A249759 -id:A249759 - cp's OEIS frontend

A256438 Numbers m such that sigma(sigma(m-1)) = 2*(m-1).

3, 5, 17, 65, 4097, 65537, 262145, 1073741825, 1152921504606846977, 309485009821345068724781057, 81129638414606681695789005144065, 85070591730234615865843651857942052865

Offset: 1

Jaroslav Krizek, Mar 29 2015

Numbers k such that A051027(k-1) = 2*(k-1).
Conjecture: numbers of the form 2^k+1 such that sigma(2^k) = prime p.
Prime terms: 3, 5, 17, 65537, ...
Supersequence of A249759.

			17 is in the sequence because sigma(sigma(17-1)) = 32 = 2*(17-1).

Cf. A000203, A051027, A019279, A249759.

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

with(numtheory): A256438:=n->`if`(sigma(sigma(n-1)) = 2*(n-1), n, NULL): seq(A256438(n), n=2..10^5); # Wesley Ivan Hurt, Mar 30 2015

Select[Range@ 1000000, DivisorSigma[1, DivisorSigma[1, # - 1]] == 2 (# - 1) &] (* Michael De Vlieger, Mar 29 2015 *)

isok(m) = sigma(sigma(m-1)) == 2*(m-1); \\ Michel Marcus, Feb 09 2020

a(n) = A019279(n) + 1. - Michel Marcus, Feb 09 2020

A270413 Numbers m such that sigma(m-1) is a prime.

Original entry on oeis.org

3, 5, 10, 17, 26, 65, 290, 730, 1682, 2402, 3482, 4097, 5042, 7922, 10202, 15626, 17162, 27890, 28562, 29930, 65537, 83522, 85850, 146690, 262145, 279842, 458330, 491402, 531442, 552050, 579122, 597530, 683930, 703922, 707282, 734450, 829922, 1190282, 1203410

Offset: 1

Jaroslav Krizek, Mar 16 2016

nonn

Prime terms are in A249759.
Corresponding values of sigma(n-1): 3, 7, 13, 31, 31, 127, 307, 1093, ...
Conjecture: supersequence of A256438.
Conjecture: 31 is the only prime p such that p = sigma(x-1) = sigma(y-1) for distinct numbers x and y; 31 = sigma(17-1) = sigma(26-1).
Supersequence of A270414 and A270415.

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

Cf. A000203, A023194, A249759, A256438, A270414, A270415.

[n: n in [2..2000000] |  IsPrime(SumOfDivisors(n-1))];

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

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

a(n) = A023194(n) + 1.

A258429 Primes p such that p - 1 = (tau(p - 1) - 1)^k for some k >= 0, where tau(n) is the number of divisors of n (A000005).

Original entry on oeis.org

2, 5, 17, 65537

Offset: 1

Jaroslav Krizek, May 29 2015

nonn

Conjecture: the sequence is finite.
Corresponding values of numbers k: 0, 2, 2, 4, ...
A Fermat prime from A019434 of the form F(n) = 2^(2^n) + 1 is a term if k = 2^n * log(2) / log(2^n) is an integer.

			65537 (prime) is in the sequence because 65537 - 1 = (tau(65536) - 1)^4 = 16^4.

Cf. A000005, A019434, A219338, A007516, A004249, A249759.

[2] cat [n+1: n in [A219338(n)] | IsPrime(n+1)];

Set(Sort([n: n in[1..1000000], k in [0..100] | IsPrime(n) and (n-1) eq (NumberOfDivisors(n-1) - 1)^k]));

listp(nn) = {print1(p=2, ", "); forprime(p=5, nn, expo = valuation(x=(p-1), y=(numdiv(p-1)-1)); if (x == y^expo, print1(p, ", ")););} \\ Michel Marcus, Jun 04 2015

A278741 Odd numbers k such that tau(k-1) is a prime.

Original entry on oeis.org

3, 5, 17, 65, 1025, 4097, 65537, 262145, 4194305, 268435457, 1073741825, 68719476737, 1099511627777, 4398046511105, 70368744177665, 4503599627370497, 288230376151711745, 1152921504606846977, 73786976294838206465, 1180591620717411303425, 4722366482869645213697

Offset: 1

Jaroslav Krizek, Nov 27 2016

nonn

tau(k) = A000005(k) = the number of divisors of k.
Conjecture: prime terms are in A249759: 3, 5, 17, 65537, ...
Supersequence of A256438 and A249759. Subsequence of {A009087(n) + 1}.

			Odd number 65 is in the sequence because tau(64) = 7 (prime).

