A063514 -id:A063514

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A333847

a(n) = A063514(A028982(n)).

+20
1

0, 0, 1, 3, 1, 7, 3, 11, 15, 7, 15, 13, 31, 3, 13, 3, 17, 23, 63, 19, 39, 27, 63, 65, 43, 69, 127, 51, 35, 91, 81, 39, 15, 1, 237, 9, 51, 255, 47, 115, 105, 281, 87, 33, 117, 121, 87, 33, 59, 207, 181, 63, 235, 511, 141, 409, 243, 517, 87, 343, 295, 249, 75, 615, 363

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Except for the first 2 terms, all terms are odd.

LINKS

Michel Marcus, Table of n, a(n) for n = 1..5000

PROG

(PARI) lista(nn) = {for (n=1, nn, if (issquare(n) || (!(n%2) && issquare(n/2)), print1(sigma(n) % eulerphi(n), ", "); ); ); }

CROSSREFS

Cf. A028982 (squares and twice squares), A063514 (sigma(n) mod phi(n)), A072808.

KEYWORD

nonn

AUTHOR

Michel Marcus, Apr 08 2020

STATUS

approved

A020492

Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203).

+10
93

1, 2, 3, 6, 12, 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The quotient A020492(n)/A002088(n) = SummatorySigma/SummatoryTotient as n increases seems to approach Pi^4/36 or zeta(2)^2 [~2.705808084277845]. - Labos Elemer, Sep 20 2004, corrected by Charles R Greathouse IV, Jun 20 2012

If 2^p-1 is prime (a Mersenne prime) then m = 2^(p-2)*(2^p-1) is in the sequence because when p = 2 we get m = 3 and phi(3) divides sigma(3) and for p > 2, phi(m) = 2^(p-2)*(2^(p-1)-1); sigma(m) = (2^(p-1)-1)*2^p hence sigma(m)/phi(m) = 4 is an integer. So for each n, A133028(n) = 2^(A000043(n)-2)*(2^A000043(n)-1) is in the sequence. - Farideh Firoozbakht, Nov 28 2005

Phi and sigma are both multiplicative functions and for this reason if m and n are coprime and included in this sequence then m*n is also in this sequence. - Enrique Pérez Herrero, Sep 05 2010

The quotients sigma(n)/phi(n) are in A023897. - Bernard Schott, Jun 06 2017

There are 544768 balanced numbers < 10^14. - Jud McCranie, Sep 10 2017

a(975807) = 419998185095132. - Jud McCranie, Nov 28 2017

REFERENCES

D. Chiang, "N's for which phi(N) divides sigma(N)", Mathematical Buds, Chap. VI pp. 53-70 Vol. 3 Ed. H. D. Ruderman, Mu Alpha Theta 1984.

LINKS

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Jud McCranie, 670314 balanced numbers (first 1000 from T. D. Noe, first 10000 from Donovan Johnson)

EXAMPLE

sigma(35) = 1+5+7+35 = 48, phi(35) = 24, hence 35 is a term.

MATHEMATICA

Select[ Range[ 4000 ], IntegerQ[ DivisorSigma[ 1, # ]/EulerPhi[ # ] ]& ]

(* Second program: *)

Select[Range@ 4000, Divisible[DivisorSigma[1, #], EulerPhi@ #] &] (* Michael De Vlieger, Nov 28 2017 *)

PROG

(Magma) [ n: n in [1..3900] | SumOfDivisors(n) mod EulerPhi(n) eq 0 ]; // Klaus Brockhaus, Nov 09 2008

(PARI) select(n->sigma(n)%eulerphi(n)==0, vector(10^4, i, i)) \\ Charles R Greathouse IV, Jun 20 2012

(Python)

from sympy import totient, divisor_sigma

print([n for n in range(1, 4001) if divisor_sigma(n)%totient(n)==0]) # Indranil Ghosh, Jul 06 2017

(Python)

from math import prod

from itertools import count, islice

from sympy import factorint

def A020492_gen(startvalue=1): # generator of terms >= startvalue

for m in count(max(startvalue, 1)):

f = factorint(m)

if not prod(p**(e+2)-p for p, e in f.items())%(m*prod((p-1)**2 for p in f)):

yield m

A020492_list = list(islice(A020492_gen(), 20)) # Chai Wah Wu, Aug 12 2024

CROSSREFS

Cf. A000010, A000043, A000203, A000668, A011257, A023897, A133028, A291565, A291566, A292422, A351114 (characteristic function).

