A010554 -id:A010554 - cp's OEIS frontend

A167766 Minimum numbers whose phi of phi are multiples of the n-th prime: the n-th term is the minimum integer x such that: prime(n) | phi(phi(x)), prime(n) being the n-th prime.

5, 19, 23, 59, 47, 107, 479, 383, 283, 467, 1367, 1187, 167, 347, 1319, 643, 2837, 2203, 2153, 3413, 587, 5693, 1997, 359, 5827, 1619, 2063, 2999, 4799, 3167, 1019, 1579, 5483, 3343, 7159, 3023, 12569, 1307, 4679, 2083, 719, 3623, 4597, 3863, 18917, 4783

Offset: 1

Fred Schneider, Nov 11 2009

These minimal integers are always prime. To be clear, the phi function referred to here is Euler's totient function.

			The first term is 5. phi(5) = 4 and phi(4)=2. 2 is a multiple of the first prime 2. 5 is the lowest such number x where 2 divides phi(phi(x)).

Donovan Johnson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Euler's Totient Function

Cf. A010554.

with(numtheory): P:=proc(n) local a,k; a:=ithprime(n);
for k from 1 to 10^3 do if frac(phi(phi(ithprime(k)))/a)=0
then RETURN(ithprime(k)); break; fi; od; end:
seq(P(i),i=1..46); # Paolo P. Lava, Oct 10 2018

a[n_] := (p=Prime[n]; k=1; While[k++; x=Prime[k]; Mod[ EulerPhi[ EulerPhi[x]], p] != 0]; x); Table[a[n], {n, 50}] (* Jean-François Alcover, Sep 14 2011 *)

/* not the most efficient implementation */ ppp(a,b)= { forprime(p=a,b, v = 2*p + 1; v2 = 1; minv = 100000000; while (v2 <= minv || v <=minv, /* print ("Checking ",v, " for ",p); */ while(!isprime(v), v += 2*p /*; print ("Checking ",v, " for ",p)*/ ); if (v%(p*p)==1, /* don't do this step if: p^2 | v-1 */ v2 = v , v2 = 2*v + 1; while (!isprime(v2) && v2 < minv, v2 += 2*v ) ); if (v2 < minv, minv = v2; ); v += 2*p ); print (p," => ",minv) ) }

A167767 First of 3 or more consecutive integers with equal values of phi(phi(n)).

Original entry on oeis.org

1, 2, 7, 8, 20, 31, 32, 33, 146, 211, 314, 384, 626, 674, 1754, 2694, 2695, 5186, 11714, 12242, 17329, 17613, 19310, 25544, 35774, 36728, 38018, 40227, 42626, 56834, 65731, 91106, 97724, 110971, 117536, 131071, 131072, 155821, 161734, 164174

Offset: 1

Fred Schneider, Nov 11 2009

Let p2(n) = phi(phi(n)). This list shows numbers n such that p2(n) = p2(n+1) = p2(n+2) = x for some x.

Here phi is Euler's totient function.

			p2(1) = p2(2) = p2(3) = 1, p2(7) = p2(8) = p2(9) = 2.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A167768.

[n: n in [1..2*10^5] | EulerPhi(EulerPhi(n)) eq EulerPhi(EulerPhi(n + 1)) and EulerPhi(EulerPhi(n)) eq EulerPhi(EulerPhi(n + 2))]; // Vincenzo Librandi, Jun 24 2016

Select[Range[100], EulerPhi[EulerPhi[#]] == EulerPhi[EulerPhi[# + 1]] && EulerPhi[EulerPhi[#]] == EulerPhi[EulerPhi[# + 2]] &] (* G. C. Greubel, Jun 23 2016 *)
SequencePosition[EulerPhi[EulerPhi[Range[170000]]],{x_,x_,x_}][[;;, 1]] (* Harvey P. Dale, Sep 02 2025 *)

isok(n) = (eulerphi(eulerphi(n)) == eulerphi(eulerphi(n+1))) && (eulerphi(eulerphi(n)) == eulerphi(eulerphi(n+2))) \\ Michel Marcus, Jul 12 2013

{n: A010554(n) = A010554(n+1) = A010554(n+2)}. - R. J. Mathar, Nov 12 2009

Edited by N. J. A. Sloane, Nov 12 2009

Extended by R. J. Mathar, Nov 12 2009

A219029 a(n) = n - 1 - phi(phi(n)).

