A045544 -id:A045544 - cp's OEIS frontend

A078168 Numbers k such that phi(k) is a perfect 8th power.

1, 2, 257, 512, 514, 544, 640, 680, 768, 816, 960, 1020, 65537, 131072, 131074, 131584, 139264, 139808, 163840, 164480, 174080, 174760, 196608, 197376, 208896, 209712, 245760, 246720, 261120, 262140, 1682227, 1683109, 1683559, 1683937

Offset: 1

Labos Elemer, Nov 27 2002

nonn

			phi of the sequence includes 1, 256, 65536, 1679616, etc.; powers arise several times; a(3) = A053576(7) = 257; in sequence smoother ranges and quite large jumps arise when power of new numbers appear as phi-values.

Amiram Eldar, Table of n, a(n) for n = 1..7500

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th, this sequence), A078169 (9th), A078170 (10th power), A001317, A053576, A045544, A000010.

k=8; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 10000000}]
Select[Range[2*10^6],IntegerQ[Surd[EulerPhi[#],8]]&] (* Harvey P. Dale, Oct 20 2014 *)

is(n)=ispower(eulerphi(n),8) \\ Charles R Greathouse IV, Apr 24 2020

A078169 Numbers k such that phi(k) is a perfect 9th power.

Original entry on oeis.org

1, 2, 771, 1024, 1028, 1088, 1280, 1360, 1536, 1542, 1632, 1920, 2040, 327685, 524288, 524296, 526336, 557056, 559232, 655360, 655370, 657920, 696320, 699040, 786432, 786444, 789504, 835584, 838848, 983040, 986880, 1044480, 1048560

Offset: 1

Labos Elemer, Nov 27 2002

nonn

			phi of the sequence includes 1, 512, 262144,.. etc.; powers arise several times; a(3) = A053576(9) = 771; in sequence smoother ranges and quite large jumps arise when power of new numbers appear as phi-values.

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

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th, this sequence), A078170 (10th power), A001317, A053576, A045544, A000010.

k=9; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 10000000}]

is(n)=ispower(eulerphi(n),9) \\ Charles R Greathouse IV, Apr 24 2020

A078170 Numbers k such that phi(k) is a perfect tenth power.

Original entry on oeis.org

1, 2, 1285, 2048, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 4080, 1114129, 2097152, 2097184, 2105344, 2228224, 2228258, 2236928, 2621440, 2621480, 2631680, 2785280, 2796160, 3145728, 3145776, 3158016, 3342336

Offset: 1

Labos Elemer, Nov 27 2002

nonn

			phi of the sequence includes 1, 1024, 1048576,.. etc.; powers emerge several times; a(3) = A053576(10) = 1285; in sequence smoother ranges and quite large jumps alternate when power of new numbers appear as phi-values.

Amiram Eldar, Table of n, a(n) for n = 1..2250

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th power, this sequence), A001317, A053576, A045544, A000010.

k=10; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 10000000}]

is(n)=ispower(eulerphi(n),10) \\ Charles R Greathouse IV, Apr 24 2020

A235040 After 1, composite odd numbers, whose prime divisors, when multiplied together without carry-bits (as codes for GF(2)[X]-polynomials, with A048720), yield the same number back.

Original entry on oeis.org

1, 15, 51, 85, 95, 111, 119, 123, 187, 219, 221, 255, 335, 365, 411, 447, 485, 511, 629, 655, 685, 697, 771, 831, 879, 959, 965, 1011, 1139, 1241, 1285, 1405, 1535, 1563, 1649, 1731, 1779, 1799, 1923, 1983, 2005, 2019, 2031, 2045, 2227, 2605, 2735, 2815, 2827

Offset: 0

Antti Karttunen, Jan 02 2014

nonn

Note: Start indexing from n=1 if you want just composite numbers. a(0)=1 is the only nonprime, noncomposite in this list.
The first term with three prime divisors is a(11) = 255 = 3*5*17.
The next terms with three prime divisors are
  255, 3855, 13107, 21845, 24415, 28527, 30583, 31215, 31611, 31695, 32691, 48059, 56283, 56797, 61935, 65365, 87805, 98005, ...
Of these 24415 (= 5*19*257) is the first one with at least one prime factor that is not a Fermat prime (A019434).
The first term with four prime divisors is a(427) = 65535 = 3*5*17*257.
The first terms which are not multiples of any Fermat prime are: 511, 959, 3647, 4039, 4847, 5371, 7141, 7231, 7679, 7913, 8071, 9179, 12179, ... (511 = 7*73, 959 = 7*137, ...)

