A007694 -id:A007694 - cp's OEIS frontend

A067606 Primes p such that p+7 == 0 (mod phi(p+7)).

5, 11, 17, 29, 41, 47, 89, 101, 137, 281, 317, 479, 569, 641, 761, 857, 1289, 1451, 1721, 2297, 2909, 3449, 3881, 8741, 9209, 11657, 12281, 17489, 23321, 26237, 36857, 39359, 46649, 62201, 73721, 98297, 147449, 157457, 331769, 393209, 839801, 944777, 1119737

Offset: 1

Benoit Cloitre, Feb 22 2002

nonn

Equivalently, these are the primes of form 2^(i+1)*3^j - 7. We empirically have log(a(n)) ~ 0.23...*n + O(1), but the constant seems difficult to compute precisely. - Matthew House, Aug 13 2024

Matthew House, Table of n, a(n) for n = 1..9949 (all primes proven via ECPP)

Cf. A007694.

Select[Prime[Range[35000]],Divisible[#+7,EulerPhi[#+7]]&] (* Harvey P. Dale, Aug 15 2016 *)
lim = 10^7 + 7; Sort[Select[Flatten[Table[2^i*3^j - 7, {i, 1, Log2[lim]}, {j, 0, Log[3, lim/2^i]}]], # > 0 && PrimeQ[#] &]] (* Matthew House, Aug 13 2024 *)

More terms from Matthew House, Aug 13 2024

A067932 Primes p such that p+3 == 0 (mod phi(p+3)).

Original entry on oeis.org

3, 5, 13, 29, 61, 509, 1021, 4093, 16381, 1048573, 4194301, 16777213, 536870909, 19807040628566084398385987581, 83076749736557242056487941267521533, 5316911983139663491615228241121378301

Offset: 1

Benoit Cloitre, Feb 22 2002

phi(n) divides n iff n=1 or n=2^w*3^u for w>=1 and u>=0 (see A007694). Such an n can only have the form p+3 if n=6 or n is a power of 2. So the terms of the sequence are 3 and the primes of the form 2^n-3, listed in A050415.

Vincenzo Librandi, Table of n, a(n) for n = 1..30

Prepend[Select[2^Range[2, 200]-3, PrimeQ], 3]

Edited and extended by Robert G. Wilson v, Feb 27 2002 and by Dean Hickerson, Mar 21 2002

A069924 Number of k, 1<=k<=n, such that phi(k) divides k.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Benoit Cloitre, May 05 2002

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/log(n)^2 for n = 1..10000

Cf. A007694, A071521, A062153.

Table[Length[Select[Range[n], Divisible[#, EulerPhi[#]] &]], {n, 1, 100}] (* Vaclav Kotesovec, Feb 16 2019 *)
Accumulate[Table[If[Divisible[n,EulerPhi[n]],1,0],{n,80}]] (* Harvey P. Dale, Jul 04 2021 *)

for(n=1,150,print1(sum(i=1,n,if(i%eulerphi(i),0,1)),","))

a(n) = Card(k: 1<=k<=n : k==0 (mod phi(k))) asymptotically: a(n) = C*log(n)^2 + o(log(n)^2) with C=0.6....

a(n) = A071521(n) - A062153(n). - Ridouane Oudra, Mar 05 2025

A145853 Numbers m such that m is a multiple of all integers smaller than the largest prime dividing m.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 54, 60, 64, 72, 96, 108, 120, 128, 144, 162, 180, 192, 216, 240, 256, 288, 300, 324, 360, 384, 420, 432, 480, 486, 512, 540, 576, 600, 648, 720, 768, 840, 864, 900, 960, 972, 1024, 1080, 1152, 1200, 1260, 1296, 1440

Offset: 1

J. Lowell, Oct 21 2008

nonn

The definition "Numbers m such that if m is a multiple of a number k, then m is a multiple of all integers less than k" produces the finite sequence 1, 2.
A007694 (numbers m such that phi(m) divides m) is a subsequence. - Klaus Brockhaus, Oct 23 2008
Numbers m such that for p prime p|m => A003418(p)|m. - David W. Wilson, Jan 05 2019

			30 does not qualify because it is divisible by prime number 5 but not by 4 < 5. However, the fact that 32 is divisible by 4 but not by 3 < 4 does not disqualify 32 from being in this sequence because 4 is not prime.