Positions of 0's in A063514.

KEYWORD

nonn

AUTHOR

David W. Wilson

EXTENSIONS

More terms from Farideh Firoozbakht, Nov 28 2005

STATUS

approved

A351114

Characteristic function of balanced numbers: a(n) = 1 if phi(n) divides sigma(n), otherwise 0.

+10
10

1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

A balanced number k is a number such that phi(k) | sigma(k).

If a(x) = 1, a(y) = 1, and gcd(x,y) = 1, then a(x*y) = 1 also. - Antti Karttunen, Jan 01 2023, based on Enrique Pérez Herrero's Sep 05 2010 comment in A020492.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100130

Index entries for characteristic functions

FORMULA

a(n) = c(sigma(n)/phi(n)), where c(n) = 1 - ceiling(n) + floor(n).

a(n) = [A063514(n) == 0], where [ ] is the Iverson bracket. - Antti Karttunen, Jan 01 2023

MATHEMATICA

a[n_] := Boole[Divisible[DivisorSigma[1, n], EulerPhi[n]]]; Array[a, 100] (* Amiram Eldar, Feb 01 2022 *)

PROG

(Python)

from math import prod

from sympy import factorint

def A351114(n):

f = factorint(n)

return int(not prod(p*(p**(e+1)-1) for p, e in f.items()) % (n*prod((p-1)**2 for p in f))) # Chai Wah Wu, Feb 01 2022

(PARI) A351114(n) = !(sigma(n)%eulerphi(n)); \\ Antti Karttunen, Jan 01 2023

CROSSREFS

Cf. A000010 (phi), A000203 (sigma), A020492 (balanced numbers), A063514.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jan 31 2022

EXTENSIONS

Data section extended up to a(105) and the name amended with a formula by Antti Karttunen, Jan 01 2023

STATUS

approved

A067193

Numbers k such that sigma(k) == 4 (mod phi(k)).

+10
3

24, 27, 44, 66, 75, 170, 944, 1200, 16064, 260864, 4189184, 17179541504, 274876596224

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(14) > 10^12. 1125899822956544 and 4611686013058678784 are also terms. - Donovan Johnson, Feb 29 2012

a(14) > 10^13. If 2^j-5 is prime (A059608) and j > 3, then 2^(j-2)*(2^j-5) is a term. - Giovanni Resta, Mar 29 2020

LINKS

Table of n, a(n) for n=1..13.

CROSSREFS

Cf. A063514, A059608, A067192.

KEYWORD

nonn,more

AUTHOR

Benoit Cloitre, Feb 19 2002

EXTENSIONS

a(11) from Donovan Johnson, Dec 14 2009

a(12)-a(13) from Donovan Johnson, Feb 29 2012

STATUS

approved

A072808

Smallest m such that sigma(m) == n (mod phi(m)) or 0 if no solution exists.

+10
2

4, 5, 8, 24, 0, 22, 16, 21, 450, 40, 25, 48, 50, 136, 32, 110, 100, 90, 144, 88, 0, 656, 121, 102, 0, 80, 169, 96, 0, 68, 64, 55, 676, 464, 289, 65, 0, 117, 162, 91, 0, 116, 225, 85, 0, 272, 529, 95, 0, 148, 288, 133, 0, 164, 0, 115, 0, 160, 841, 147, 0, 333, 128, 247

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Warning: It is only conjectured that there are no solutions for n such that a(n) = 0. The search for solutions tested all m <= 10^10 for these n.

For odd remainders a(n) is a square or twice a square. See A028982, except terms 1 and 2.

All zeros corresponding to odd terms a(n) with n < 64 confirmed up to m <= 10^24. - Giovanni Resta, Apr 02 2020

LINKS

Table of n, a(n) for n=1..64.

