39585, 49742, 50065, 58058, 58435, 64285, 70686, 71145, 74613, 78210, 87087

Offset: 1

Labos Elemer, Oct 15 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)

Cf. A000010, A000203, A055234.

f[x_] := DivisorSigma[1, x]/EulerPhi[x] Do[s=f[n]; s1=f[2*n]; If[IntegerQ[s]&&IntegerQ[s1], Print[n]], {n, 1, 100000}]

A342104 Balanced numbers (A020492) that are not arithmetic numbers (A003601).

Original entry on oeis.org

2, 12, 18630, 27000, 443394, 6242022, 14412720, 22315419, 26744100, 44630838, 50496960, 106034880, 128710944, 148536990, 162907584, 212072880, 218470770, 296259930, 349444530, 397253968, 535267776, 641250900, 641418960, 666274653, 684165552, 688208724, 709639408

Offset: 1

Bernard Schott, Feb 28 2021

nonn

Equivalently, numbers m such that phi(m) divides sigma(m) but tau(m) does not divide sigma(m), the corresponding quotients sigma(m)/phi(m) = A023897(m).

The only prime in the sequence is 2, because sigma(2)/phi(2) = 3 and sigma(2)/tau(2) = 3/2; then, if p odd prime, sigma(p)/phi(p) = (p+1)/(p-1) is an integer iff p = 3, but for p = 3, tau(3) divides sigma(3) with sigma(3)/tau(3) = 4/2 = 2.

			Sigma(12) = 28, phi(12) = 4 and tau(12) = 6, hence phi(12) divides sigma(12), but tau(12) does not divide sigma(12), so 12 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..5620

Equals A020492 \ A003601.
Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A023897 (sigma/phi).
Cf. A342103, A342105, A342106.

with(numtheory): filter:= q -> (sigma(q) mod phi(q) = 0) and (sigma(q) mod tau(q) <> 0) : select(filter, [$1..500000]);

Select[Range[500000], Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] == {False, True} &] (* Amiram Eldar, Feb 28 2021 *)

isok(m) = my(s=sigma(m)); !(s % eulerphi(m)) && (s % numdiv(m)); \\ Michel Marcus, Mar 01 2021

a(5)-a(27) from Amiram Eldar, Feb 28 2021

A342105 Arithmetic numbers (A003601) that are not balanced numbers (A020492).

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 20, 21, 22, 23, 27, 29, 31, 33, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 71, 73, 77, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102, 103, 107, 109, 110, 111, 113, 114, 115, 116

Offset: 1

Bernard Schott, Mar 05 2021

nonn

Equivalently, numbers m such that tau(m) divides sigma(m) but phi(m) does not divide sigma(m), the corresponding quotients sigma(m)/tau(m) = A102187(m).

Primes in the sequence are primes >= 5; proof: 2 is in A342104 and 3 is in A342103, then for p prime >= 5, phi(p) = p-1 >= 4, tau(p) = 2, sigma(p) = p+1 >= 6; hence 2 divides p+1 but p-1 does not divide p+1.

			Sigma(21) = 32, tau(21) = 4 and phi(21) = 12, hence tau(21) divides sigma(21), but phi(21) does not divide sigma(21), so 21 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Equals A003601 \ A020492.
Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A102187 (sigma/tau).
Cf. A342103, A342104, A342106.

with(numtheory): filter:= q -> (sigma(q) mod tau(q) = 0) and (sigma(q) mod phi(q) <> 0) : select(filter, [$1..120]);

Select[Range[120], Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] == {True, False} &] (* Amiram Eldar, Mar 05 2021 *)

isok(m) = my(s=sigma(m)); !(s % numdiv(m)) && (s % eulerphi(m)); \\ Michel Marcus, Mar 05 2021

A076376 Numbers k such that k, 2k and 4k are balanced numbers (A020492).

Original entry on oeis.org

3, 35, 105, 210, 1045, 1485, 2970, 3135, 6270, 24871, 29029, 35343, 39105, 50065, 58435, 64285, 70686, 71145, 74613, 78210, 87087, 87685, 124605, 137885, 140335, 142290, 149226, 150195, 174174, 175305, 176715, 192855, 249210, 263055, 300390, 350610, 373065

Offset: 1

Labos Elemer, Oct 15 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)

Cf. A000010, A000203, A055234, A020492, A076375.

f[x_] := DivisorSigma[1, x]/EulerPhi[x] Do[s=f[n]; s1=f[2*n]; s2=f[4*n]; If[IntegerQ[s]&&IntegerQ[s1]&&IntegerQ[s2], Print[n]], {n, 1, 1000000}]

A078540 Least non-balanced x (i.e., not in A020492) such that sigma(p(n),x)/phi(x) is an integer, where p(n) = n-th prime.

