A033632 -id:A033632 - cp's OEIS frontend

A092584 Numbers k such that sigma(phi(k)) == phi(sigma(k)) (mod k), that is, A033632(k)/k is an integer.

1, 5, 9, 157, 225, 242, 516, 729, 3872, 13932, 14406, 17672, 18225, 20124, 21780, 29262, 29616, 45996, 65025, 76832, 92778, 95916, 106092, 106308, 114630, 114930, 121872, 125652, 140130, 140625, 145794, 149124, 160986, 179562, 185100, 234876

Offset: 1

Labos Elemer, Mar 01 2004

nonn

			Includes but is not identical with A033632.
Below 10^7 only a(2) = 5 and a(4) = 157 give A033632(n)/n nonzero.

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A033632, A092585, A092586, A092587, A092588, A065395.

Select[Range[250000], Divisible[DivisorSigma[1, EulerPhi[#]] - EulerPhi[DivisorSigma[1, #]] , #] &]  (* Amiram Eldar, Mar 12 2020 *)

is(n)=sigma(eulerphi(n))==Mod(eulerphi(sigma(n)),n) \\ Charles R Greathouse IV, Nov 27 2013

A062401 a(n) = phi(sigma(n)).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 4, 8, 12, 6, 4, 12, 6, 8, 8, 30, 6, 24, 8, 12, 16, 12, 8, 16, 30, 12, 16, 24, 8, 24, 16, 36, 16, 18, 16, 72, 18, 16, 24, 24, 12, 32, 20, 24, 24, 24, 16, 60, 36, 60, 24, 42, 18, 32, 24, 32, 32, 24, 16, 48, 30, 32, 48, 126, 24, 48, 32, 36, 32, 48, 24, 96, 36, 36, 60

Offset: 1

Jason Earls, Jul 08 2001

nonn

			a(9) = 12 because sigma(9) = 13 and phi(13) = 12.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 14.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A000010, A062402, A096852, A096857, A096994, A096995, A033632.

Cf. A001229 (fixed points).

a062401 = a000010 . a000203  -- Reinhard Zumkeller, Jan 04 2013

with(numtheory); A062401:=n->phi(sigma(n)); seq(A062401(n), n=1..50); # Wesley Ivan Hurt, Apr 07 2014

Table[EulerPhi[DivisorSigma[1, n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

PARI
```
vector(150, n, eulerphi(sigma(n)))
```

for (n=1, 10000, write("b062401.txt", n, " ", eulerphi(sigma(n))) ) \\ Harry J. Smith, Aug 07 2009

sigma(a(n)) = A062402(sigma(n)) or phi(A062402(n)) = a(phi(n)). - Labos Elemer, Jul 22 2004

A028491 Numbers k such that (3^k - 1)/2 is prime.

Original entry on oeis.org

3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, 7973131, 8530117

Offset: 1

N. J. A. Sloane, Jean-Yves Perrier (nperrj(AT)ascom.ch)

If k is in the sequence and m=3^(k-1) then m is a term of A033632 (phi(sigma(m)) = sigma(phi(m))), so 3^(A028491-1) is a subsequence of A033632. For example since 9551 is in the sequence, phi(sigma(3^9550)) = sigma(phi(3^9550)). - Farideh Firoozbakht, Feb 09 2005
Salas lists these, except 3, in "Open Problems" p. 6 [March 2012], and proves that the Cantor primes > 3 are exactly the prime-valued cyclotomic polynomials of the form Phi_s(3^{s^j}) == 1 (mod 4).
Also, k such that 3^k-1 is a semiprime - see also A080892. - M. F. Hasler, Mar 19 2013

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antal Bege and Kinga Fogarasi, Generalized perfect numbers, arXiv:1008.0155 [math.NT], 2010. See p. 81.
Paul Bourdelais, A Generalized Repunit Conjecture, Posting in NMBRTHRY@LISTSERV.NODAK.EDU, Jun 25, 2009.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field
Christian Salas, Cantor Primes as Prime-Valued Cyclotomic Polynomials, arXiv:1203.3969 [math.NT], 2012.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
Index to primes in various ranges, form ((k+1)^n-1)/k

Cf. A076481, A033632, A112646.

Do[If[PrimeQ[(3^n-1)/2], Print[n]], {n, 10000}] (* Farideh Firoozbakht, Feb 09 2005 *)

forprime(p=2,1e5,if(ispseudoprime(3^p\2),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(13) from Farideh Firoozbakht, Mar 27 2005
a(14)-a(16) from Robert G. Wilson v, Apr 11 2005
All larger terms only correspond to probable primes.
a(17) from Paul Bourdelais, Feb 08 2010
a(18) from Paul Bourdelais, Jul 06 2010
a(19) from Paul Bourdelais, Feb 05 2019
a(20) and a(21) from Ryan Propper, Dec 29 2021
a(22) from Ryan Propper, Nov 06 2023
a(23) from Ryan Propper, Nov 09 2023

A062402 a(n) = sigma(phi(n)).