FORMULA

a(n) = Min{x; Mod(A000203(x), A000010(x))=n} or 0 if apparently no solutions.

EXAMPLE

For n=4: a(4)=24 since sigma(24)=60, phi(24)=8 and Mod(60, 8)=4.

MATHEMATICA

f[x_] := Mod[DivisorSigma[1, x], EulerPhi[x]] t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10000000000}];

CROSSREFS

Cf. A000203, A000010, A028982, A063514, A067192, A067193, A071390.

KEYWORD

nonn

AUTHOR

Labos Elemer, Jul 12 2002

STATUS

approved

A302172

Distance from sigma(n) to nearest multiple of phi(n).

+10
1

0, 0, 0, 1, 2, 0, 2, 1, 1, 2, 2, 0, 2, 0, 0, 1, 2, 3, 2, 2, 4, 4, 2, 4, 9, 6, 4, 4, 2, 0, 2, 1, 8, 6, 0, 5, 2, 6, 8, 6, 2, 0, 2, 4, 6, 6, 2, 4, 15, 7, 8, 2, 2, 6, 8, 0, 8, 6, 2, 8, 2, 6, 4, 1, 12, 4, 2, 2, 8, 0, 2, 3, 2, 6, 4, 4, 24, 0, 2, 6, 13, 6, 2, 8, 20, 6, 8, 20, 2, 6, 32, 8, 8, 6, 24, 4, 2, 3, 24, 17

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

Numbers n such that a(n) <> A063514(n) are 8, 16, 21, 22, 25, 28, 32, 36, 40, 48, 50, 54, 55, 63, 64, 65, 68, 76, 77, 80, ...

Numbers n such that a(n) = 1 are 4, 8, 9, 16, 32, 64, 128, 256, 400, 512, 1024, 2048, 4096, 8192, 16384, ...

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

FORMULA

a(A020492(n)) = 0.

a(2^k) = 1 for k > 1.

a(p) = 2 for prime p > 3.

EXAMPLE

a(21) = 4 because sigma(21) = 32 and phi(21) = 12; 12*3 - 32 = 4 is the smallest corresponding distance.

MATHEMATICA

dsp[n_]:=Module[{s=DivisorSigma[1, n], p=EulerPhi[n], m}, m=Floor[s/p]; Abs[ Nearest[ {m*p, (m+1)p}, s]-s]]; Array[dsp, 100][[All, 1]] (* Harvey P. Dale, Apr 29 2018 *)

PROG

(PARI) a(n) = {my(k=0, s=sigma(n), p=eulerphi(n)); while((s+k) % p != 0 && (s-k) % p != 0, k++); k; }

CROSSREFS

Cf. A020492, A051612, A063514.

KEYWORD

nonn,easy

AUTHOR

Altug Alkan, Apr 03 2018

STATUS

approved

A333633

Smallest m such that sigma(m) == 2*n (mod phi(m)) or 0 if no solution exists.

+10
1

1, 5, 24, 22, 21, 40, 48, 136, 110, 90, 88, 656, 102, 80, 96, 68, 55, 464, 65, 117, 91, 116, 85, 272, 95, 148, 133, 164, 115, 160, 147, 333, 247, 212, 145, 243968, 155, 244, 217, 405, 230, 11072, 185, 292, 259, 1184, 205, 237824, 215, 657, 301, 356, 189, 343, 329, 388, 559, 404

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(221) <= 288230257234804736 = 2^(k-2)*(2^k-443) for k=30. - Michel Marcus, Apr 02 2020

a(221) > 10^13. - Giovanni Resta, Apr 12 2020

LINKS

Michel Marcus, Table of n, a(n) for n = 0..220

PROG

(PARI) g(n) = my(f=factor(n)); sigma(f) % eulerphi(f);

a(n) = {n *= 2; my(k=1); while (g(k) != n, k++); k; } \\ Michel Marcus, Mar 30 2020

CROSSREFS

Bisection of A072808.

Cf. A000203, A000010, A063514, A067192, A067193.

KEYWORD

nonn

AUTHOR

Michel Marcus, Mar 30 2020

EXTENSIONS

a(0) prepended by Jinyuan Wang, Mar 30 2020

STATUS

approved

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