David W. Wilson, Table of n, a(n) for n = 1..1000 (terms 1..200 from Vincenzo Librandi)

Cf. A007694, A006530 (largest prime dividing n). - Klaus Brockhaus, Oct 23 2008

[ n: n in [1..1450] | forall{ x: x in [2..p] | n mod x eq 0 } where p is #f eq 0 select 1 else f[ #f][1] where f is Factorization(n) ]; // Klaus Brockhaus, Oct 23 2008

a = {1}; For[n = 2, n < 2000, n++, b = FactorInteger[n][[ -1, 1]]; If[Length[Select[Range[b], Mod[n, # ] == 0 &]] == b, AppendTo[a, n]]]; a (* Stefan Steinerberger, Oct 25 2008 *)

More terms from Klaus Brockhaus and Stefan Steinerberger, Oct 23 2008

Better definition from Stefan Steinerberger, Oct 23 2008

A167211 Numbers k such that number of perfect partitions of k-1 divides k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 13, 15, 16, 17, 19, 21, 23, 29, 31, 32, 33, 37, 39, 40, 41, 43, 47, 48, 51, 53, 57, 59, 61, 64, 67, 69, 71, 73, 78, 79, 83, 87, 89, 93, 97, 101, 103, 107, 109, 111, 113, 123, 127, 128, 129, 130, 131, 132, 137, 139, 141, 149, 151, 157

Offset: 1

Juri-Stepan Gerasimov, Oct 30 2009

nonn

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

Cf. A002033, A007694, A033950.

f[1] = 1; f[n_] := f[n] = DivisorSum[n, f[#] &, # < n &]; Select[Range[100], Divisible[#, f[#]] &] (* Amiram Eldar, Dec 30 2019 *)

{n: A002033(n-1) | n}.

Definition corrected by R. J. Mathar, May 21 2010

More terms from Amiram Eldar, Dec 30 2019

A245047 Numbers n where phi(n)|n or tau(n)|n.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 144, 152, 156, 162, 180, 184, 192, 204, 216, 225, 228, 232, 240, 248, 252, 256, 276, 288, 296, 324, 328, 344, 348, 360, 372, 376, 384, 396, 424, 432, 441, 444, 448, 450

Offset: 1

Reinhard Muehlfeld, Jul 13 2014

nonn

Phi(n) is Euler totient function (A000010); tau(n) is the number of divisors of n (A000005).

Union of A007694 and A033950. - Michel Marcus, Jul 15 2014

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

Cf. A000005, A000010, A007694, A033950, A235353.

select(t -> (t mod numtheory:-phi(t) = 0) or (t mod numtheory:-tau(t) = 0), [$1..1000]); # Robert Israel, Jul 15 2014

Select[Range[500],AnyTrue[{#/EulerPhi[#],#/DivisorSigma[0,#]},IntegerQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 12 2020 *)

isok(n) = !((n % eulerphi(n)) && (n % numdiv(n))); \\ Michel Marcus, Jul 15 2014

A263811 Numbers k such that k = tau(k) * phi(k-1) + 1.

Original entry on oeis.org

3, 5, 17, 25, 49, 257, 289, 65537

Offset: 1

Jaroslav Krizek, Nov 04 2015

Numbers k such that k = A000005(k) * A000010(k-1) + 1.
The first 5 known Fermat primes from A019434 are in this sequence.
The next term, if it exists, must be greater than 2*10^7.
A prime p is in the sequence iff p is a Fermat prime (A019434) - see proof in A171271.
Observation: the known composite terms are squares of primes. - Omar E. Pol, Nov 04 2015
From Charlie Neder, Mar 02 2019: (Start)
Rearranging the definition gives (k-1)/phi(k-1) = tau(k), which means k-1 is in A007694. Since k-1 is thus 3-smooth, there are two possibilities:
1) k-1 is a power of 2 and tau(k) = 2, i.e., k is a Fermat prime,
2) k-1 is a 3-smooth number divisible by 6 and tau(k) = 3, i.e., k is a Pierpont number and the square of a prime.
In the second case, k-1 factors as (p-1)(p+1) for some p, and both parts are 3-smooth if and only if p is in {2,3,5,7,17} (2 and 3 are excluded since in those cases k-1 is not divisible by 6). Therefore, this sequence is complete if and only if there are no more Fermat primes. (End)

			17 is in this sequence because 17 = tau(17)*phi(16) + 1 = 2*8 + 1.

Cf. A000005, A000010, A019434.

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

[n: n in [2..1000000] |  n eq NumberOfDivisors(n) * EulerPhi(n-1) + 1];

Select[Range[10^5], # == DivisorSigma[0, #] EulerPhi[# - 1] + 1 &] (* Michael De Vlieger, Nov 05 2015 *)

for(n=1, 1e5, if( n-1 == numdiv(n)*eulerphi(n-1) , print1(n, ", "))) \\ Altug Alkan, Nov 05 2015

A275245 Numbers k such that phi(k) divides k^2 while phi(k) does not divide k.

Original entry on oeis.org

10, 20, 40, 42, 50, 60, 80, 84, 100, 114, 120, 126, 136, 156, 160, 168, 180, 200, 220, 228, 240, 250, 252, 272, 294, 300, 312, 320, 336, 342, 360, 378, 400, 440, 444, 456, 468, 480, 500, 504, 540, 544, 588, 600, 624, 640, 672, 684, 720, 756, 800, 816

Offset: 1

Altug Alkan, Jul 21 2016

nonn

			10 is a term because phi(10) = 4; 10 mod 4 = 2 and 10^2 mod 4 = 0.