Cf. A000005, A000040, A001348, A009087, A061286, A249759, A256438.

[n: n in[2..10000000] |  IsOdd(n) and IsPrime(NumberOfDivisors(n-1))];

isok(n) = (n % 2) && isprime(numdiv(n-1)); \\ Michel Marcus, Nov 27 2016

a(n) = A061286(n) + 1.
sigma(a(n)-1) = A001348(n), i.e., Mersenne numbers.
tau(a(n)-1) = A000040(n), i.e., all primes; a(n) = the smallest odd number k such that tau(a(n)-1) = prime(n) = A000040(n).

A249760 Numbers k such that k+1 and sigma(k) are both primes.

Original entry on oeis.org

2, 4, 16, 65536

Offset: 1

Jaroslav Krizek, Nov 13 2014

4 is the only number k such that k-1 and sigma(k) are both primes.
Corresponding values of k+1 and sigma(k) are in A249759 and A249761.
Conjectures: (1) sequence is finite; (2) a(n) + 1 is a Fermat prime (A019434); (3) sigma(a(n)) is a Mersenne prime (A000668).
Subsequence of A023194, and by a comment in that entry it follows that each term is a prime power. From that conjectures (2) and (3) above easily follow. - Jeppe Stig Nielsen, Jan 13 2015

			16 is a term because 16+1=17 and sigma(16)=31 are both primes.

Cf. A000203, A000668, A019434, A249759, A249761.

[n: n in [1..10^8] | IsPrime(n+1) and IsPrime(SumOfDivisors(n))];

Select[Range[10^5], PrimeQ[# + 1]&& PrimeQ[DivisorSigma[1, #]] &] (* Vincenzo Librandi, Nov 14 2014 *)

a(n) = A249759(n) - 1.

A249761 Primes p such that there is prime q with sigma(q-1) = p.

Original entry on oeis.org

3, 7, 31, 131071

Offset: 1

Jaroslav Krizek, Nov 13 2014

a(n) = corresponding values of sigma(n) of numbers n from A249760(n); a(n) = A000203(A249760(n)).
Corresponding values of primes q are in A249759.
Conjectures: 1) sequence is finite; 2) (a(n)+3)/2 is a Fermat prime (A019434); 3) a(n) is subsequence of Mersenne primes (A000668); 4) a(n) = 2*A249759(n)-3.

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

Cf. A000203, A000668, A019434, A249759, A249760.

[a: p in PrimesUpTo(5000000) | IsPrime(a) where a is  SumOfDivisors(p-1)]

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

A270415 Numbers n such that sigma(n-1) and sigma(n) - 1 are both primes.

Original entry on oeis.org

3, 5, 10, 17, 26, 65, 65537, 146690, 703922, 1481090, 1885130, 2036330, 2211170, 2430482, 2505890, 5470922, 9840770, 11607650, 17783090, 24137570, 74425130, 76615010, 77563250, 133379402, 138697730, 138980522, 142396490, 155575730, 177715562, 181899170

Offset: 1

Jaroslav Krizek, Mar 16 2016

nonn

Numbers n such that A000203(n-1) and A039653(n) are both primes.
Intersection of A270413 and A248792.
Prime terms are in A249759.
Corresponding values of sigma(n-1): 3, 7, 13, 31, 31, 127, 131071, ...
Corresponding values of sigma(n) - 1: 3, 5, 17, 17, 41, 83, 65537, ...

			17 is in the sequence because sigma(17-1) = sigma(16) = 31 and sigma(10) - 1 = 18 - 1 = 17 (both primes).

Cf. A000203, A039653, A248792, A249759, A270413.

[n: n in [2..10000000] | IsPrime(SumOfDivisors(n-1)) and  IsPrime(SumOfDivisors(n)-1)];

Select[Range[10^7], And[PrimeQ@ DivisorSigma[1, # - 1], PrimeQ[DivisorSigma[1, #] - 1]] &] (* Michael De Vlieger, Mar 17 2016 *)

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

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

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

A256438 Numbers m such that sigma(sigma(m-1)) = 2*(m-1).

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A270413 Numbers m such that sigma(m-1) is a prime.

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A258429 Primes p such that p - 1 = (tau(p - 1) - 1)^k for some k >= 0, where tau(n) is the number of divisors of n (A000005).

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A278741 Odd numbers k such that tau(k-1) is a prime.

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A249760 Numbers k such that k+1 and sigma(k) are both primes.

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Magma

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A249761 Primes p such that there is prime q with sigma(q-1) = p.

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

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A270415 Numbers n such that sigma(n-1) and sigma(n) - 1 are both primes.