cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A203966 Numbers n such that phi(n) divides n+1, where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 2, 3, 15, 255, 65535, 83623935, 4294967295, 6992962672132095
Offset: 1

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Keywords

Comments

Numbers k such that A060473(k) = 1. - Robert G. Wilson v, Jul 06 2014
Except for a(2), all terms are odd. - Chai Wah Wu, Jun 06 2017
Since gcd(phi(n),n) = 1, all terms are squarefree. Then, for n = p1 * ... * pk with primes p1 < ... < pk, (n+1)/phi(n) is very close to p1/(p1-1)*...*pk/(p1-1). Since p/(p-1) is decreasing as p grows, having (n+1)/phi(n) = 3 implies that p1 >= 5 and further that n >= 2.4*10^56 is a product of at least 33 primes. Similarly, having (n+1)/phi(n) >= 4 implies that n >= 1.6*10^30 is a product of at least 21 primes. Hence, the terms of this sequence below 1.6*10^30 have (n+1)/phi(n) = 2 and thus coincide with A050474. - Max Alekseyev, Jan 30 2022

Examples

			15 is in the sequence because phi(15) = 8, and 8 divides 16 = 15 + 1 evenly.
		

Crossrefs

Superset of A050474.

Programs

  • Mathematica
    Select[Range[100000], Divisible[#+1, EulerPhi[#]]&]

Extensions

a(8) from Donovan Johnson, Jan 13 2012
a(9) confirmed by Max Alekseyev, Jan 30 2022

A202855 Numbers n such that phi(n) - 1 divides n, where phi is Euler's totient function.

Original entry on oeis.org

3, 4, 6, 12, 60, 1020, 262140, 334495740, 17179869180, 27971850688528380
Offset: 1

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Comments

The sequence b(n) = 4*A050474(n) is a subsequence of this sequence, and comprises solutions of n/(phi(n) - 1) = 4, accounting for all terms up to a(9) except a(1) and a(3). Proof: suppose n/(phi(n) - 1) = 4. With n = 4*x, x/(phi(4*x) - 1) = 1, or phi(4*x) = x + 1. Since phi(k) is even for k > 2, x is odd, and phi(4*x) = 2*phi(x) = x + 1, the definition of A050474. It follows that 4*A050474(8) = 27971850688528380 is a term of this sequence. - Chris Boyd, Mar 22 2015
Similarly, the terms with n/(phi(n) - 1) = 3 are given by 3 * terms of A050474 coprime to 3; n/(phi(n) - 1) = 6 are given by 6 * terms of A050474 coprime to 6. Also, the terms of n/(phi(n) - 1) = 5 are given by 5 * terms t of A203966 coprime to 5 and having (t+1)/phi(t) = 4. Note that n/(phi(n) - 1) = 2 is impossible. - Max Alekseyev, Oct 26 2023

Crossrefs

Programs

  • Mathematica
    Select[1 + Range[1000000], Divisible[#, EulerPhi[#] - 1] &]
  • PARI
    for(n=3,1e7,if(n%(eulerphi(n)-1)==0,print1(n", "))) \\ Charles R Greathouse IV, Dec 26 2011

Extensions

a(8) from Charles R Greathouse IV, Dec 27 2011
a(9) from Donovan Johnson, Dec 29 2011
a(10) from Chris Boyd confirmed by Max Alekseyev, Oct 26 2023

A097029 Fixed points when the function f(x) = phi(x) + floor(x/2) is iterated, i.e., solutions to f(x) = x.

Original entry on oeis.org

1, 2, 3, 4, 8, 15, 16, 32, 64, 128, 255, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65535, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 83623935, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967295
Offset: 1

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Author

Labos Elemer, Aug 27 2004

Keywords

Comments

Trivial fixed points are the powers of 2. How many nontrivial cases exist like 3, 15, 255, 65535: the first 5 terms of A051179. More?
83623935 is the next such term (see also A050474 and A203966). - Michel Marcus, Nov 13 2015

Examples

			For fixed points the cycle lengths are A097026(n=fix)=1, but the reverse is not true because long transients may also lead to 1-cycles.
So, e.g., 1910 is not here because its terminal 1-cycle is prefixed by a long transient: {1910, 1715, 2033, 2924, 2806, 2723, 3689, 4724, 4722, 3933, 4342, 4163, 6041, 8192, 8192}.
		

Crossrefs

Programs

  • PARI
    isok(n) = eulerphi(n) + n\2 == n; \\ Michel Marcus, Nov 13 2015

Extensions

a(30)-a(35) from Michel Marcus, Nov 13 2015
a(36)-a(38) from Jinyuan Wang, Jul 22 2021

A232720 Numbers such that 2*phi(n-2) = n-1.