Original entry on oeis.org

1, 1, 3, 3, 7, 3, 12, 7, 12, 7, 18, 7, 28, 12, 15, 15, 31, 12, 39, 15, 28, 18, 36, 15, 42, 28, 39, 28, 56, 15, 72, 31, 42, 31, 60, 28, 91, 39, 60, 31, 90, 28, 96, 42, 60, 36, 72, 31, 96, 42, 63, 60, 98, 39, 90, 60, 91, 56, 90, 31, 168, 72, 91, 63, 124, 42, 144, 63, 84, 60, 144

Offset: 1

Jason Earls, Jul 08 2001

nonn

Makowski and Schinzel conjectured in 1964 that a(n) = sigma(phi(n)) >= n/2 for all n (B42 of Guy Unsolved Problems...). This has been verified for various classes of numbers and proved to be true in general if it is true for squarefree integers (see Cohen's paper). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

Atanassov proves the above conjecture. - Charles R Greathouse IV, Dec 06 2016

			a(9)= 12 because phi(9)= 6 and sigma(6)= 12.

Krassimir T. Atanassov, One property of φ and σ functions, Bull. Number Theory Related Topics 13 (1989), pp. 29-37.
A. Makowski and A. Schinzel, On the functions phi(n) and sigma(n), Colloq. Math. 13, 95-99 (1964).
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 13.

T. D. Noe, Table of n, a(n) for n = 1..10000
G. L. Cohen, On a conjecture of Makowski and Schinzel. Colloq. Math. 74, No. 1, 1-8 (1997).
A. Grytczuk, F. Luca and M. Wojtowicz, On a conjecture of Makowski and Schinzel concerning the composition of the arithmetic functions sigma and phi, Colloq. Math. 86, No. 1, 31-36 (2000).
F. Luca and C. Pomerance, On some problems of Makowski-Schinzel and Erdos concerning the arithmetical functions phi and sigma, Colloq. Math. 92, No. 1, 111-130 (2002).

Cf. A000203, A000010, A062401, A096852, A096857, A096994, A096995, A033632.

a062402 = a000203 . a000010  -- Reinhard Zumkeller, Jan 04 2013

[SumOfDivisors(EulerPhi(n)): n in [1..100]] //  Marius A. Burtea, Jan 19 2019

with(numtheory); A062402:=n->sigma(phi(n)); seq(A062402(k), k=1..100); # Wesley Ivan Hurt, Nov 01 2013

Table[DivisorSigma[1, EulerPhi[n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

a(n)=sigma(eulerphi(n));
vector(150,n,a(n))

from sympy import divisor_sigma, totient
print([divisor_sigma(totient(n)) for n in range(1, 101)]) # Indranil Ghosh, Mar 18 2017

sigma(A062401(x)) = a(sigma(x)) or phi(a(x)) = A062401(phi(x)). - Labos Elemer, Jul 22 2004

A006872 Numbers k such that phi(k) = phi(sigma(k)).

Original entry on oeis.org

1, 3, 15, 26, 39, 45, 74, 104, 111, 117, 122, 146, 175, 183, 195, 219, 296, 314, 333, 357, 386, 471, 488, 549, 554, 555, 579, 584, 585, 608, 626, 646, 657, 794, 831, 842, 914, 915, 939, 962, 1071, 1082, 1095, 1191, 1226, 1256, 1263, 1292, 1322, 1346

Offset: 1

Jeffrey Shallit, N. J. A. Sloane

nonn

S. W. Golomb, Equality among number-theoretic functions, Abstract 882-11-16, Abstracts Amer. Math. Soc., 14 (1993), 415-416.
R. K. Guy, Unsolved Problems in Number Theory, B42.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe, terms 1001..12394 from Marius A. Burtea)
S. W. Golomb, Letter to N. J. A. Sloane, Oct. 1992
S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)

Cf. A000010, A000203, A062401, A353637 (characteristic function).
Positions of zeros in A353636.
Setwise difference of A353684 and A353683, and also of A353685 and A353686.
Intersection of A353684 and A353685.
Subsequences: A260021, A353634, A353635, A353679 (odd terms).
Cf. also A033631, A033632, A067880, A137815, A260020.

a006872 n = a006872_list !! (n-1)
a006872_list = filter (\x -> a000010' x == a000010' (a000203' x)) [1..]
-- Reinhard Zumkeller, Jul 14 2015

[n:n in [1..2000]| EulerPhi(SumOfDivisors(n)) eq EulerPhi(n)]; // Marius A. Burtea, Jan 01 2019

Select[Range@ 1350, EulerPhi@ # == EulerPhi@ DivisorSigma[1, #] &] (* Michael De Vlieger, Jan 01 2019 *)

lista(nn) = {for (i=1, nn, if (eulerphi(i)==eulerphi(sigma(i)), print1(i, ", ")););} \\ Michel Marcus, May 25 2013

More terms from Jud McCranie

A065395 Commutator of sigma and phi functions.

