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?(#<5&), has authored 3 sequences.

A354156 Primes p == 1 (mod 4) which are not Lagrange primes.

37, 61, 89, 101, 109, 113, 149, 157, 173, 181, 193, 197, 233, 269, 277, 293, 317, 337, 349, 353, 373, 389, 401, 421, 433, 509, 557, 569, 577, 593, 601, 613, 641, 673, 701, 709, 757, 761, 773, 797, 821, 829, 877, 881, 937, 941, 977, 1009, 1013, 1033, 1049, 1061

Offset: 1

N. J. A. Sloane, May 29 2022, based on Section 18.5 of Cosgrave (2022)

nonn

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.5.

Michael S. Branicky, Table of n, a(n) for n = 1..5383

This is the complement of A354155 in A002144.

from itertools import islice
from sympy import factorial, nextprime
def agen(): # generator of terms
    p = 5
    while True:
        X = (p-1)//2
        Xf = factorial(X)**2
        if any(pow(factorial(Y), 2, p)+1 == p for Y in range(X-1, 0, -1)):
            yield p
        p = nextprime(p)
        while p%4 != 1:
            p = nextprime(p)
print(list(islice(agen(), 5))) # Michael S. Branicky, May 30 2022

a(26) and beyond from Michael S. Branicky, May 30 2022

A354155 Lagrange primes: primes p == 1 (mod 4) such that X = (p-1)/2 is the least solution in the interval [1, (p-1)/2] of the congruence (X!)^2 == -1 (mod p).

Original entry on oeis.org

5, 13, 17, 29, 41, 53, 73, 97, 137, 229, 241, 257, 281, 313, 397, 409, 449, 457, 461, 521, 541, 617, 653, 661, 677, 733, 769, 809, 853, 857, 929, 953, 997, 1021, 1069, 1109, 1201, 1213, 1217, 1249, 1277, 1361, 1373, 1409, 1489, 1553, 1597, 1609, 1621, 1697

Offset: 1

N. J. A. Sloane, May 29 2022, based on Section 18.5 of Cosgrave (2022)

nonn

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.5.

Michael S. Branicky, Table of n, a(n) for n = 1..3100

Cf. A002144, A354156.

is(n)=if(n%4 != 1 || !isprime(n), return(0)); my(t1=lift(sqrt(Mod(-1,n))), t2=n-t1, t=Mod(1,n)); for(k=2,n\2, if(t==t1 || t==t2, return(0)); t*=k); 1 \\ Charles R Greathouse IV, Aug 03 2023

list(lim)=my(v=List()); forprimestep(p=5,lim\1,4, my(t1=lift(sqrt(Mod(-1,p))),t2=p-t1, t=Mod(1,p)); for(k=2,p\2, if(t==t1 || t==t2, next(2)); t*=k); listput(v,p)); Vec(v) \\ Charles R Greathouse IV, Aug 03 2023

from itertools import islice
from sympy import factorial, nextprime
def agen(): # generator of terms
    p = 5
    while True:
        X = (p-1)//2
        Xf = factorial(X)**2
        if all(pow(factorial(Y), 2, p)+1 != p for Y in range(X-1, 0, -1)):
            yield p
        p = nextprime(p)
        while p%4 != 1:
            p = nextprime(p)
print(list(islice(agen(), 5))) # Michael S. Branicky, May 30 2022

a(20) and beyond from Michael S. Branicky, May 30 2022

A330339 Boustrophedon primes: write the numbers 0, 1, 2, 3, ... in a triangle on a square grid in the boustrophedon manner, ending a row when a prime is reached; sequence lists primes that appear in the zeroth column.

Original entry on oeis.org

37, 53, 89, 113, 3821, 3989, 4657, 28661, 29021, 41641, 41669, 44249, 50909, 56053, 57041, 57301, 133981, 16501361, 46178761, 47633441, 47633477, 47722049, 47736121, 47774621, 47803477, 47810209, 47835013, 47835341, 47854969, 47862413, 47865017, 49448573, 49448617

Offset: 1

N. J. A. Sloane, Dec 17 2019, following a suggestion from Eric Angelini. a(5) and a(6) were found by Walter Trump. a(7)-a(17) from N. J. A. Sloane, Dec 17 2019

nonn

Eric Angelini's illustration shows the first 19 rows of the triangle. Each row ends when a prime is reached, and the next row starts directly under this prime but moves in the opposite direction.
The extended illustration from Walter Trump resembles a giant ski run.
Hans Havermann's plots of A330545, linked here, extend Walter Trump's graph to 4*10^8 rows (probably the longest ski run in the world). Only the turns are shown, and the illustration has been turned sideways.
A330545(k) = 0 iff prime(k) is a term of the present sequence. In a sense A330545 and the simpler A330547 are the more fundamental sequences and show the connection between the present problem and the ordinary primes and their alternating sums.
Note that because primes > 2 are odd, a prime can only appear in column 0 at the end of a row that is moving towards the left. A prime appearing in a row moving to the right will always appear in an odd-numbered column (and in particular, not in the zero column).
Furthermore the column number mod 4 uniquely determines the residue class of primes mod 4 in that column. If the column number is 0,1,2,3 mod 4 then the primes in that column are 1,3,3,1 respectively (see the "Notes" link). In particular, a(n) == 1 mod 4. - N. J. A. Sloane, Jan 04 2020
Note that the primes > 2 in column one and two are the primes in A282178.
Note on the links: The illustrations from Angelini and Trump show all the terms 0,1,2,3,4,..., while those of Havermann and Sloane just show the primes (as in A330545).

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 516 terms from Hans Havermann)
Eric Angelini, Illustration of beginning of the triangle in A330339.
Hans Havermann, Plot of 4*10^8 terms of A330545, sampled every 1000 terms, points joined.
Hans Havermann, More detailed view of terms of A330545 from 290 million to 310 million, sampled every 10 terms, points joined.
N. J. A. Sloane, Illustration of first 16 rows of A330545.
N. J. A. Sloane, Notes on the sequence of Bostrophedon primes (A330339) and the "ski-run" A330545.
N. J. A. Sloane, State diagram for columns of A330545.
Walter Trump, An extended picture of the triangle in A330339, showing the first 550 rows, down to the row that starts 3989. [The zeroth column is just to the right of the vertical red line. Note that after a while the rows extend to the left of the red line. The digits are too small to be read.]
Walter Trump, An extended picture of the triangle in A330339, showing the first 550 rows, down to the row that starts 3989. [Same picture as the previous one, but with 6 red dots added to show the primes in column 0.]

Cf. A282178, A330545, A330547.
A330546 gives the list of indices i such that a(n) = prime(i).
A127596 is another sequence with a similar flavor.
Not to be confused with A000747 = Boustrophedon transform of primes.

More terms from Hans Havermann, Dec 17 2019

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User: ?(#<5&),

A354156 Primes p == 1 (mod 4) which are not Lagrange primes.

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A354155 Lagrange primes: primes p == 1 (mod 4) such that X = (p-1)/2 is the least solution in the interval [1, (p-1)/2] of the congruence (X!)^2 == -1 (mod p).

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A330339 Boustrophedon primes: write the numbers 0, 1, 2, 3, ... in a triangle on a square grid in the boustrophedon manner, ending a row when a prime is reached; sequence lists primes that appear in the zeroth column.

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