A062514 -id:A062514 - cp's OEIS frontend

A249759 Primes p such that sigma(p-1) is a prime q.

3, 5, 17, 65537

Offset: 1

Jaroslav Krizek, Nov 13 2014

Subsequence of {A023194(n)+1}.
Conjectures: 1) sequence is finite; 2) sequence is a subsequence of A019434 (Fermat primes); 3) sequence consists of Fermat primes p such that sigma(p-1) is a Mersenne prime; 4) a(n) = (A249761(n)+3)/2.
3 is the only prime p such that sigma(p+1) is prime, i.e., 3 is the only prime p such that sigma(p-1) and sigma(p+1) are both primes.
Conjecture: 3 is the only number n such that n and sigma(n+1) are both prime.
Primes p such that A051027(p-1) = sigma(sigma(p-1)) = 2*(p-1). Subsequence of A256438. - Jaroslav Krizek, Mar 29 2015
From Jaroslav Krizek, Mar 17 2016: (Start)
Primes p such that A000203(A000010(p)) = sigma(phi(p)) is a prime.
Prime terms from A062514 and A270413, A270414, A270415 and A270416. (End)
From Jaroslav Krizek, Nov 27 2016: (Start)
Corresponding values of primes q are in A249761: 3, 7, 31, 131071, ...
Conjecture: subsequence of A256438 and A278741.
Conjecture: also primes p such that tau(p-1) is a prime q; corresponding values of primes q are 2, 3, 5, 17. (End)

			Prime 17 is in the sequence because sigma(17-1) = sigma(16) = 31 (prime).

Cf. A000203, A000668, A019434, A023194, A249760, A249761.

[p: p in PrimesUpTo(1000000) | IsPrime(SumOfDivisors(p-1))]

with(numtheory): A249759:=n->`if`(isprime(n) and isprime(sigma(n-1)), n, NULL): seq(A249759(n), n=1..6*10^5); # Wesley Ivan Hurt, Nov 14 2014

Select[Range[10^5], PrimeQ[#]&& PrimeQ[DivisorSigma[1, # - 1]] &] (* Vincenzo Librandi, Nov 14 2014 *)
Select[Prime[Range[7000]],PrimeQ[DivisorSigma[1,#-1]]&] (* Harvey P. Dale, Jun 14 2020 *)

lista(nn) = {forprime(p=1, nn, if (isprime(sigma(p-1)), print1(p, ", ")););} \\ Michel Marcus, Nov 14 2014

a(n) = A249760(n) + 1.

Sigma(a(n)-1) = A249761(n).

A270414 Numbers m such that sigma(m-1) and sigma(phi(m)) are both primes.

Original entry on oeis.org

3, 5, 10, 17, 65537

Offset: 1

Jaroslav Krizek, Mar 16 2016

Numbers n such that A000203(n-1) and A062402(n) are both primes.
There are no other terms <= 10^7.
Intersection of A270413 and A062514.
Prime terms are in A249759.
Corresponding values of sigma(n-1): 3, 7, 13, 31, 131071, ...
Corresponding values of sigma(phi(n)): 3, 7, 7, 31, 131071, ...
Conjecture: union of number 10 and A249759.

			10 is in the sequence because sigma(10-1) = sigma(9) = 13 and sigma(phi(10)) = sigma(4) = 7 (both primes).

Cf. A000010, A000203, A062514, A249759, A256438, A270413.

[n: n in [2..100000] |  IsPrime(SumOfDivisors(n-1)) and IsPrime(SumOfDivisors(EulerPhi(n)))];

Select[Range[10^6], And[PrimeQ@ DivisorSigma[1, # - 1], PrimeQ@ DivisorSigma[1, EulerPhi@ #]] &] (* Michael De Vlieger, Mar 17 2016 *)

isok(n) = isprime(sigma(n-1)) && isprime(sigma(eulerphi(n))); \\ Michel Marcus, Mar 17 2016

A065875 Numbers k such that usigma(phi(k)) is a prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 12, 17, 32, 34, 40, 48, 60, 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

Offset: 1

Robert G. Wilson v, Dec 07 2001

nonn

The only odd terms below 10^7 are 3, 5, 17, 257 and 65537.

Numbers k such that phi(k) = 2^(2^m) where 2^(2^m)+1 is a Fermat prime (A019434). a(42) >= 2^(2^33) + 1, if a 6th Fermat prime exists. - Amiram Eldar, Dec 14 2024

Cf. A062514, A034448, A000010, A019434.

u(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d));
for(n=1,10^5,if(isprime(u(eulerphi(n))),print1(n,", "))); \\ Joerg Arndt, Sep 17 2023

Deleted incorrect MMA program. - N. J. A. Sloane, Sep 17 2023

A270416 Numbers n such that sigma(n) - 1 and sigma(phi(n)) are both primes.

Original entry on oeis.org

3, 5, 6, 10, 17, 34, 40, 60, 85, 136, 204, 240, 4369, 8224, 8704, 8738, 10880, 12336, 13056, 65537, 131074, 131584, 139264, 163840, 164480, 174760, 208896, 245760, 262140, 524296, 526336, 559232, 835584, 838848, 2281736192, 2694881440, 2852170240, 2863267840, 3221274624, 3233857728, 4026593280

Offset: 1

Jaroslav Krizek, Mar 16 2016

nonn

Numbers n such that A039653(n) and A062402(n) are both primes.
Intersection of A248792 and A062514.
Prime terms are in A249759.
Corresponding values of sigma(n) - 1: 3, 5, 11, 17, 17, 53, 89, 167, ...
Corresponding values of sigma(phi(n)): 3, 7, 3, 7, 31, 31, 31, 31, 127, ...

