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 1-3 of 3 results.

A247822 Corresponding values of primes p from A247821 and A247838.

Original entry on oeis.org

7, 8191, 8191, 131071, 524287, 524287, 2147483647, 2147483647, 2305843009213693951, 2305843009213693951, 2305843009213693951, 2305843009213693951
Offset: 1

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Author

Jaroslav Krizek, Sep 28 2014

Keywords

Comments

Conjecture: all terms are Mersenne primes (A000668).
Conjecture: next terms are 2305843009213693951, 2305843009213693951, 2305843009213693951, 2305843009213693951 and 618970019642690137449562111. - Jaroslav Krizek, Mar 25 2015

Examples

			a(2) = 8191 because sigma(sigma(2*A247821(2)-1)) = sigma(sigma(A247838(2))) = 8191.

Crossrefs

Cf. A000203, A008438, A247790, A247791, A247820, A247821, A247823, A247954.

Programs

  • Magma
    [SumOfDivisors(SumOfDivisors(n)): n in [A247838(n)]];

Formula

a(n) = sigma(sigma(2*A247821(n)-1)) = A000203(A000203(2*A247821(n)-1)) = A051027(2*A247821(n)-1).
a(n) = sigma(sigma(A247838(n))) = A000203(A000203(A247838(n))) = A051027(A247838(n)).

Extensions

a(7)-a(8) from Jaroslav Krizek, Mar 25 2015
a(9)-a(12) from Giovanni Resta, Feb 14 2020

A247821 Numbers k such that sigma(sigma(2k-1)) is a prime p.

Original entry on oeis.org

2, 1334, 1969, 28669, 86006, 126961, 338603654, 536801281, 366479720500691270, 375344017599431990, 500461553802019261, 554079264075351985
Offset: 1

Views

Author

Jaroslav Krizek, Sep 28 2014

Keywords

Comments

Numbers n such that A000203(A000203(2n-1)) = A000203(A008438(n-1)) = A051027(2n-1) is a prime p.
Corresponding values of primes p are 7, 8191, 8191, 131071, 524287, 524287, ... (= A247822). Conjecture: The primes p are Mersenne primes (A000668).
sigma(sigma(2*a(9)-1)) > 10^16.
If the above conjecture is true, the next terms are 366479720500691270, 375344017599431990, 500461553802019261, 554079264075351985, 98375588019240949991670086, ... . - Hiroaki Yamanouchi, Oct 01 2014
a(13) > 5*10^18. - Giovanni Resta, Feb 14 2020

Examples

			Number 1334 is in sequence because sigma(sigma(2*1334-1)) = sigma(sigma(2667)) = sigma(4096) = 8191, i.e., prime.

Crossrefs

Cf. A000203, A008438, A247790, A247791, A247820, A247822, A247823, A247954.

Programs

  • Magma
    [n: n in [1..10000000] | IsPrime(SumOfDivisors(SumOfDivisors(2*n-1)))]
  • Mathematica
    Select[Range[10^6], PrimeQ[DivisorSigma[1, DivisorSigma[1, 2 # - 1]]] &] (* Robert Price, May 17 2019 *)
  • PARI
    for(n=1,10^7,if(ispseudoprime(sigma(sigma(2*n-1))),print1(n,", "))) \\ Derek Orr, Sep 29 2014

Formula

a(n) = (A247838(n) +1) / 2.
a(n)-1 = numbers n such that sigma(sigma(2n+1)) is a prime p: 1, 1333, 1968, 28668, 86005, 126960, ...

Extensions

a(7)-a(8) from Hiroaki Yamanouchi, Oct 01 2014
a(9)-a(12) from Giovanni Resta, Feb 14 2020

A247823 Mersenne primes p such that there is a number k with sigma(sigma(2k-1)) = p.

Original entry on oeis.org

7, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111
Offset: 1

Views

Author

Jaroslav Krizek, Sep 28 2014

Keywords

Comments

Mersenne primes p such that there is a number m such that sigma(sigma(m)) = p.
Distinct values attained by the A247822(n) function, in ascending order.
Mersenne primes p such that there are a numbers n and m such that sigma(sigma(2n-1)) = sigma(sigma(2*A247821(n)-1)) = A000203(A000203(2*A247821(n)-1)) = A051027(2*A247821(n)-1) = sigma(sigma(A247838(m))) = A000203(A000203(A247838(m))) = A051027(A247838(m)) where m = 2n-1.
The Mersenne prime 7 is the only prime p such that there is a prime q with sigma(sigma(q)) = p.

Examples

			Mersenne prime 8191 is in sequence because there are numbers n = 1334 and 1969 with sigma(sigma(2*n-1)) = 8191.

Crossrefs

Cf. A000203, A008438, A247790, A247791, A247820, A247821, A247822, A247954.
Cf. A000668 (Mersenne primes).

Programs

  • Magma
    Set(Sort([SumOfDivisors(SumOfDivisors(n)): n in [1..10000000] | IsPrime(SumOfDivisors(SumOfDivisors(n)))])) // Jaroslav Krizek, Mar 25 2015

Extensions

a(5)-a(7) from Jaroslav Krizek, Mar 25 2015
Showing 1-3 of 3 results.

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