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

Cf. A000010, A007694, A090778.

Select[Range[10^3], Function[k, And[Divisible[#^2, k], ! Divisible[#, k]]]@ EulerPhi@ # &] (* Michael De Vlieger, Jul 21 2016 *)

isok(n) = (n % eulerphi(n) != 0) && (n^2 % eulerphi(n) == 0)

A280990 Least prime p such that n divides phi(p*n).

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 31, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 31, 31, 2, 67, 17, 71, 3, 37, 19, 13, 5, 41, 7, 43, 11, 31, 23, 47, 3, 7, 5, 103, 13, 53, 3, 11, 7, 19, 29, 59, 31, 61, 31, 7, 2, 131, 67, 67, 17, 139, 71, 71, 3, 73, 37, 31, 19, 463

Offset: 1

Altug Alkan, Jan 12 2017

nonn

n*a(n) are 2, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 36, 169, 98, 465, 32, 289, ...

a(n) <= A034694(A007947(n)). If n is in A050384 then a(n) = A034694(n). - Robert Israel, Jan 12 2017

			a(15) = 31 because 15 does not divide phi(p*15) for p < 31 where p is prime and phi(31*15) = 2*4*30 is divisible by 15.

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

Cf. A000010, A007694, A007947, A034694, A050384, A109395.

Cf. A000079, A065119, A086761: for those n such that a(n)=2,3,5. - Michel Marcus, Jan 20 2017

f:= proc(n) local p;
    p:= 2;
    while numtheory:-phi(p*n) mod n <> 0 do p:= nextprime(p) od:
    p
end proc:
map(f, [$1..100]); # Robert Israel, Jan 12 2017

lpp[n_]:=Module[{p=2},While[Mod[EulerPhi[p*n],n]!=0,p=NextPrime[p]];p]; Array[lpp,80] (* Harvey P. Dale, Sep 26 2020 *)

a(n)=my(k = 1); while (eulerphi(prime(k)*n) % n != 0, k++); prime(k);

a(n)=my(t=n/gcd(eulerphi(n),n)); if(t==1, return(2)); forstep(p=if(t%2,2*t,t)+1, if(isprime(t), t, oo),lcm(t,2), if(isprime(p), return(p))); t \\ Charles R Greathouse IV, Jan 20 2017

a(p^k) = p for all primes p and k >= 1. - Robert Israel, Jan 12 2017

a(n) << n^5 by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Jan 20 2017

A283808 Numbers k such that phi(phi(k)) divides k, where phi(k) is A000010(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 32, 36, 40, 48, 54, 56, 64, 72, 80, 96, 108, 112, 128, 144, 160, 162, 192, 216, 224, 256, 288, 320, 324, 384, 432, 448, 486, 512, 576, 640, 648, 768, 864, 896, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1728, 1792, 1944, 2048, 2304, 2560

Offset: 1

Giovanni Resta, Mar 17 2017

nonn

M. Hausman has proved (see Links) that a number belongs to this sequence if and only if it is of one of the following forms: 2^s, 2^s * 3^t, 5 * 2^t, or 7 * 2^t , where s >= 0 and t >= 1.

			56 is in the sequence because phi(phi(56)) = 8 divides 56.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Miriam Hausman, The solution of a special arithmetic equation, Canad. Math. Bull., 25(1) (1982), 114-117.

Cf. A010554, A007694, A000010, A019278.

Select[Range[1000], Mod[#, EulerPhi@ EulerPhi@ #] == 0 &]

alias(e, eulerphi);
for(n = 1, 1000, if(!Mod(n,e(e(n))), print1(n,", "))) \\ Indranil Ghosh, Mar 18 2017

from sympy import totient as e
print([n for n in range(1, 1001) if n%e(e(n))==0]) # Indranil Ghosh, Mar 18 2017

Sum_{n>=1} 1/a(n) = 667/210. - Amiram Eldar, Dec 13 2024

cp's OEIS Frontend

A067606 Primes p such that p+7 == 0 (mod phi(p+7)).

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A067932 Primes p such that p+3 == 0 (mod phi(p+3)).

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A069924 Number of k, 1<=k<=n, such that phi(k) divides k.

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A145853 Numbers m such that m is a multiple of all integers smaller than the largest prime dividing m.

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Magma

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A167211 Numbers k such that number of perfect partitions of k-1 divides k.

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A245047 Numbers n where phi(n)|n or tau(n)|n.

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Maple

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A263811 Numbers k such that k = tau(k) * phi(k-1) + 1.

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Magma

Mathematica

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A275245 Numbers k such that phi(k) divides k^2 while phi(k) does not divide k.

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