Original entry on oeis.org

3, 5, 17, 257, 65537, 83623937, 4294967297, 6992962672132097
Offset: 1

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Author

Michel Marcus, Nov 28 2013

Keywords

Comments

Numbers such that A074057(n) = 0.
Note that the first 5 terms are Fermat primes (A019434), while a(6) is prime but is not a Fermat number, and a(7) is (F5 from A000215) but is not prime. Note also that F6, F7, F8 and F9 do not belong to this sequence.

Crossrefs

Programs

  • PARI
    is_A232720(n) = 2*eulerphi(n-2) == n-1

Formula

a(n) = A050474(n) + 2. - Jaroslav Krizek, Feb 25 2015

Extensions

a(8) from Jaroslav Krizek, Feb 25 2015

A238232 Composite numbers n such that the sum of numbers x<=n not coprime to n divides the sum of numbers y<=n coprime to n.

Original entry on oeis.org

15, 35, 95, 119, 143, 209, 255, 287, 319, 323, 377, 527, 559, 779, 899, 923, 989, 1007, 1189, 1199, 1295, 1343, 1349, 1763, 1919, 2159, 2507, 2759, 2911, 3239, 3599, 3827, 4031, 4607, 5183, 5207, 5249, 5459, 5543, 6439, 6479, 6887, 7067, 7279, 7739, 8159, 8639
Offset: 1

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Author

Paolo P. Lava, Feb 21 2014

Keywords

Comments

Also numbers n such that n+1-phi(n) | phi(n).
A203966 lists the numbers n such that the sum of numbers x<=n coprime to n divides the sum of numbers y<=n not coprime to n. This is equivalent to numbers n such that phi(n) | n+1. [suggested by Giovanni Resta]

Examples

			The numbers coprime to 15 are 1, 2, 4, 7, 8, 11, 13, 14 and their sum is 60. In fact 15*phi(15)/2 = 60.
The sum of the numbers from 1 to 15 is 15*(15+1)/2 = 120 and therefore the sum of the numbers not coprime to 15 is 120 - 60 = 60. At the end we have that 60/60 = 1.
		

Crossrefs

Programs

  • Maple
    with(numtheory);P:=proc(q) local i,n;
    for n from 2 to q do if not isprime(n) then
    if type(phi(n)/(n+1-phi(n)),integer) then print(n);
    fi; fi; od; end: P(10^6);

A250405 Numbers k such that all values of Euler phi (A000010) of all divisors of k are pairwise distinct and represent all proper divisors of k+1.

Original entry on oeis.org

1, 3, 15, 255, 65535, 4294967295
Offset: 1

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Author

Jaroslav Krizek, Nov 22 2014

Keywords

Comments

Numbers k such that {phi(d) : d|k} = {d : d|(k+1), d<(k+1)} as multisets.
Conjecture: last term is 4294967295.
First six terms coincide with A051179. - Omar E. Pol, Apr 12 2025

Examples

			15 is in the sequence because {phi(d) : d|15} = {1, 2, 4, 8} = {d : d|16, d<16}.
2 is not in the sequence because {phi(d) : d|2} = {1, 1}, but {d : d|2, d<2} = {1}.
		

Crossrefs

Subsequence of A250404 and A203966.
Sequence differs from A051179, A050474 and A116518.
Cf. A000010.

Programs

  • Magma
    [n: n in [1..100000] | ([EulerPhi(d): d in Divisors(n)]) eq ([d: d in Divisors(n+1) | d lt n+1 ])];

Extensions

Edited and a(6) added by Max Alekseyev, May 04 2024

A254576 Primes p such that phi(p-2) divides p-1 where phi is Euler's totient function (A000010).

Original entry on oeis.org

3, 5, 17, 257, 65537, 83623937
Offset: 1

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Author

Jaroslav Krizek, Feb 25 2015

Keywords

Comments

The first 5 known Fermat primes from A019434 are terms.
Conjecture: also primes p such that 2*phi(p-2) = p-1 (i.e., primes in A232720).
a(7) > 10^25. - Max Alekseyev, Feb 02 2024

Crossrefs

Subsequence of A249541.

Programs

  • Magma
    [n: n in [3..10000000] | IsPrime(n) and (n-1) mod EulerPhi(n-2) eq 0];

A177012 Numbers k such that k^k == -1 (mod phi(k)).