			15 = 3*5. When these factors (with binary representations '11' and '101') are multiplied as:
   101
  1010
  ----
  1111 = 15
we see that the intermediate products 1*5 and 2*5 can be added together without producing any carry-bits (as they have no 1-bits in the same columns/bit-positions), so A048720(3,5) = 3*5 and thus 15 is included in this sequence.

Odd nonprimes in A235034. A235039 is a subsequence.
The composite terms in A045544 (A004729) all occur also here.
Cf. also A019434, A048720, A235045, A235050, A115857, A115872.

A293231 a(n) = Product_{d|n, dA019565(A193231(d)).

Original entry on oeis.org

1, 2, 2, 12, 2, 36, 2, 120, 6, 60, 2, 5400, 2, 360, 30, 25200, 2, 56700, 2, 21000, 180, 840, 2, 23814000, 10, 504, 630, 50400, 2, 661500, 2, 554400, 420, 132, 300, 392931000, 2, 792, 252, 242550000, 2, 24948000, 2, 2772000, 22050, 1980, 2, 605113740000, 60, 4851000, 66, 3880800, 2, 720373500, 700, 4889808000, 396, 2772, 2, 588305025000, 2, 1848

Offset: 1

Antti Karttunen, Oct 03 2017

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences related to binary expansion of n

Cf. A019565, A193231, A290090, A293214, A293232 (rgs-version of this sequence).

Cf. also A001317, A045544, A053576.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ This function from Franklin T. Adams-Watters
A293231(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A193231(d)))); m; };

a(n) = Product_{d|n, dA019565(A193231(d)).
For all n >= 1, A007814(a(n)) = A290090(n).
For n = 0..5, a(A001317((2^n)-1)) = A002110((2^n)-1).

A058213 Triangular arrangement of solutions of phi(x) = 2^n (n >= 0), where phi=A000010 is Euler's totient function. Each row corresponds to a particular n and its length is n+2 for 0 <= n <= 31, 32 for n >= 32. (This assumes that there are only 5 Fermat primes.)

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 10, 12, 15, 16, 20, 24, 30, 17, 32, 34, 40, 48, 60, 51, 64, 68, 80, 96, 102, 120, 85, 128, 136, 160, 170, 192, 204, 240, 255, 256, 272, 320, 340, 384, 408, 480, 510, 257, 512, 514, 544, 640, 680, 768, 816, 960, 1020, 771, 1024, 1028, 1088

Offset: 0

Labos Elemer, Nov 30 2000

phi(x) is a power of 2 if and only if x is a power of 2 multiplied by a product of distinct Fermat primes. So if, as is conjectured, there are only 5 Fermat primes, then there are only 32 possibilities for the odd part of x, namely the divisors of 2^32-1, given in A004729.
The same numbers, in increasing order, are given in A003401.
The first entry in row n is the n-th divisor of 2^32-1 for 0 <= n <= 31 (A004729) and is 2^(n+1) for n >= 32. The last entry in row n is given in A058215.

			Triangle begins:
  { 1,   2},
  { 3,   4,   6},
  { 5,   8,  10,  12},
  {15,  16,  20,  24,  30},
  {17,  32,  34,  40,  48,  60},
  {51,  64,  68,  80,  96, 102, 120},
  {85, 128, 136, 160, 170, 192, 204, 240},
  ...

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A000010, A001317, A003401, A004729, A019434, A045544, A047999, A053576, A054432, A058214, A058215.

phiinv[ n_, pl_ ] := Module[ {i, p, e, pe, val}, If[ pl=={}, Return[ If[ n==1, {1}, {} ] ] ]; val={}; p=Last[ pl ]; For[ e=0; pe=1, e==0||Mod[ n, (p-1)pe/p ]==0, e++; pe*=p, val=Join[ val, pe*phiinv[ If[ e==0, n, n*p/pe/(p-1) ], Drop[ pl, -1 ] ] ] ]; Sort[ val ] ]; phiinv[ n_ ] := phiinv[ n, Select[ 1+Divisors[ n ], PrimeQ ] ]; Join@@(phiinv[ 2^# ]&/@Range[ 0, 10 ]) (* phiinv[ n, pl ] = list of x with phi(x)=n and all prime divisors of x in list pl. phiinv[ n ] = list of x with phi(x)=n *)

Edited by Dean Hickerson, Jan 25 2002

A058214 Sum of solutions of phi(x) = 2^n.

