Robert Dougherty-Bliss - cp's OEIS frontend

A365469 Number of n X n nonnegative integer arrays with upper left entry 0, lower right entry n - 6, every value within 5 of its king-move distance from the upper left, and every value increasing by 0 or 1 with every step right, diagonally southeast, or down.

0, 0, 0, 0, 0, 1, 923, 414638, 141905658, 44926520946, 14490345222595, 4982468227997168, 1856518533812384105, 750167381136816327009, 326628155352701098746754, 151933161224857150407326456, 74864085012461648054019799968, 38785037369022058785365519932029

Offset: 1

Robert Dougherty-Bliss, Sep 04 2023

nonn

			Example for n = 8:
  00011112
  11111122
  22222222
  22222222
  22222222
  22222222
  22222222
  22222222

Robert Dougherty-Bliss and Manuel Kauers, Hardinian Arrays, arXiv:2309.00487 [math.CO], 2023.

Cf. A253026, A253217, A252998.

Conjecture: a(n) ~ (2^4 / (3^23 * Pi^2)) * 1024^n / n^10.

A365468 Number of n X n nonnegative integer arrays with upper left entry 0, lower right entry n - 5, every value within 4 of its king-move distance from the upper left, and every value increasing by 0 or 1 with every step right, diagonally southeast, or down.

Original entry on oeis.org

0, 0, 0, 0, 1, 251, 35176, 4105312, 466029138, 54944041476, 6886218018445, 920364154307990, 130498580283240363, 19482086451147555715, 3039784702880850466554, 492560752706827973431360, 82451674043232981553367317, 14196926753609833253834234271

Offset: 1

Robert Dougherty-Bliss, Sep 04 2023

nonn

			Example for n = 7:
  0011112
  0111222
  0111222
  0112222
  0112222
  1111222
  2222222

Robert Dougherty-Bliss and Manuel Kauers, Hardinian Arrays, arXiv 2309.00487 [math.CO], 2023.

Cf. A253026, A253217, A252998.

Conjecture: a(n) ~ (2^2 / (3^16 * Pi^2)) * 256^n / n^6.

A364701 Pseudoprimes corresponding to a Perrin-like primality test.

Original entry on oeis.org

1531398, 114009582, 940084647, 4206644978, 7962908038, 20293639091, 41947594698

Offset: 1

Robert Dougherty-Bliss, Aug 03 2023

The sequence b(n) defined by the generating function (3*x^4+5*x^2+6*x-7)/(4*x^7+x^4+x^2+x-1) has the property that b(p) == 1 (mod p) if p is a prime. A pseudoprime for b(n) is a composite number k such that b(k) == 1 (mod k).

The first seven pseudoprimes are the only ones up to 10^12.

			The value of b(1531398) is a 399290-digit number which is congruent to 1 modulo 1531398 = 2 * 3 * 11 * 23203.

Robert Dougherty-Bliss and Doron Zeilberger, Lots and Lots of Perrin-Type Primality Tests and Their Pseudo-Primes, arXiv:2307.16069 [math.NT], 2023.

b(n) is A362923.

Cf. A001608, A013998, A018187.

A351916 a(1) = a(2) = 1; for n >= 2, a(n+1) = (a(n)^7 + 1)/a(n-1).

Original entry on oeis.org

1, 1, 2, 129, 297233651245505, 1588898389043626055434220300433167237829218942966252641093888571632886068535351219199489258571766594

Offset: 1

Robert Dougherty-Bliss, Feb 25 2022

nonn

a(7) has 680 digits.

Cf. A076839, A003819, A003818, A003821.

a:=proc(n) option remember: if n <= 2 then 1: else (a(n-1)^7+1)/a(n-2): fi: end:

A346601 Record values in A346599.

Original entry on oeis.org

1, 3, 4, 5, 15, 45, 91, 187, 703, 1891, 2701, 3337, 12403, 18721, 38503, 49141, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653, 1537381, 1755001

Offset: 1

Robert Dougherty-Bliss and N. J. A. Sloane, Jul 28 2021

nonn

These are the values of m (see A344005) that take a record number of steps to appear. See A346600 for their indices.

Chai Wah Wu, Table of n, a(n) for n = 1..195

max=0;lst={};Do[g=(k=1;n=1;While[m=1;While[!Divisible[m(m+1),n],m++];m!=i,n++];n);If[g>max,AppendTo[lst,n];max=g],{i,100}];lst (* Giorgos Kalogeropoulos, Jul 29 2021 *)

More terms from Chai Wah Wu, Jul 29 2021

A346600 Indices of records in A346599.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 13, 33, 37, 61, 73, 141, 157, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061

Offset: 1

Robert Dougherty-Bliss and N. J. A. Sloane, Jul 28 2021

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..195

Cf. A344005, A346598, A346599, A346601.

max=0;Monitor[Do[g=(k=1;n=1;While[m=1;While[!Divisible[m(m+1),n],m++];m!=s,n++];n);If[g>max,Print@s;max=g],{s,3000}],s] (* Giorgos Kalogeropoulos, Jul 29 2021 *)

More terms from Chai Wah Wu, Jul 29 2021

A346599 Smallest k such that A344005(k) = n.