Original entry on oeis.org

-1, 0, 1, 2, 2, 4, 4, 5, 6, 7, 6, 9, 8, 11, 10, 11, 8, 15, 12, 15, 16, 17, 12, 19, 16, 21, 20, 23, 16, 25, 22, 23, 24, 25, 26, 31, 24, 31, 30, 31, 24, 37, 30, 35, 36, 35, 24, 39, 36, 41, 34, 43, 28, 47, 38, 47, 44, 45, 30, 51, 44, 53, 50, 47, 48, 57, 46, 51, 48

Offset: 1

V. Raman, Nov 10 2012

sign

There are exactly n - 1 - phi(phi(n)) non-primitive roots for n, less than n, if n is prime.

a(n) will be the same as A219027(n) except when n is a member of A033949 or n = 1, i.e., n is not 2, 4, prime, power of a prime, twice a prime, or twice a prime power.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A008330 (number of primitive roots for the n-th prime).
Cf. A046144 (number of primitive roots for n).
Cf. A010554 (value of phi(phi(n))).
Cf. A219027, A219428.

[(n - 1 - EulerPhi(EulerPhi(n))): n in [1..70] ]; // Vincenzo Librandi, Jan 26 2013

Table[n - (EulerPhi[EulerPhi[n]] + 1), {n, 75}] (* Alonso del Arte, Nov 17 2012 *)

for(n=1,100,print1(n-1-eulerphi(eulerphi(n))","))

a(n) = n - 1 - A010554(n). - V. Raman, Nov 22 2012

A231772 Smallest positive number which has exactly n primitive roots, or 0 if no such number exists.

Original entry on oeis.org

8, 1, 5, 0, 11, 0, 19, 0, 17, 0, 23, 0, 29, 0, 0, 0, 41, 0, 81, 0, 67, 0, 47, 0, 53, 0, 0, 0, 59, 0, 0, 0, 97, 0, 0, 0, 109, 0, 0, 0, 83, 0, 0, 0, 139, 0, 0, 0, 113, 0, 0, 0, 107, 0, 163, 0, 0, 0, 0, 0, 199, 0, 0, 0, 137, 0, 0, 0, 0, 0, 0, 0, 149, 0, 0, 0, 0, 0

Offset: 0

Arkadiusz Wesolowski, Nov 13 2013

nonn

If n >= 3 and n is odd, then a(n) = 0.

T. D. Noe, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Primitive Root

Cf. A007617, A010554, A046144, A231773.

nn = 100; t = Join[{1}, Table[p = PrimitiveRoot[n]; If[IntegerQ[p], EulerPhi[EulerPhi[n]], 0], {n, 2, 2*nn}]]; Table[s = Position[t, n, 1, 1]; If[s == {}, 0, s[[1, 1]]], {n, 0, nn}] (* T. D. Noe, Nov 14 2013 *)

r=77; print1(8, ", ", 1, ", "); for(n=2, r, m=0; for(c=2*n+1, n^2+1, if(n%2==1, break); e=eulerphi(c); if(e==lcm(znstar(c)[2])&&eulerphi(e)==n, m=1; print1(c, ", "); break)); if(m==0, print1(0, ", ")));

A163379 a(n) = phi(phi(tau(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2

Offset: 1

Jaroslav Krizek, Jul 25 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A010554, A163109.