			10 is in the sequence because sigma(10) - 1 = 18 - 1 = 17 and sigma(phi(10)) = sigma(4) = 7 (both primes).

Cf. A000203, A039653, A062402, A062514, A248792, A249759.

Select[Range[10^6], And[PrimeQ[DivisorSigma[1, #] - 1], PrimeQ@ DivisorSigma[1, EulerPhi@ #]] &] (* Michael De Vlieger, Mar 17 2016 *)

isok(n) = isprime(sigma(n)-1) && isprime(sigma(eulerphi(n))); \\ Michel Marcus, Mar 17 2016

a(35)-a(41) from Giovanni Resta, Apr 10 2016

A281627 a(n) is the smallest number k such that sigma(phi(k)) = A062402(k) is the n-th Mersenne prime (A000668(n)), or 0 if no such k exists.

Original entry on oeis.org

3, 5, 17, 85, 4369, 65537, 327685, 1431655765, 2305843009213693952, 618970019642690137449562112, 162259276829213363391578010288128, 170141183460469231731687303715884105728

Offset: 1

Jaroslav Krizek, Feb 11 2017

nonn

Conjecture 1: If A062402(n) = A000203(A000010(n)) = sigma(phi(n)) is a prime p for some n, then p is Mersenne prime (A000668); a(n) > 0 for all n.
Conjecture 2: a(n) = the smallest number k such that phi(k) has exactly A000043(n)-1 divisors; see A276044.
Conjecture 3: a(n) = the smallest number k such that phi(k) has exactly A000043(n)-1 prime factors (counted with multiplicity); see A275969.
a(n) <= A000668(n) for n = 1-18; conjecture: a(n) <= A000668(n) for all n.
Equals A002181 (index in A002202 of (intersection of A023194 and A002202)). - Michel Marcus, Feb 12 2017

Amiram Eldar, Table of n, a(n) for n = 1..13
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A000043, A000203, A000668, A062402, A062514, A275969, A276044.
Cf. A002181, A002202, A023194.
Cf. A053576 (includes the first 13 known terms of this sequence).

A281627:=func; Set(Sort([A281627(n): n in [SumOfDivisors(EulerPhi(n)): n in[1..1000000] | IsPrime(SumOfDivisors(EulerPhi(n)))]]));

terms() = {v = readvec("b023194.txt"); for(i=1, #v, if (istotient(v[i], &n), print1(n/2, ", ")););} \\ Michel Marcus, Feb 12 2017

f(p) = {my(s = invsigma(p), t, minv = 0); for(i = 1 ,#s, t = invphi(s[i]); for(j = 1, #t, if(minv == 0, minv = t[j]); if(t[j] < minv, minv = t[j]))); minv;} \\ using Max Alekseyev's invphi.gp
list(pmax) = forprime(p = 1, pmax, if(isprime(2^p-1), print1(f(2^p-1), ", "))); \\ Amiram Eldar, Dec 23 2024

a(8) from Michel Marcus, Feb 12 2017

a(9)-a(12) from Amiram Eldar, Dec 23 2024

A300217 Numbers k such that tau(phi(k)) is a prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 12, 17, 32, 34, 40, 48, 60, 85, 128, 136, 160, 170, 192, 204, 240, 1285, 2048, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 4080, 4369, 8192, 8224, 8704, 8738, 10240, 10280, 10880, 12288, 12336, 13056, 15360, 15420, 16320, 65537

Offset: 1

Jaroslav Krizek, Feb 28 2018

nonn

Numbers k such that A062821(k) = A000005(A000010(k)) is a prime.
Supersequence of A062514.
From Robert Israel, Mar 18 2018: (Start)
Numbers k such that A000010(k) = 2^(p-1) where p is prime.
Numbers of the form 2^m*Product_{i=1..k} (2^(2^(e_i))+1) where 2^(2^(e_i)+1) are distinct Fermat primes (A019434) and m + 1 + Sum_i 2^(e_i) is prime.  In particular the prime terms are A249759.
(End)
According to a comment in A009087, if the sum of divisors is prime, then the number of divisors is also prime. - Michael B. Porter, Mar 23 2018

			17 is a term because phi(17) = 16, tau(16) = 5 (prime).

Jaroslav Krizek, Table of n, a(n) for n = 1..100

Cf. A000005, A000010, A062514, A062821, A019434, A249759.

[n: n in[1..10^6] | IsPrime(NumberOfDivisors(EulerPhi(n)))];

select(isprime @ numtheory:-tau @ numtheory:-phi, [$1..10^5]); # Robert Israel, Mar 18 2018

Select[Range[2^16 + 1], PrimeQ@ DivisorSigma[0, EulerPhi@ #] &] (* Michael De Vlieger, Mar 01 2018 *)

isok(k) = isprime(numdiv(eulerphi(k))); \\ Altug Alkan, Mar 04 2018

cp's OEIS Frontend

A249759 Primes p such that sigma(p-1) is a prime q.

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A270414 Numbers m such that sigma(m-1) and sigma(phi(m)) are both primes.

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A065875 Numbers k such that usigma(phi(k)) is a prime.

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A270416 Numbers n such that sigma(n) - 1 and sigma(phi(n)) are both primes.

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A281627 a(n) is the smallest number k such that sigma(phi(k)) = A062402(k) is the n-th Mersenne prime (A000668(n)), or 0 if no such k exists.

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A300217 Numbers k such that tau(phi(k)) is a prime.

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