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.

Previous Showing 11-18 of 18 results.

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

Views

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.
Cf. A000010, A019434, A050474, A203966, A232720.

Programs

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

A306531 Composite numbers k such that the sum of their aliquot parts divides k-1.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 77, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 611, 625, 729, 841, 961, 1024, 1073, 1331, 1369, 1681, 1849, 2033, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5293, 5329, 6031, 6241
Offset: 1

Views

Author

Paolo P. Lava, Feb 22 2019

Keywords

Examples

			Aliquot parts of 77 are 1, 7, 11 and 78/(1+7+11) = 76/19 = 4.

Links

Crossrefs

Union of A059047 and A246547.
Cf. A203966, A306532.

Programs

  • Maple
    with(numtheory): P:=proc(n) if not isprime(n) and frac((n-1)/(sigma(n)-n))=0 then n; fi; end: seq(P(i),i=2..6241);
  • Mathematica
    q[k_] := !PrimeQ[k] && Divisible[k-1, DivisorSigma[1, k]-k]; Select[Range[2, 6500], q] (* Amiram Eldar, Jul 26 2025 *)
  • PARI
    isok(n) = (n!=1) && !isprime(n) && !frac((n-1)/(sigma(n)-n)); \\ Michel Marcus, Feb 28 2019

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

Views

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

Cf. A000010, A177006.
Cf. A050474, A203966.

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

Views

Author

José María Grau Ribas, Feb 19 2012

Keywords

Comments

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

Crossrefs

Cf. A000010, A050474, A207574, A202855, A203966.

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

A226105 Numbers k such that phi(k)+3 divides k+3, excluding numbers of the form 6*p for a prime p.

Original entry on oeis.org

1, 195, 5187, 1141967133868035, 3658018932844533311864835
Offset: 1

Views

Author

José María Grau Ribas, May 26 2013

Keywords

Comments

Terms having (k+3)/(phi(k)+3) = 2 are shared with A350777. - Max Alekseyev, Oct 26 2023

Crossrefs

Set difference of A226104 and 6 * A000040.
Cf. A202855, A203966, A207575, A207667, A350777.

Programs

  • Mathematica
    Select[Range[10000000], !PrimeQ[#/6] && IntegerQ[(# + 3)/(EulerPhi[#] + 3)] &]
  • PARI
    for(n=1,10^8, if( (n+3)%(eulerphi(n)+3)==0 && (n%6 || !isprime(n\6)), print(n)));

Extensions

Edited and a(4)-a(5) added by Max Alekseyev, Nov 05 2023

A250404 Numbers k such that the set of all distinct values of phi of all divisors of k equals the set of all proper divisors of k+1 where phi is the Euler totient function (A000010).

Original entry on oeis.org

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

Views

Author

Jaroslav Krizek, Nov 22 2014

Keywords

Comments

Numbers k such that {phi(d) : d|k} = {d : d|(k+1), d<(k+1)} as sets.
Conjecture: last term is 4294967295.
Sequence differs from A203966 because 83623935 is not in this sequence.

Examples

			2 is a term since {phi(d) : d|2} = {1} = {d; d|2, d<2}.
15 is a term since {phi(d) : d|15} = {1, 2, 4, 8} = {d : d|16, d<16}.

Crossrefs

Subsequence of A203966.
Cf. A000010, A203966, A250405.

Programs

  • Magma
    [n: n in [1..100000] | Set([EulerPhi(d): d in Divisors(n)]) eq Set([d: d in Divisors(n+1) | d lt n+1 ])]
  • PARI
    isok(n) = {sphi = []; fordiv(n, d, sphi = Set(concat(sphi, eulerphi(d)))); sphi == setminus(Set(divisors(n+1)), Set(n+1));} \\ Michel Marcus, Nov 23 2014

Extensions

Edited and a(7) added by Max Alekseyev, May 04 2024

A263810 Numbers k such that k = tau(k) * phi(k-2) + 1.

Original entry on oeis.org

3, 4, 5, 17, 257, 65537, 83623937
Offset: 1

Views

Author

Jaroslav Krizek, Oct 27 2015

Keywords

Comments

Numbers k such that k = A000005(k) * A000010(k-2) + 1.
Sequence deviates from A249541; numbers 4294967297 and 6992962672132097 are not terms of this sequence.
The first 5 known Fermat primes from A019434 are in this sequence.
Conjecture: primes from this sequence are in A254576.
a(8) > 10^25. If k = tau(k) * phi(k-2) + 1 then phi(k-2) must divide k-1, thus k-2 must be a term of A203966, which has already been searched up to 10^25. - Giovanni Resta, Feb 21 2020; updated by Max Alekseyev, Feb 21 2025

Examples

			17 is in this sequence because 17 = tau(17)*phi(15) + 1 = 2*8 + 1.

Crossrefs

Cf. A000005, A000010, A019434, A249541, A254576, A203966.
Cf. A263811 (numbers k such that k = tau(k) * phi(k-1) + 1).

Programs

  • Magma
    [n: n in [3..1000000] |  n eq NumberOfDivisors(n) * EulerPhi(n-2) + 1];
  • Mathematica
    Select[Range@ 100000, # == DivisorSigma[0, #] EulerPhi[# - 2] + 1 &] (* Michael De Vlieger, Oct 27 2015 *)
  • PARI
    for(n=3, 1e8, if(numdiv(n)*eulerphi(n-2) == n-1, print1(n ", "))) \\ Altug Alkan, Oct 28 2015
  • PARI
    lista(na, nb) = {my(f1 = factor(na-2), f2 = factor(na-1), f3); for(n=na, nb, f3 = factor(n); if (numdiv(f3)*eulerphi(f1) == n-1, print1(n ", ")); f1 = f2; f2 = f3;);}; \\ Michel Marcus, Feb 21 2020

A306532 Composite numbers k such that the sum of their aliquot parts divides k+1.

Original entry on oeis.org

21, 115, 329, 731, 2133, 2171, 6821, 7379, 8357, 13987, 19521, 24331, 24881, 29491, 46001, 50579, 56789, 79421, 103729, 117409, 125159, 137881, 174109, 176661, 226859, 235721, 257291, 357769, 492071, 499091, 560537, 584021, 704791, 776341, 822857, 850607, 908981
Offset: 1

Views

Author

Paolo P. Lava, Feb 22 2019

Keywords

Examples

			Aliquot parts of 21 are 1, 3, 7 and 22/(1+3+7) = 22/11 = 2.

Crossrefs

Cf. A203966, A306531.

Programs

  • Maple
    with(numtheory): P:=proc(n) if not isprime(n) and frac((n+1)/(sigma(n)-n))=0 then n; fi; end: seq(P(i),i=2..100000);
  • Mathematica
    Select[Range[10^6],CompositeQ[#]&&Mod[#+1,Total[Most[Divisors[#]]]]==0&] (* Harvey P. Dale, Jul 28 2024 *)
  • PARI
    isok(n) = (n!=1) && !isprime(n) && !frac((n+1)/(sigma(n)-n)); \\ Michel Marcus, Feb 28 2019
Previous Showing 11-18 of 18 results.

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