EulerPhi[EulerPhi[DivisorSigma[0, Range[100]]]] (* G. C. Greubel, Dec 20 2016 *)

vector(100, n, eulerphi(eulerphi(numdiv(n)))) \\ G. C. Greubel, Dec 20 2016

a(n) = A000010(A000010(A000005(n))) = A000010(A163109(n)) = A010554(A000005(n)).

More terms added by G. C. Greubel, Dec 20 2016

A216321 phi(delta(n)), n >= 1, with phi = A000010 (Euler's totient) and delta = A055034 (degree of minimal polynomials with coefficients given in A187360).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 4, 2, 6, 4, 2, 4, 10, 4, 4, 4, 6, 4, 6, 4, 8, 8, 4, 8, 4, 4, 6, 6, 4, 8, 8, 4, 12, 8, 4, 10, 22, 8, 12, 8, 8, 8, 12, 6, 8, 8, 6, 12, 28, 8, 8, 8, 6, 16, 8, 8, 20, 16, 10, 8, 24, 8, 12, 12, 8, 12, 8, 8, 24, 16, 18, 16, 40, 8, 16, 12

Offset: 1

Wolfdieter Lang, Sep 21 2012

nonn

If n belongs to A206551 (cyclic multiplicative group Modd n) then there exist precisely a(n) primitive roots Modd n. For these n values the number of entries in row n of the table A216319 with value delta(n) (the row length) is a(n). Note that a(n) is also defined for the complementary n values from A206552 (non-cyclic multiplicative group Modd n) for which no primitive root Modd n exists.

See also A216322 for the number of primitive roots Modd n.

			a(8) = 2 because delta(8) = 4 and phi(4) = 2. There are 2 primitive roots Modd 8, namely 3 and 5 (see the two 4s in row n=8 of A216320). 8 = A206551(8).
a(12) = 2 because delta(12) = 4 and phi(4) = 2. But there is no primitive root Modd 12, because 4 does not show up in row n=12 of A216320. 12 = A206552(1).

Cf. A000010, A055034, A216319, A216320, A216322, A010554 (analog in modulo n case).

a(n)=eulerphi(ceil(eulerphi(2*n)/2)) \\ Charles R Greathouse IV, Feb 21 2013

a(n) = phi(delta(n)), n >= 1, with phi = A000010 (Euler's totient) and delta = A055034 with delta(1) = 1 and delta(n) = phi(2*n)/2 if n >= 2.

A219028 Number of non-primitive roots for prime(n), less than prime(n).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 8, 12, 12, 16, 22, 24, 24, 30, 24, 28, 30, 44, 46, 46, 48, 54, 42, 48, 64, 60, 70, 54, 72, 64, 90, 82, 72, 94, 76, 110, 108, 108, 84, 88, 90, 132, 118, 128, 112, 138, 162, 150, 114, 156, 120, 142, 176, 150, 128, 132, 136, 198, 188, 184, 190

Offset: 1

V. Raman, Nov 10 2012

nonn

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

Cf. A008330 (number of primitive roots for the n-th prime, less than n-th prime).
Cf. A046144 (number of primitive roots for n, less than n).
Cf. A010554 (value of phi(phi(n))).

Table[c=Prime[n];c-1-EulerPhi[EulerPhi[c]],{n,70}] (* Harvey P. Dale, Feb 12 2013 *)

forprime(i=2,600,p=0;for(q=1,i-1,if(znorder(Mod(q,i))!=eulerphi(i),p++));print1(p","))

a(n) = p - 1 - phi(phi(p)), where p is the n-th prime.

a(n) = p - 1 - A008330(n) = p - 1 - A010554(p), where p is the n-th prime. - V. Raman, Nov 22 2012

A231773 Numbers k such that there are no numbers with k primitive roots.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95

Offset: 1

Arkadiusz Wesolowski, Nov 13 2013

nonn

Numbers k such that A231772(k) = 0.

