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: Felix Fröhlich

Felix Fröhlich's wiki page.

Felix Fröhlich has authored 734 sequences. Here are the ten most recent ones:

A365514 Lucas-V pseudoprimes: composites c such that V_{c+1} == 2Q (mod c), where V_k is a Lucas sequence with parameters P and Q.

Original entry on oeis.org

913, 150267335403, 430558874533, 14760229232131, 936916995253453
Offset: 1

Views

Author

Felix Fröhlich, Sep 07 2023

Keywords

Comments

The values are given in Table 2 in Baillie et al., 2021.
There are only five terms less than 10^15.

Links

Crossrefs

Cf. A217120.

A365098 Primes p such that Sum_{k=1..p-1} q^2_p(k) == 0 (mod p), with q_p(k) a Fermat quotient.

Original entry on oeis.org

2, 11, 971
Offset: 1

Views

Author

Felix Fröhlich, Aug 21 2023

Keywords

Comments

The congruence in the definition is given in Gy, 2018, Eq. 16.
The terms, except for the prime 2, satisfy the congruence B_{p-1} - 1 + 1/p == (B_{2p-2} - 1 + 1/p)/2 (mod p^2), with B_i a Bernoulli number (cf. Gy, 2018, Eq. 18).
Any odd prime that is a term of both A007540 and A197632, i.e., that is simultaneously a Wilson prime and a Lerch prime, is in this sequence (cf. Gy, 2018, Theorem 5).
An equivalent definition, better suited for computational purposes, is: "Primes p such that Sum_{k=1..p-1} (k^(p-1) - 1)^2 == 0 (mod p^3)." - John Blythe Dobson, Apr 30 2024
a(4) > 427000, if it exists (Gy, 2018). - Amiram Eldar, Aug 22 2023
a(4) > 39540000, if it exists. - John Blythe Dobson, Apr 30 2024

Links

Crossrefs

Cf. A007540, A197632.

Programs

  • Mathematica
    Join[{2}, Select[Prime[Range[2, 200]], Divisible[Numerator[BernoulliB[# - 1] - 1 + 1/# - (BernoulliB[2*# - 2] - 1 + 1/#)/2], #^2] &]] (* Amiram Eldar, Aug 22 2023 *)
  • PARI
    forprime(p = 2, 10000, if(sum(j=1, p-1, (Mod(j, p^3)^(p-1) - 1)^2) % p^3 == 0, print1(p, ", "))) /* John Blythe Dobson, Apr 30 2024 */

A357365 Primes q such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where p = A151799(A151799(A151799(A151799(q)))).

Original entry on oeis.org

19, 67, 349, 2011, 22307, 13699249, 2018905087, 9809844767
Offset: 1

Views

Author

Felix Fröhlich, Sep 25 2022

Keywords

Crossrefs

Cf. A151799, A151800, A357362, A357363, A357364.

Programs

  • PARI
    is(n) = my(b=precprime(precprime(precprime(precprime(n-1)-1)-1)-1)); Mod(b, n^2)^(n-1)==1 || Mod(n, b^2)^(b-1)==1
forprime(p=11, , if(is(p), print1(p, ", ")))
  • Python
    from sympy import nextprime
from itertools import islice
def agen():
    p, m1, m2, m3, q = 2, 3, 5, 7, 11
    while True:
        if pow(p, q-1, q*q) == 1 or pow(q, p-1, p*p) == 1: yield q
        p, m1, m2, m3, q = m1, m2, m3, q, nextprime(q)
print(list(islice(agen(), 5))) # Michael S. Branicky, Sep 30 2022

Extensions

a(7)-a(8) from Michael S. Branicky, Sep 26 2022

A357362 Primes q such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where p = A151799(q).

Original entry on oeis.org

7, 53, 59, 151057, 240733, 911135857
Offset: 1

Views

Author

Felix Fröhlich, Sep 25 2022

Keywords

Crossrefs

Cf. A151799, A151800, A357363, A357364, A357365.

Programs

  • PARI
    is(n) = my(b=precprime(n-1)); Mod(b, n^2)^(n-1)==1 || Mod(n, b^2)^(b-1)==1
forprime(p=3, , if(is(p), print1(p, ", ")))
  • Python
    from sympy import nextprime
from itertools import islice
def agen():
    p, q = 2, 3
    while True:
        if pow(p, q-1, q*q) == 1 or pow(q, p-1, p*p) == 1: yield q
        p, q = q, nextprime(q)
print(list(islice(agen(), 5))) # Michael S. Branicky, Sep 30 2022

Extensions

a(6) from Michael S. Branicky, Sep 26 2022

A357364 Primes p such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where q = A151800(A151800(A151800(p))).