Original entry on oeis.org

0, -1, 1, -3, 5, -1, 8, -1, 0, 1, 14, -5, 22, 4, 7, -15, 25, -12, 31, 3, 12, 6, 28, -1, 12, 16, 23, 4, 48, -9, 56, -5, 26, 13, 44, -44, 73, 23, 36, 7, 78, -4, 76, 18, 36, 12, 56, -29, 60, -18, 39, 18, 80, 7, 66, 28, 59, 32, 74, -17, 138, 40, 43, -63, 100, -6

Offset: 1

Labos Elemer, Nov 05 2001

Golomb (1993) proved that the terms are both positive and negative infinitely often. - Amiram Eldar, Feb 27 2024

			n = 13: sigma(13) = 14, phi(14) = 6, phi(13) = 12, sigma(12) = 28, a(13) = 28-6 = 22.

Solomon W. Golomb, Equality among number-theoretic functions, Abstracts Amer. Math. Soc., Vol. 14 (1993), pp. 415-416.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Jean-Marie De Koninck and Florian Luca, On the composition of the Euler function and the sum of divisors function, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.
Solomon W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)

Cf. A000010, A000203, A033632 (positions of 0's), A062401, A062402.

[DivisorSigma(1, EulerPhi(n))-EulerPhi(DivisorSigma(1, n)): n in [1..70]]; // Bruno Berselli, Oct 20 2015

with(numtheory); A065395:=n->sigma(phi(n))-phi(sigma(n)); seq(A065395(n), n=1..100); # Wesley Ivan Hurt, Dec 26 2013

Table[DivisorSigma[1, EulerPhi[n]] - EulerPhi[DivisorSigma[1, n]], {n, 100}] (* T. D. Noe, Nov 04 2013 *)

a(n) = { sigma(eulerphi(n)) - eulerphi(sigma(n)) } \\ Harry J. Smith, Oct 18 2009

a(n) = sigma(phi(n)) - phi(sigma(n)) = A000203(A000010(n)) - A000010(A000203(n)).

a(n) = A062402(n) - A062401(n). - Amiram Eldar, Feb 27 2024

A092588 Numbers k such that sigma(phi(k)) - phi(sigma(k)) is nonzero and divisible by sigma(k), that is A065395(k)/A000203(k) is a nonzero integer.

Original entry on oeis.org

7, 327, 463, 497, 617, 691, 751, 1207, 1633, 2451, 2643, 3143, 3337, 3503, 4939, 5609, 7093, 7597, 10327, 14987, 20427, 21103, 22345, 22481, 24739, 26491, 27193, 28077, 37753, 37915, 42711, 42717, 47647, 48043, 49243, 50071, 51727, 54823, 57478

Offset: 1

Labos Elemer, Mar 01 2004

nonn

			(sigma(phi(x))-phi(sigma(x)))/sigma(x) quotient equals 1 for x=7, 2 for x=327, 3 for x=5609.

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

Cf. A000010, A000203, A033632, A065395, A092584-A092588.

fs[x_] := EulerPhi[DivisorSigma[1, x]] sf[x_] := DivisorSigma[1, EulerPhi[x]] {t=Table[0, {100}], j=1}; Do[s=(sf[n]-fs[n])/DivisorSigma[1, n]; If[ !Equal[s, 0]&&IntegerQ[s], Print[n];t[[j]]=n;j=j+1], {n, 2, 1000000}] t

is(n)=my(s=sigma(n),t=sigma(eulerphi(n))-eulerphi(s)); t && t%s==0 \\ Charles R Greathouse IV, Feb 14 2013

A290002 Numbers k such that psi(phi(k)) = phi(psi(k)).

Original entry on oeis.org

1, 10, 18, 20, 36, 40, 54, 70, 72, 78, 80, 108, 110, 140, 144, 156, 160, 162, 174, 198, 216, 220, 222, 230, 234, 246, 280, 288, 294, 312, 320, 324, 348, 396, 414, 426, 432, 438, 440, 444, 450, 460, 468, 470, 486, 492, 534, 560, 576, 588, 594, 624, 640, 648, 666, 696, 702, 770, 792, 828, 846, 852