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

Cf. A007617, A010554, A046144, A231772.

r=95; for(n=2, r, m=0; for(c=2*n+1, n^2+1, if(n%2==1, break); e=eulerphi(c); if(e==lcm(znstar(c)[2])&&eulerphi(e)==n, m=1; break)); if(m==0, print1(n, ", ")));

a(n) = A007617(n) for n <= 16.

A307308 Self-composition of the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 6, 15, 42, 106, 280, 702, 1778, 4398, 10910, 26678, 65172, 157656, 380524, 912846, 2185906, 5216588, 12433166, 29564544, 70189672, 166245574, 392909240, 926290066, 2178881218, 5114469170, 11985221654, 28049398284, 65588182636, 153277006212, 358073997608

Offset: 1

Ilya Gutkovskiy, Apr 02 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Totient Function

Cf. A000010, A008683, A010554, A065093, A307309.

g[x_] := g[x] = Sum[MoebiusMu[k] x^k/(1 - x^k)^2, {k, 1, 31}]; a[n_] := a[n] = SeriesCoefficient[g[g[x]], {x, 0, n}]; Table[a[n], {n, 31}]

G.f.: g(g(x)), where g(x) = Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2 is the g.f. of A000010.

A378507 The smallest number k such that the equation phi(phi(x)) = k has exactly n solutions.

Original entry on oeis.org

10, 56, 6, 1, 84, 312, 2, 200, 464, 36, 108, 4, 12, 88, 816, 264, 440, 360, 552, 120, 224, 8, 3696, 1320, 928, 176, 624, 1472, 832, 5728, 24, 4560, 1080, 2000, 16, 2848, 72, 1312, 1872, 80, 1120, 216, 880, 336, 23360, 448, 3808, 10608, 648, 528, 352, 9280, 32

Offset: 2

Amiram Eldar, Nov 29 2024

nonn

The smallest number k such that A378506(k) = n.

If phi(phi(x)) = k has a solution, then according to Carmichael's totient function conjecture there is at least one another number y != x such that phi(y) = phi(x) and then y is also a solution. Therefore, according to this conjecture, a(1) does not exist.

Amiram Eldar, Table of n, a(n) for n = 2..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
David M. Bressoud, A Course in Computational Number Theory (web page), CNT.m, Computational Number Theory Mathematica package.
Eric Weisstein's World of Mathematics, Carmichael's Totient Function Conjecture.
Wikipedia, Carmichael's totient function conjecture.

Cf. A000010, A010554, A007374, A378506.

s[n_] := Sum[PhiMultiplicity[k], {k, PhiInverse[n]}]; seq[len_] := Module[{v = Table[0, {len+1}], c = 0, k = 1, ns}, While[c < len, ns = s[k]; If[0 < ns <= len + 1 && v[[ns]] == 0, v[[ns]] = k; c++]; k++]; Rest[v]]; seq[30] (* using David M. Bressoud's CNT.m *)

s(n) = vecsum(apply(x -> invphiNum(x), invphi(n))); \\ using Max Alekseyev's invphi.gp
lista(len) = {my(v = vector(len+1), c = 0, k = 1, ns); while(c < len, ns = s(k); if(ns > 0 && ns <= len + 1 && v[ns] == 0, c++; v[ns] = k); k++); vecextract(v,"^1");}

cp's OEIS Frontend

A167766 Minimum numbers whose phi of phi are multiples of the n-th prime: the n-th term is the minimum integer x such that: prime(n) | phi(phi(x)), prime(n) being the n-th prime.

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A167767 First of 3 or more consecutive integers with equal values of phi(phi(n)).

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Magma

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A219029 a(n) = n - 1 - phi(phi(n)).

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A231772 Smallest positive number which has exactly n primitive roots, or 0 if no such number exists.

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Mathematica

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A163379 a(n) = phi(phi(tau(n))).

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A216321 phi(delta(n)), n >= 1, with phi = A000010 (Euler's totient) and delta = A055034 (degree of minimal polynomials with coefficients given in A187360).

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A219028 Number of non-primitive roots for prime(n), less than prime(n).

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