Original entry on oeis.org

11, 23, 41, 107, 389, 1987673, 35603983
Offset: 1

Views

Author

Felix Fröhlich, Sep 25 2022

Keywords

Crossrefs

Cf. A151800, A357362, A357363, A357365.

Programs

  • PARI
    is(n) = my(b=precprime(precprime(precprime(n-1)-1)-1)); Mod(b, n^2)^(n-1)==1 || Mod(n, b^2)^(b-1)==1
forprime(p=7, , if(is(p), print1(p, ", ")))

A357363 Primes p such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where q = A151800(A151800(p)).

Original entry on oeis.org

5, 19, 263, 1667
Offset: 1

Views

Author

Felix Fröhlich, Sep 25 2022

Keywords

Crossrefs

Cf. A151800, A357362, A357364, A357365.

Programs

  • PARI
    is(n) = my(b=precprime(precprime(n-1)-1)); Mod(b, n^2)^(n-1)==1 || Mod(n, b^2)^(b-1)==1
forprime(p=5, , if(is(p), print1(p, ", ")))

A357361 Smallest number k such that A345112(k) = n.

Original entry on oeis.org

1, 5, 19, 118, 89, 123, 102, 145, 104, 777, 1133, 1012, 858, 942, 651, 150, 453, 132, 39, 112551, 39782, 23244, 81914, 43810, 40346
Offset: 1

Views

Author

Felix Fröhlich, Sep 25 2022

Keywords

Crossrefs

Cf. A345112.

Programs

  • PARI
    eva(n) = subst(Pol(n), x, 10)
rot(vec) = if(#vec < 2, return(vec)); my(s=concat(Str(2), ".."), v=[]); s=concat(s, Str(#vec)); v=vecextract(vec, s); v=concat(v, vec[1]); v
a345112(n, bound) = my(x=n, i=0); while(1, x=x+eva(rot(digits(x))); i++; if(i > bound, return(-1), if(digits(x)==Vecrev(digits(x)), break))); i
a(n) = if(n==0, return(0)); for(k=1, oo, if(a345112(k, n)==n, return(k)))

A355965 Primes p such that (p+10)^(p-1) == 1 (mod p^2).

Original entry on oeis.org

13, 41, 97, 809, 1151, 1657, 800011
Offset: 1

Views

Author

Felix Fröhlich, Jul 21 2022

Keywords

Comments

A computer search taking less than 3 seconds shows there are no further terms below the one millionth prime. - Harvey P. Dale, Mar 04 2024
I ran the PARI program below for 8.5 hours and it did not find any further terms. (I do not know how far it searched.) - N. J. A. Sloane, Mar 05 2024

Crossrefs

(p+k)^(p-1) == 1 (mod p^2): A355959 (k=2), A355960 (k=5), A355961 (k=6), A355962 (k=7), A355963 (k=8), A355964 (k=9).

Programs

  • Mathematica
    Select[Prime[Range[70000]],PowerMod[#+10,#-1,#^2]==1&] (* Harvey P. Dale, Mar 04 2024 *)
  • PARI
    forprime(p=1, , if(Mod(p+10, p^2)^(p-1)==1, print1(p, ", ")))

A355964 Primes p such that (p+9)^(p-1) == 1 (mod p^2).

Original entry on oeis.org

13, 19, 2207, 26041, 332698495781
Offset: 1

Views

Author

Felix Fröhlich, Jul 21 2022

Keywords

Comments

a(6) > 10^13 if it exists. - Jason Yuen, May 12 2024

Crossrefs

(p+k)^(p-1) == 1 (mod p^2): A355959 (k=2), A355960 (k=5), A355961 (k=6), A355962 (k=7), A355963 (k=8), A355965 (k=10).

Programs

  • PARI
    forprime(p=1, , if(Mod(p+9, p^2)^(p-1)==1, print1(p, ", ")))

Extensions

a(5) from Jason Yuen, May 12 2024

A355963 Primes p such that (p+8)^(p-1) == 1 (mod p^2).

Original entry on oeis.org

1531, 7445287
Offset: 1

Views

Author

Felix Fröhlich, Jul 21 2022

Keywords

Comments

a(3) > 34294200797 if it exists.

Crossrefs

(p+k)^(p-1) == 1 (mod p^2): A355959 (k=2), A355960 (k=5), A355961 (k=6), A355962 (k=7), A355964 (k=9), A355965 (k=10).

Programs

  • PARI
    forprime(p=1, , if(Mod(p+8, p^2)^(p-1)==1, print1(p, ", ")))

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

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