Original entry on oeis.org

3, 13, 35, 105, 231, 581, 1315, 3225, 6711, 15221, 32755, 74505, 154407, 339397, 718115, 1589145, 3243831, 6946421, 14482675, 31259145, 63894567, 135588037, 281203235, 601400985, 1219907127, 2557715317, 5267017715, 11123540745, 22600784679, 47205887429

Offset: 0

Labos Elemer, Nov 30 2000

nonn

			For n = 6, 2^n = 64; the solutions of phi(x) = 64 are {85,128,136,160,170,192,204,240}, whose sum is a(6) = 1315.

Amiram Eldar, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A001317, A003401, A004729, A019434, A045544, A047999, A053576, A054432, A058213, A058215.

phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl=={}, Return[If[n==1, {1}, {}]]]; val={}; p=Last[pl]; For[e=0; pe=1, e==0||Mod[n, (p-1)pe/p]==0, e++; pe*=p, val=Join[val, pe*phiinv[If[e==0, n, n*p/pe/(p-1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1+Divisors[n], PrimeQ]]; Table[Plus@@phiinv[2^n], {n, 0, 30}] (* phiinv[n, pl] = list of x with phi(x)=n and all prime divisors of x in list pl. phiinv[n] = list of x with phi(x)=n *)

a(n) = vecsum(invphi(2^n)); \\ Amiram Eldar, Nov 11 2024, using Max Alekseyev's invphi.gp

If there are only five Fermat primes, then a(n) = 2^(n-30) * 99852066765 for n > 31. - T. D. Noe, Jun 21 2012

Edited by Dean Hickerson, Jan 25 2002

a(28)-a(29) from Donovan Johnson, Oct 22 2011

A058215 Largest solution of phi(x) = 2^n.

Original entry on oeis.org

2, 6, 12, 30, 60, 120, 240, 510, 1020, 2040, 4080, 8160, 16320, 32640, 65280, 131070, 262140, 524280, 1048560, 2097120, 4194240, 8388480, 16776960, 33553920, 67107840, 134215680, 268431360, 536862720, 1073725440, 2147450880, 4294901760, 8589934590

Offset: 0

Labos Elemer, Nov 30 2000

nonn

The ratio of adjacent terms is 2 except for five terms (if there are exactly five Fermat primes). - T. D. Noe, Jun 21 2012

			For n = 6, 2^n = 64; the solutions of phi(x) = 64 are {85,128,136,160,170,192,204,240}; the largest is a(6) = 240.

Amiram Eldar, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A001317, A003401, A004729, A019434, A045544, A047999, A053576, A054432, A058213, A058214.

phiinv[ n_, pl_ ] := Module[ {i, p, e, pe, val}, If[ pl=={}, Return[ If[ n==1, {1}, {} ] ] ]; val={}; p=Last[ pl ]; For[ e=0; pe=1, e==0||Mod[ n, (p-1)pe/p ]==0, e++; pe*=p, val=Join[ val, pe*phiinv[ If[ e==0, n, n*p/pe/(p-1) ], Drop[ pl, -1 ] ] ] ]; Sort[ val ] ]; phiinv[ n_ ] := phiinv[ n, Select[ 1+Divisors[ n ], PrimeQ ] ]; Table[ phiinv[ 2^n ][ [ -1 ] ], {n, 0, 30} ] (* phiinv[ n, pl ] = list of x with phi(x)=n and all prime divisors of x in list pl. phiinv[ n ] = list of x with phi(x)=n *)

a(n) = invphiMax(2^n); \\ Amiram Eldar, Nov 11 2024, using Max Alekseyev's invphi.gp

Assuming there are only 5 Fermat primes (A019434), a(n) = 2^(n-30)*(2^32-1) for n >= 31.

Edited by Dean Hickerson, Jan 25 2002

A058321 Number of x such that phi(x) = 2^n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32

Offset: 0

Labos Elemer, Dec 11 2000

nonn

If there are only 5 Fermat primes (A019434), then a(n) = 32 for n > 31. - T. D. Noe, Jun 21 2012 [Corrected by Jeppe Stig Nielsen, Oct 02 2021.]

The first unknown term is a(8589934592) which depends on whether A000215(33) is composite or prime. - Jeppe Stig Nielsen, Oct 02 2021

			For n = 0, a(0) = 2 because phi(1) = phi(2) = 1.
For n = 5, invphi(32) gives 7 values as follows: phi({51,64,68,80,96,102,120}) = {32,32,32,32,32,32,32}.