Original entry on oeis.org

22, 38, 46, 295, 235, 749, 3687, 6128, 1415, 4254, 10451, 8351, 334, 4511, 3398, 1286, 148870, 11015, 35519, 10239, 14072, 76088, 5991, 718, 11654, 30761, 7431, 20993, 700654, 22169, 5095, 4198, 27415, 26744, 14318, 48368, 180878, 16991, 173123, 4166, 5033, 7246

Offset: 1

Labos Elemer, Dec 02 2002

nonn

			n=6: prime(6)=13, cases of sigma(13,x)/phi(x) is an integer are listed in A015771: 1, 2, 3, 6, 12, etc.; the first term which is non-balanced, i.e., not in A020492, is a(6) = 749 = A020492(31); a(29) = 700854 and a(45) = 510759 are remarkably large.

Amiram Eldar, Table of n, a(n) for n = 1..129

Cf. A000005, A000010, A000040, A015759-A015774, A020492, A078538, A078539.

Table[fl=1; Do[s1=DivisorSigma[1, n]/EulerPhi[n]; sk=DivisorSigma[Prime[k], n]/EulerPhi[n]; If[ !IntegerQ[s1]&&IntegerQ[sk]&&Equal[fl, 1], Print[{n, Prime[k]}]; fl=0], {n, 1, 1000000}], {k, 1, 100}]

lista(nmax) = {my(ps = primes(nmax), pmax = ps[#ps], v = vector(pmax), c = 0, k = 2, f, e, p); while(c < nmax, f = factor(k); e = eulerphi(f); if(sigma(f) % e > 0, for(i = 1, nmax, p = ps[i]; if(!(sigma(f, p) % e) && v[p] == 0, c++; v[p] = k))); k++); for(i = 1, pmax, if(v[i] > 0, print1(v[i], ", ")));} \\ Amiram Eldar, Aug 29 2024

a(n) = min{x; A000203(x) mod A000005(x) = 0 but sigma(A000040(n), x) mod phi(x) is not 0}.

a(18) corrected and more terms added by Amiram Eldar, Aug 29 2024

A076377 Numbers k such that k, 2k, 4k and 8*k are balanced numbers (A020492).

Original entry on oeis.org

105, 1485, 3135, 35343, 39105, 71145, 74613, 87087, 124605, 150195, 175305, 192855, 263055, 413655, 421005, 697851, 930699, 1404765, 1873485, 2471931, 2576115, 2965599, 3281265, 3398625, 3937635, 4172259, 4532625, 4589949, 4975965, 5218521, 5474115

Offset: 1

Labos Elemer, Oct 15 2002

nonn

The quotients q = Sigma(u)/phi(u) for u = {n, 2n, 4n, 8n} are integers and for all terms, and equal 4, 12, 14, 15 respectively. For u = 16n, q = 31/2, i.e. no integer was found for u < 6000000.

The comment above is true for terms up to a(238) and true for 985 of the first 1000 terms. - Donovan Johnson, Mar 03 2013

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)

Cf. A000010, A000203, A055234, A020492, A076375, A076376.

f[x_] := DivisorSigma[1, x]/EulerPhi[x] Do[s=f[n]; s1=f[2*n]; s2=f[4*n]; s3=f[8*n] If[IntegerQ[s]&&IntegerQ[s1]&&IntegerQ[s2]&& IntegerQ[s3], Print[n]], {n, 1, 10000000}]

Missing term added by Donovan Johnson, Mar 03 2013

cp's OEIS Frontend

A023897 a(n) = sigma_1(k) / phi(k) where k = A020492(n) is the n-th balanced number.

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A078539 Least non-balanced x (i.e., not in A020492) such that sigma(2n-1,x)/phi(x) is an integer.

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A078557 Squarefree balanced numbers (i.e., squarefree members of A020492).

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A342103 Balanced numbers (A020492) that are also arithmetic numbers (A003601).

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A076375 Numbers k such that both k and 2*k are balanced numbers (A020492).

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A342104 Balanced numbers (A020492) that are not arithmetic numbers (A003601).

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A342105 Arithmetic numbers (A003601) that are not balanced numbers (A020492).

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A076376 Numbers k such that k, 2*k and 4*k are balanced numbers (A020492).

A076376 Numbers k such that k, 2k and 4k are balanced numbers (A020492).

A076377 Numbers k such that k, 2k, 4k and 8*k are balanced numbers (A020492).