A046277 -id:A046277 - cp's OEIS frontend

A046276 Largest palindromic substring in n! without an initial zero.

1, 1, 2, 6, 4, 2, 7, 5, 4, 88, 88, 99, 9, 22, 87178, 767, 898, 55, 737, 121, 66, 909, 7777, 25852, 484, 1551, 66, 8888, 888, 93739, 848, 82228, 353, 881188, 414, 6666, 999, 59795, 1111, 99, 69596, 61316, 4260624, 383, 8448, 7337, 89498, 979, 67776, 828

Offset: 0

Patrick De Geest, Jun 15 1998

			14! = {87178}291200.

T. D. Noe, Table of n, a(n) for n = 0..300

Cf. A046277.

isPal[d_List] := d[[1]] != 0 && d == Reverse[d]; Table[d = IntegerDigits[n!]; k = Length[d]; While[s = Select[Partition[d, k, 1], isPal]; s == {}, k--]; Max[FromDigits /@ s], {n, 0, 100}] (* T. D. Noe, Mar 25 2011 *)

Corrected by D. S. McNeil, Dec 10 2010

A379944 Smallest number of leading digits of n! that form a prime (or 0 if none exist).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 2, 0, 0, 2, 1, 1, 0, 7, 1, 1, 2, 1, 0, 5, 0, 0, 1, 8, 1, 0, 1, 6, 1, 3, 1, 2, 1, 1, 0, 1, 8, 38, 1, 2, 1, 1, 5, 34, 1, 5, 6, 0, 1, 0, 1, 6, 1, 2, 2, 1, 1, 2, 8, 9, 1, 1, 1, 2, 2, 0, 2, 5, 1, 1, 0, 4, 2, 2, 1, 1, 2, 1, 1, 1, 1

Offset: 0

Carson R. Smith, Jan 07 2025

It appears that as n gets large, a(n) can become arbitrarily large.

It appears that values of n such that a(n) = 0 exist for arbitrarily large n.

			For n = 3, 3! = 6, 6 is not prime, a(3) = 0.
For n = 19, 19! = 121645100408832000, 1216451 is the smallest prime, a(19) = 7.

Carson R. Smith, Table of n, a(n) for n = 0..2000

Cf. A000040, A000142, A046277.

A379944[n_] := Catch[Do[If[PrimeQ[FromDigits[#[[;; k]]]], Throw[k]],{k,Length[#]}] & [IntegerDigits[n!]]; 0];
Array[A379944, 100, 0] (* Paolo Xausa, Jan 16 2025 *)

a(n) = my(d=digits(n!)); for (k=1, #d, if (isprime(fromdigits(Vec(d, k))), return(k))); \\ Michel Marcus, Jan 08 2025

import math
from sympy import isprime
def a(n):
    factorial = str(math.factorial(n))
    for d in range(1, len(factorial)+1):
        if isprime(int(factorial[:d])):
            return d
    return 0

More terms from Jinyuan Wang, Jan 07 2025

A052056 Numbers k such that k! starts with its largest prime substring.

Original entry on oeis.org

2, 4, 6, 7, 9, 10, 15, 16, 20, 21, 23, 25, 30, 35, 43, 78, 102, 105, 132, 138, 151, 189, 202, 215, 219, 233, 241, 264, 320, 334, 349, 352, 367, 386, 433, 458, 520, 583, 779, 885, 905, 1068, 1078, 1131, 1149, 1198, 1271, 1276, 1314, 1503, 1623, 1646, 1903, 1962, 2053

Offset: 1

Patrick De Geest, Jan 15 2000

			16 is a term because 16! = {209227}89888000 and its largest prime substring 209227 starts from the left.

Michael S. Branicky, Python program

Cf. A000142, A046277.

from sympy import isprime
def starts_with_lps(n): # see link for faster version
    s = str(n)
    ss = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1))
    lps = max((u for u in (int(t) for t in ss) if isprime(u)), default=0)
    return lps > 0 and s.startswith(str(lps))
def afind():
    k, fk = 1, 1
    while True:
        if starts_with_lps(fk):
            print(k, end=", ")
        k += 1
        fk *= k
afind() # Michael S. Branicky, Dec 31 2021

More terms from Sean A. Irvine, Feb 16 2011
Offset changed to 1 by Jon E. Schoenfield, Oct 17 2019
a(38)-a(49) from Michael S. Branicky, Dec 31 2021
a(50)-a(55) from Michael S. Branicky, May 31 2023

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A046276 Largest palindromic substring in n! without an initial zero.

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A379944 Smallest number of leading digits of n! that form a prime (or 0 if none exist).

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A052056 Numbers k such that k! starts with its largest prime substring.

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Author

Keywords

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Crossrefs

Programs

Python

Extensions