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
R. D. Carmichael, On Euler's phi-function, Bull. Amer. Math. Soc. 13 (1907), 241-243.
R. D. Carmichael, Erratum: On Euler's phi-function, Bull. Amer. Math. Soc. 54 (1948), 1192.
R. D. Carmichael, Erratum: Erratum: On Euler's phi-function, Bull. Amer. Math. Soc. 55 (1949), 212.
Mathematics Stack Exchange, Empirical Observation on number of solutions to phi(n) = m.
Eric W. Weisstein, MathWorld: Fermat prime.
Wikipedia, Euler's totient function.

Cf. A000010, A000079, A003401, A004729, A014197, A019434, A045544, A058213.

with(numtheory):[seq(nops(invphi(2^i)),i=1..100)];

a(n) = invphiNum(1 << n); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

a(n) = A014197(2^n) = A014197(A000079(n)).

Added a(0) and corrected a(31) - T. D. Noe, Jun 21 2012

Correction of a(31) reverted; true value is a(31) = 33. - Jeppe Stig Nielsen, Oct 02 2021

A262534 Numbers k such that phi(k-2) = phi(k-1) = (k-1) / 2.

Original entry on oeis.org

3, 5, 17, 257, 65537, 4294967297

Offset: 1

Jaroslav Krizek, Sep 24 2015

No more terms below 10^8; 4294967297 is a term of this sequence.
First 5 terms are Fermat primes (A019434).
Conjecture: next term is 4294967297.
Subsequence of A232720 and A000051.
Sequence is different from A232720 and A000215; A232720(6) = 83623937 and A000215(7) = 18446744073709551617 are not terms of this sequence.
From Jeppe Stig Nielsen, Nov 19 2016: (Start)
Since n-1 is a solution to phi(x)=x/2, it is clear from the formula for phi that x=n-1 is a nontrivial power of two (in A000079 and even). Then y=n-2 is an odd number such that phi(y) is a power of two, and again recalling the formula for phi, this can only happen when y is a product of distinct Fermat primes (y in A045544).
Question: Is the above comment, together with Euler's demonstration that the Fermat number 2^32 + 1 =  A000215(5) is composite, enough to prove that A262534 has only these six terms?
With the above observations, we need only search next to powers of two (A000051), so it is quick to determine that there are no terms between 65537 and 4294967297. (End)
From Robert Israel, Dec 08 2016: (Start)
The j-th binary digit (i.e. coefficient of 2^j) of the product of a set of distinct Fermat numbers, say y=Product_{k in T} (2^(2^k)+1), is 1 iff j = Sum_{k in S} 2^k for some subset S of T.  In order for x = y+1 to be a power of 2, all of y's binary digits must be 1.  Since 2^(2^5)+1 is composite, T cannot contain 5, so digit 32 is 0, and y < 2^32.  Thus we do have only these 6 terms. (End)

			17 is in this sequence because phi(15) = phi(16) = 8 = (17 - 1) / 2.

Cf. A000010, A000215, A232720, A000079, A045544, A000051, A019434.

[n: n in [3..10000000] | n-1 eq 2*EulerPhi(n-1) and n-1 eq 2*EulerPhi(n-2)];

Select[Range@ 100000, EulerPhi[# - 2] == EulerPhi[# - 1] == (# - 1)/2 &] (* Michael De Vlieger, Sep 25 2015 *)

for(n=1, 1e8, if(eulerphi(n-2) == eulerphi(n-1) && 2*eulerphi(n-1) == (n-1), print1(n ", "))) \\ Altug Alkan, Oct 11 2015

a(6) from Jeppe Stig Nielsen, Nov 19 2016

cp's OEIS Frontend

A078168 Numbers k such that phi(k) is a perfect 8th power.

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A078169 Numbers k such that phi(k) is a perfect 9th power.

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A078170 Numbers k such that phi(k) is a perfect tenth power.

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A235040 After 1, composite odd numbers, whose prime divisors, when multiplied together without carry-bits (as codes for GF(2)[X]-polynomials, with A048720), yield the same number back.

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A293231 a(n) = Product_{d|n, dA019565(A193231(d)).

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A058213 Triangular arrangement of solutions of phi(x) = 2^n (n >= 0), where phi=A000010 is Euler's totient function. Each row corresponds to a particular n and its length is n+2 for 0 <= n <= 31, 32 for n >= 32. (This assumes that there are only 5 Fermat primes.)

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A058214 Sum of solutions of phi(x) = 2^n.

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A058215 Largest solution of phi(x) = 2^n.

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