Offset: 1

Altug Alkan, Sep 03 2017

Squarefree terms are 1, 10, 70, 78, 110, 174, 222, 230, 246, 426, 438, ...
Common terms of this sequence and A033632 are 1, 14406, 544500, 141118050, ...
From Robert Israel, Sep 03 2017: (Start)
Includes 2^i*3^j if i >= 1 and j >= 2, i.e., 3*A033845, and A020714(n) for n >= 1.
If an even number m is in the sequence, then so is 2*m.
Are there any odd terms other than 1? (End)
a(1) = 1 is the only odd term. LHS of equation allows for 1 and 3 but only for k <= 6. RHS allows for 1 and only for k = 1. - Torlach Rush, Jul 28 2023

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

Cf. A000010, A001615, A020714, A033632, A033845.

psi:= proc(n)  n*mul((1+1/i[1]), i=ifactors(n)[2]) end:
select(psi @ numtheory:-phi = numtheory:-phi @ psi, [$1..1000]); # Robert Israel, Sep 03 2017

f[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@ n}]; Select[Range[10^3], f[EulerPhi@ #] == EulerPhi[f@ #] &] (* Michael De Vlieger, Sep 03 2017 *)

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
isok(n) = eulerphi(a001615(n))==a001615(eulerphi(n)); \\ after Charles R Greathouse IV at A001615

A078148 Numbers k such that d(phi(k)) = phi(d(k)), where d=A000005 and phi=A000010.

Original entry on oeis.org

1, 2, 4, 6, 16, 24, 30, 36, 64, 384, 408, 480, 510, 1024, 1296, 1560, 1680, 2304, 2640, 3480, 4096, 5440, 5520, 6360, 9240, 11280, 14040, 14160, 14400, 15120, 15960, 17880, 19320, 19920, 20760, 22848, 24480, 25680, 26880, 30360, 32280, 35160

Offset: 1

Labos Elemer, Nov 26 2002

nonn

2^m is in the sequence iff m=0 or m+1 is prime (the proof is easy). Also all numbers of the form 3*2^(2^m-1) are in the sequence because d(phi(3*2^(2^m-1))) - phi(d(3*2^(2^m-1)))= d(2*2^(2^m-2)) - phi(2*2^m) = d(2^(2^m-1)) - phi(2^(m+1)) = 2^m - 2^m = 0. So this sequence is infinite. - Farideh Firoozbakht, Jan 25 2006

Next odd term after 1 is 15230439315 (cf. A378315). - Max Alekseyev, Jan 09 2025

			k = 24: d(24) = 8, phi(8) = 4, phi(24) = 8, d(8) = 4, so 24 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from T. D. Noe)

The odd terms are listed in A378315.

Cf. A000005, A000010, A033632.

cm[x_] := DivisorSigma[0, EulerPhi[x]]-EulerPhi[DivisorSigma[0, x]] Do[s=cm[n]; If[Equal[s, 0], Print[n]], {n, 1, 100000}]
Select[Range[36000],DivisorSigma[0,EulerPhi[#]]==EulerPhi[ DivisorSigma[ 0,#]]&] (* Harvey P. Dale, Sep 02 2013 *)

is(n)=numdiv(eulerphi(n))==eulerphi(numdiv(n)) \\ Charles R Greathouse IV, Feb 21 2013

A092586 Numbers k such that sigma(phi(k))-phi(sigma(k)) is nonzero and is divisible by (k+1), that is A065395(k)/(k+1) = (phi(sigma(k))-sigma(phi(k)))/(k+1) is a nonzero integer.

Original entry on oeis.org

7, 87, 231, 463, 617, 691, 751, 855, 1059, 1127, 2795, 4819, 11999, 18527, 22481, 75311, 121939, 232901, 256751, 288883, 313919, 371519, 845831, 1285841, 1762799, 1815167, 7195199, 9096191, 40324121, 93070943, 99388823, 113140151, 238072223, 487394063

Offset: 1

Labos Elemer, Mar 01 2004

nonn

			(sigma(phi(x))-phi(sigma(x)))/(x+1) equals 1 if x=7; is 2 if x=463; is 3 if x=4819.

Cf. A033632, A092584-A092588, A000203, A000010, A065395.

f[ x_] := EulerPhi[ DivisorSigma[1, x]] - DivisorSigma[1, EulerPhi[x]]; t = {}; Do[ s = f[n]; If[ s != 0 && Mod[ s, n + 1] == 0, Print[n]; AppendTo[t, n]], {n, 2*10^8}]; t

Edited and extended by Robert G. Wilson v, Mar 03 2004

a(33)-a(34) from Donovan Johnson, Mar 04 2013

cp's OEIS Frontend

A092584 Numbers k such that sigma(phi(k)) == phi(sigma(k)) (mod k), that is, A033632(k)/k is an integer.

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A062401 a(n) = phi(sigma(n)).

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A028491 Numbers k such that (3^k - 1)/2 is prime.

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A062402 a(n) = sigma(phi(n)).

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A006872 Numbers k such that phi(k) = phi(sigma(k)).

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A065395 Commutator of sigma and phi functions.

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