Original entry on oeis.org

1, 3, 4, 5, 15, 7, 8, 9, 45, 11, 33, 13, 91, 35, 16, 17, 51, 19, 76, 84, 77, 23, 69, 25, 65, 27, 63, 29, 87, 31, 32, 88, 187, 85, 180, 37, 703, 247, 104, 41, 123, 43, 172, 99, 115, 47, 141, 49, 175, 255, 204, 53, 159, 135, 280, 133, 551, 59, 177, 61, 1891, 217, 64, 160, 143, 67

Offset: 1

Robert Dougherty-Bliss and N. J. A. Sloane, Jul 28 2021

nonn

			A344005(15) = 5 is the first appearance of 5 in A344005, so a(5) = 15.

Chai Wah Wu, Table of n, a(n) for n = 1..20000 (terms 1..156 from N. J. A. Sloane, terms 157..420 from Michel Marcus).

Cf. A344005, A346598.

Array[(k=1;n=1;While[m=1;While[!Divisible[m(m+1),n],m++];m!=#,n++];n)&,70] (* Giorgos Kalogeropoulos, Jul 29 2021 *)

f(n) = my(m=1); while (m*(m+1) %n, m++); m; \\ A344005
a(n) = my(k=1); while (f(k) !=n, k++); k; \\ Michel Marcus, Jul 29 2021

A346598 a(n) is the number of terms in A344005 that are equal to n.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 3, 5, 2, 4, 4, 6, 2, 4, 7, 5, 4, 6, 4, 3, 4, 4, 6, 8, 4, 6, 7, 6, 6, 8, 5, 7, 4, 6, 4, 9, 2, 4, 8, 8, 6, 8, 4, 10, 8, 4, 8, 13, 6, 6, 7, 6, 6, 6, 8, 11, 4, 4, 10, 12, 2, 8, 13, 8, 8, 8, 4, 6, 8, 8, 10, 12, 2, 8, 9, 6, 6, 8, 8, 17, 6, 4, 10, 10, 6, 4, 11, 8, 10, 10, 8, 7, 4, 6, 11, 12

Offset: 1

Robert Dougherty-Bliss and N. J. A. Sloane, Jul 27 2021

nonn

If A344005(n) = m then n <= m*(m+1).

			A344005(n) is equal to 4 just for n = 5, 10, and 20, so a(4) = 3.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A344005, A346599, A346600, A346601.

# uses[python code from A344005]
def A346598(n):
    return sum(1 for m in range(1, n*(n + 1) + 1) if A344005(m) == n)
# Chai Wah Wu, Jul 29 2021

A346596 Let m = A344005(n) = smallest m such that n divides m*(m+1); a(n) = max(gcd(n,m), gcd(n,m+1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 8, 25, 13, 27, 7, 29, 6, 31, 32, 11, 17, 7, 9, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 27, 11, 8, 19, 29, 59, 15, 61, 31, 9, 64, 13, 11, 67, 17, 23, 14, 71, 9, 73, 37, 25, 19

Offset: 1

Robert Dougherty-Bliss and N. J. A. Sloane, Jul 15 2021

nonn

This is the maximum of A345992 and A345993.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A344005, A345992, A345993, A345994, A345995.

Cf. also A034699, A324388.

f(n) = my(m=1); while ((m*(m+1)) % n, m++); m; \\ A344005
a(n) = my(m=f(n)); max(gcd(n,m), gcd(n,m+1)); \\ Michel Marcus, Aug 06 2021
(Python 3.8+)
from math import gcd, prod
from itertools import combinations
from sympy import factorint
from sympy.ntheory.modular import crt
def A346596(n):
    if n == 1:
        return 1
    plist = tuple(p**q for p, q in factorint(n).items())
    return n if len(plist) == 1 else max(gcd(n,s:=int(min(min(crt((m, n//m), (0, -1))[0], crt((n//m, m), (0, -1))[0]) for m in (prod(d) for l in range(1, len(plist)//2+1) for d in combinations(plist, l))))),gcd(n,s+1)) # Chai Wah Wu, Jun 17 2022

A345999 a(n) = (m+1)/gcd(m+1,n), where m = A344005(n).

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 2, 2, 1, 1, 8, 3, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 1, 1, 7, 1, 10, 3, 1, 1, 2, 6, 1, 1, 4, 11, 1, 1, 5, 1

Offset: 1

Robert Dougherty-Bliss and N. J. A. Sloane, Jul 18 2021

nonn

This is (A344005(n)+1)/A345993(n).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A344005, A345992-A345998.

cp's OEIS Frontend

User: Robert Dougherty-Bliss

A365469 Number of n X n nonnegative integer arrays with upper left entry 0, lower right entry n - 6, every value within 5 of its king-move distance from the upper left, and every value increasing by 0 or 1 with every step right, diagonally southeast, or down.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A365468 Number of n X n nonnegative integer arrays with upper left entry 0, lower right entry n - 5, every value within 4 of its king-move distance from the upper left, and every value increasing by 0 or 1 with every step right, diagonally southeast, or down.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A364701 Pseudoprimes corresponding to a Perrin-like primality test.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A351916 a(1) = a(2) = 1; for n >= 2, a(n+1) = (a(n)^7 + 1)/a(n-1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

A346601 Record values in A346599.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Extensions

A346600 Indices of records in A346599.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A346599 Smallest k such that A344005(k) = n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A346598 a(n) is the number of terms in A344005 that are equal to n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

A346596 Let m = A344005(n) = smallest m such that n divides m*(m+1); a(n) = max(gcd(n,m), gcd(n,m+1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs