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 1-7 of 7 results.

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

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

Views

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

A247790 Primes p such that sigma(sigma(2p-1)) is a prime.

Original entry on oeis.org

2, 28669, 126961, 500461553802019261
Offset: 1

Views

Author

Jaroslav Krizek, Sep 28 2014

Keywords

Comments

The next term, if it exists, must be greater than 5*10^7.
Primes p such that A247954(p) = A000203(A000203(2p-1)) = A000203(A008438(p-1)) = A051027(2p-1) is a prime q. The corresponding values of the primes q are: 7, 131071, 524287, ... (A247791). Conjecture: the primes q are Mersenne primes (A000668).
Conjecture: the next term is 500461553802019261 (see comment from Hiroaki Yamanouchi in A247821). - Jaroslav Krizek, Oct 08 2014
These are the primes in A247821. - M. F. Hasler, Oct 14 2014
No other terms up to 5*10^10. - Michel Marcus, Feb 11 2020
a(5) > 5*10^18. - Giovanni Resta, Feb 14 2020

Examples

			Prime 2 is in the sequence because sigma(sigma(2*2-1)) = sigma(sigma(3)) = sigma(4) = 7, i.e., prime.

Crossrefs

Cf. A000203, A008438, A247791, A247820, A247821, A247822, A247823, A247954.

Programs

  • Magma
    [p: p in PrimesUpTo(50000000) | IsPrime(SumOfDivisors(SumOfDivisors(2*p-1)))]
  • Maple
    with(numtheory): A247790:=n->`if`(isprime(n) and isprime(sigma(sigma(2*n-1))),n,NULL): seq(A247790(n), n=1..130000); # Wesley Ivan Hurt, Oct 17 2014
  • PARI
    forprime(p=1,10^7,if(ispseudoprime(sigma(sigma(2*p-1))),print1(p,", "))) \\ Derek Orr, Sep 29 2014

Extensions

a(4) from Giovanni Resta, Feb 14 2020

A247791 Primes p such that there is a prime q for which sigma(sigma(2*q-1)) = p.

Original entry on oeis.org

7, 131071, 524287
Offset: 1

Views

Author

Jaroslav Krizek, Sep 28 2014

Keywords

Comments

The next term, if it exists, must be greater than 5*10^7.
Primes p such that there is prime q for which sigma(sigma(2*q-1)) = A247954(q) = A000203(A000203(2*q-1)) = A000203(A008438(q-1)) = A051027(2*q-1) = p.
Corresponding values of primes q: 2, 28669, 126961, ... (A247790).
Conjecture: Subsequence of Mersenne primes.
Conjecture: the next term is 2305843009213693951 when 2305843009213693951 = sigma(sigma(2*500461553802019261-1)) where 500461553802019261 is prime (see comment of Hiroaki Yamanouchi in A247821). - Jaroslav Krizek, Oct 08 2014

Examples

			Prime 7 is in sequence because there is prime 2 such that sigma(sigma(2*2-1)) = sigma(sigma(3)) = sigma(4) = 7.

Crossrefs

Cf. A000203, A008438, A247790, A247820, A247821, A247822, A247823, A247954.

Programs

  • Magma
    [SumOfDivisors(SumOfDivisors(2*n-1)): n in [A247790(n)]];
  • Magma
    [SumOfDivisors(SumOfDivisors(2*n-1)): n in[1..1000000] | IsPrime(SumOfDivisors(SumOfDivisors(2*n-1))) and IsPrime(n)];
  • PARI
    forprime(p=1,10^7,if(ispseudoprime(sigma(sigma(2*p-1))),print1(sigma(sigma(2*p-1)),", "))) \\ Derek Orr, Sep 29 2014

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

A247838 Numbers k such that sigma(sigma(k)) is prime.

Original entry on oeis.org

3, 2667, 3937, 57337, 172011, 253921, 677207307, 1073602561, 732959441001382539, 750688035198863979, 1000923107604038521, 1108158528150703969
Offset: 1

Views

Author

Jaroslav Krizek, Sep 28 2014

Keywords

Comments

Numbers k such that A051027(k) is a prime p.
Prime 3 is the only prime p such that sigma(sigma(p)) is a prime q.
Conjecture: Subsequence of A046528 (numbers that are a product of distinct Mersenne primes).
Corresponding values of primes p: 7, 8191, 8191, 131071, 524287, 524287, ... (A247822). Conjecture: values of primes p is equal to Mersenne primes (A000668).
732959441001382539, 750688035198863979, 1000923107604038521, 1108158528150703969 and 196751176038481899983340171 are terms. - Jaroslav Krizek, Mar 25 2015
a(9) > 10^10. - Michel Marcus, Feb 13 2020
a(13) > 10^19. - Giovanni Resta, Feb 14 2020

Examples

			2667 is a term because sigma(sigma(2667)) = sigma(4096) = 8191 (i.e., prime).

Crossrefs

Cf. A000203, A023194, A063103, A000668, A046528, A051027, A247821, A247822, A247954.

Programs

  • Magma
    [n: n in [1..10000000] | IsPrime(SumOfDivisors(SumOfDivisors(n)))];
  • Maple
    with(numtheory): A247838:=n->`if`(isprime(sigma(sigma(n))),n,NULL): seq(A247838(n), n=1..10^5); # Wesley Ivan Hurt, Oct 02 2014
  • Mathematica
    Select[Range[260000],PrimeQ[DivisorSigma[1,DivisorSigma[1,#]]]&] (* The program generates the first six terms of the sequence. *) (* Harvey P. Dale, Jan 18 2024 *)
  • PARI
    isok(n) = isprime(sigma(sigma(n))); \\ Michel Marcus, Oct 01 2014

Formula

a(n) = 2*A247821(n)-1.

Extensions

a(7)-a(8) from Michel Marcus, Oct 02 2014
a(9)-a(12) from Giovanni Resta, Feb 14 2020

A247955 Primes p such that there is prime q with sigma(q+2) = p.

Original entry on oeis.org

7, 13, 31, 1093, 2801, 5113, 8011, 17293, 30103, 30941, 86143, 459007, 552793, 579883, 732541, 1191373, 3500201, 3730693, 4534771, 5168803, 5333791, 7450171, 10378063, 25646167, 25882657, 28792661, 30266503, 43553401, 48037081, 52265671, 56964757, 62433703, 65504743, 67856407, 76413823, 77572057
Offset: 1

Views

Author

Jaroslav Krizek, Sep 28 2014

Keywords

Comments

Primes p such that there is prime q such that A000203(q+2) = p.
Primes p of the form sigma(A171130(n)) in increasing order.

Crossrefs

Cf. A000203, A008438, A171130, A247791, A247820, A247821, A247822, A247823, A247954.

Programs

  • Mathematica
    Sort[Select[DivisorSigma[1,#+2]&/@Prime[Range[5200000]],PrimeQ]] (* Harvey P. Dale, Apr 27 2022 *)
  • PARI
    v=[];forprime(p=1,10^8,if(ispseudoprime(sigma(p+2)),v=concat(v,sigma(p+2))));v \\ Derek Orr, Oct 26 2014

Extensions

More terms from Michel Marcus, Oct 02 2014
Corrected and extended by Harvey P. Dale, Apr 27 2022
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.