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.

User: Jean Luc Garambois

Jean Luc Garambois's wiki page.

Jean Luc Garambois has authored 4 sequences.

A365351 Exponents e such that the aliquot sequence starting with 2^e ends with a prime number at index 2.

Original entry on oeis.org

6, 11, 18, 27, 41, 74, 157, 197, 294, 549, 581
Offset: 1

Author

Jean Luc Garambois, Sep 02 2023

Keywords

Comments

That is, exponents e such that s(s(2^e)) is prime, where s(n) = sigma(n)-n (A001065).
Note that exponents e such that aliquot sequences starting with 2^e end with a prime number at index 1 (exponents e such that s(2^e) is prime) are called "Mersenne exponents" (see A000043).
From Amiram Eldar, Sep 02 2023: (Start)
Numbers k such that 2^k - 1 is a term of A037020.
1206 < a(12) <= 2351 (2351 is a term). (End)

Crossrefs

Cf. A000043 (Mersenne exponents), A001065, A037020.

Programs

  • Mathematica
    Select[Range[100], PrimeQ[DivisorSigma[1, 2^# - 1] - 2^# + 1] &] (* Amiram Eldar, Sep 02 2023 *)
  • PARI
    f(n) = sigma(n) - n; \\ A001065
    isok(k) = ispseudoprime(f(f(2^k))); \\ Michel Marcus, Sep 02 2023
  • Sage
    def s(n):
        sn = sigma(n) - n
        return sn
    e = 1
    exponents_list = []
    while e<=200:
        m = 2^e
        index = 0
        if is_prime(s(s(m))):
            exponents_list.append(e)
        e+=1
    print (exponents_list)
    

A358266 Numbers k such that the aliquot sequence of 2^k ends with the prime 7.

Original entry on oeis.org

3, 10, 12, 141, 278, 387, 421
Offset: 1

Author

Jean Luc Garambois, Nov 06 2022

Keywords

Examples

			a(4)=141 because the aliquot sequence that starts with the integer 2^141 ends with the prime number 7 and there are only three smaller powers of 2 that do the same: 2^3, 2^10 and 2^12.
		

Crossrefs

Programs

  • PARI
    f(n) = if (n==1, return(2)); my(list = List(), s=2^n); for (i=1, oo, s = sigma(s) - s; if (#select(x->(x==s), list), return(0)); if (isprime(s), return (s)); listput(list, s); );
    isok(m) = f(m) == 7;

Formula

Define s(i) = sigma(i) - i = A000203(i) - i. Then k is a term of this sequence if the aliquot sequence obtained by repeatedly applying the mapping i->s(i) taking as initial value 2^k terminates in the prime 7.

A358239 Numbers k such that the aliquot sequence of 2^k ends with the prime 3.

Original entry on oeis.org

2, 4, 55, 164, 305, 317
Offset: 1

Author

Jean Luc Garambois, Nov 04 2022

Keywords

Examples

			a(3)=55 because the aliquot sequence that starts with the integer 2^55 ends with the prime number 3 and there are only two smaller powers of 2 that do the same: 2^2 and 2^4.
		

Crossrefs

Programs

  • PARI
    f(n) = if (n==1, return(2)); my(list = List(), s=2^n); for (i=1, oo, s = sigma(s) - s; if (#select(x->(x==s), list), return(0)); if (isprime(s), return (s)); listput(list, s););
    isok(m) = f(m) == 3; \\ Michel Marcus, Nov 05 2022

Formula

Define s(i) = sigma(i) - i = A000203(i) - i. Then k is a term of this sequence if the aliquot sequence obtained by repeatedly applying the mapping i->s(i) taking as initial value 2^k terminates in the prime 3.

A238382 Untouchable amicable numbers: amicable pairs which cannot be reached by any aliquot sequence starting from a number that does not belong to this pair.

Original entry on oeis.org

356408, 399592, 643336, 652664, 5232010, 5799542, 9363584, 9437056, 10596368, 11199112, 15363832, 16517768, 31818952, 32205616, 34352624, 34860248, 46237730, 48641584, 48852176, 49215166, 52695376, 55349570, 56208368, 61319902, 91996816, 93259184
Offset: 1

Author

Jean Luc Garambois, Mar 03 2014

Keywords

Comments

A pair of numbers x and y is called an untouchable amicable pair if x and y are amicable numbers (see A063990) and if x and y each have only one aliquot antecedent: the other number of their pair. In other words, they are amicable pairs which no aliquot sequence starting on a number that does not belong to this pair can reach.
The sequence lists the untouchable amicable numbers in increasing order. Note that the pairs x, y are not always adjacent to each other in the list.
Numbers that are the smaller number of their untouchable amicable pair are 356408, 643336, 5232010, 9363584, 10596368, 15363832, 31818952, 32205616, ... (subsequence of A002025).
The remaining numbers in the sequence are the larger number of their untouchable amicable pair: 399592, 652664, 5799542, 9437056, 11199112, 16517768, 34860248, 34352624, ... (subsequence of A002046).
We can call those pairs "untouchable amicable pairs", "isolated amicable pairs", or "isolated amicable cycles with two links" ... .

Crossrefs