Original entry on oeis.org

1, 2, 3, 15, 87, 255, 11759, 26279, 39455, 43919, 65535, 112895, 443807, 1347455, 1464911, 1568255, 1604559, 1968095, 2441559, 5948799, 16210655, 39624767, 39839039, 59187455, 81624279, 83623935, 251009695, 256685439, 338979839, 434357967, 455345855, 471783935, 487722815, 518291135, 596835839
Offset: 1

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Author

Farideh Firoozbakht, May 19 2010

Keywords

Comments

3 is the largest prime term of this sequence.
All terms are squarefree. There is no further term up to 2*10^8.
If phi(k) divides k+1 then k is in the sequence. This implies A050474 and A203966 are subsequences of this sequence. - Jahangeer Kholdi, Dec 10 2014

Examples

			phi(15)=8 and 15^15 == -1 (mod 8), so 15 is in the sequence.
		

Crossrefs

Programs

  • Mathematica
    v={};Do[If[PowerMod[n,n,EulerPhi[n]]==EulerPhi[n]-1,AppendTo[v,n];
    Print[v]],{n,200000000}]

Extensions

a(27)-a(29) from Jahangeer Kholdi, Dec 10 2014
a(30)-a(35) from Farideh Firoozbakht, Dec 10 2014

A207667 Numbers n such that phi(n) divides n+2.

Original entry on oeis.org

1, 2, 4, 6, 10, 30, 70, 510, 2590, 131070, 3359230, 167247870, 8589934590, 13985925344264190
Offset: 1

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Author

Keywords

Comments

Contains 2 * A203966 as a subsequence. - Max Alekseyev, Oct 27 2023

Crossrefs

Programs

  • Mathematica
    Select[Range[50000000],Divisible[#+2,EulerPhi[#]]&]

Extensions

a(12)-a(13) from Donovan Johnson, Mar 01 2012
a(14) from Max Alekseyev, Oct 27 2023

A092591 Exponents m such that 1-A065395(2^m) is a power of 2, where A065395(n) = sigma(phi(n)) - phi(sigma(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 30, 31, 60, 88, 106, 126, 520, 606, 1278, 2202, 2280, 3216, 4252, 4422, 9688, 9940, 11212, 19936, 21700, 23208, 44496, 86242, 110502, 132048, 216090, 756838, 859432, 1257786, 1398268, 2976220, 3021376, 6972592, 13466916, 20996010, 24036582, 25964950, 30402456, 32582656, 37156666, 42643800, 43112608, 57885160
Offset: 1

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Author

Labos Elemer, Mar 03 2004

Keywords

Comments

A000043(k) - 1 is a term for all k >= 1. - Amiram Eldar, Aug 22 2019
Let 2^k = 1-A065395(2^m) = phi(2^(m+1)-1) - 2^m + 2. If k = 0, then phi(2^(m+1)-1) is odd, implying extraneous m = 0. If k = 1, then phi(2^(m+1)-1) = 2^m, meaning that 2^(m+1)-1 is a product of distinct Fermat primes (A019434), which also a term of A050474. The five known Fermat primes give m in {0, 1, 3, 7, 15, 31}. If k >= 2, then phi(2^(m+1)-1) == 2 (mod 4), implying that 2^(m+1)-1 is a prime power, and by Mihăilescu's theorem, 2^(m+1)-1 must be just a prime, that is, m+1 is a term of A000043 and k = m. Hence, unless there exist other Fermat primes, this sequence is the union of {0, 1, 3, 7, 15, 31} and terms of A000043 decreased by 1. - Max Alekseyev, Jun 14 2025

Examples

			At exponents m=1, 3, 7, 15, 31: 1-A065395(2^m)=2.
While at m=2, 4, 6, 12, 16, 18, 30, 60, 88, 106, 126: 1-A065395(2^m)=2^m.
		

Crossrefs

Programs

  • Mathematica
    f[n_] := DivisorSigma[1, EulerPhi[n]] - EulerPhi[DivisorSigma[1, n]]; pow2Q[n_] := n == 2^IntegerExponent[n, 2]; aQ[n_] := pow2Q[1 - f[2^n]]; Select[Range[0, 130], aQ] (* Amiram Eldar, Aug 22 2019 *)
  • PARI
    f(n) = sigma(eulerphi(n)) - eulerphi(sigma(n)); \\ A065395
    ispp2(k) = k == 2^valuation(k,2);
    isok(n) = ispp2(1-f(2^n)); \\ Michel Marcus, Aug 22 2019, Jun 16 2025

Formula

If there are only 5 Fermat primes (A019434), then for n >= 14, a(n) = A000043(n-5) - 1. - Max Alekseyev, Jun 14 2025

Extensions

Name and example edited by Michel Marcus, Aug 22 2019
a(18)-a(19) from Amiram Eldar, Aug 23 2019
a(1)=0 inserted and terms a(20) onward added by Max Alekseyev, Jun 14 2025
Showing 1-10 of 11 results. Next