A108574 -id:A108574 - cp's OEIS frontend

A326610 Least k such that A000790(k) = A108574(n).

0, 3, 26, 11, 14, 59, 83, 23, 443, 338, 263, 578, 38, 662, 47, 227, 3467, 1823, 842, 4898, 983, 4622, 4847, 2747, 4127, 11567, 347, 542, 17483, 2867, 22367, 43067, 18527, 5042, 5063, 12422, 66047, 2858, 87302, 11702, 11147, 3062, 24602, 158, 94763, 247838, 1202

Offset: 1

Richard N. Smith, Jul 14 2019

The largest term is a(106) = 10009487 (for the primary pretender 453).

Richard N. Smith, Table of n, a(n) for n = 1..132
N. J. A. Sloane, The primary pretenders
Richard N. Smith, Least positive and negative bases such that the primary pretender is n (format: n (in A108574), least positive base, least negative base)

Cf. A000790, A108574.

A000790 Primary pretenders: least composite c such that n^c == n (mod c).

Original entry on oeis.org

4, 4, 341, 6, 4, 4, 6, 6, 4, 4, 6, 10, 4, 4, 14, 6, 4, 4, 6, 6, 4, 4, 6, 22, 4, 4, 9, 6, 4, 4, 6, 6, 4, 4, 6, 9, 4, 4, 38, 6, 4, 4, 6, 6, 4, 4, 6, 46, 4, 4, 10, 6, 4, 4, 6, 6, 4, 4, 6, 15, 4, 4, 9, 6, 4, 4, 6, 6, 4, 4, 6, 9, 4, 4, 15, 6, 4, 4, 6, 6, 4, 4, 6, 21, 4, 4, 10, 6, 4

Offset: 0

N. J. A. Sloane

It is remarkable that this sequence is periodic with period 19568584333460072587245340037736278982017213829337604336734362\ 294738647777395483196097971852999259921329236506842360439300 = 2^2 * 3^2 * 5^2 * 7^2 * 11^2 * 13^2 * 17^2 * 19^2 * 23^2 * 29 * 31 * 37 * 41 * 43 * 47 * 53 * 59 * 61 * 67 * 71 * 73 * 79 * 83 * 89 * 97 * 101 * 103 * 107 * 109 * 113 * 127 * 131 * 137 * 139 * 149 * 151 * 157 * 163 * 167 * 173 * 179 * 181 * 191 * 193 * 197 * 199 * 211 * 223 * 227 * 229 * 233 * 239 * 241 * 251 * 257 * 263 * 269 * 271 * 277.
Note that the period is 277# * 23# (where as usual # is the primorial). - Charles R Greathouse IV, Feb 23 2014
Records are 4, 341, 382 & 561, and they occur at indices of 0, 2, 383 & 10103. - Robert G. Wilson v, Feb 22 2014
Andrzej Schinzel (1961) proved that a(n) > 6 if and only if n == {2, 11} (mod 12). - Thomas Ordowski and Krzysztof Ziemak, Jan 21 2018
We have a(n) <= A090086(n), with equality iff gcd(a(n),n) = 1. - Thomas Ordowski, Feb 13 2018
Sequence b(n) = gcd(a(n), n) is also periodic with period P = 23# * 277#, because this is the LCM of all terms, cf. A108574. - M. F. Hasler, Feb 16 2018

			a(2) = 341 because 2^341 == 2 (mod 341) and there is no smaller composite number c such that 2^c == 2 (mod c).
a(3) = 6 because 3^6 == 3 (mod 6) (whereas 3^4 == 1 (mod 4)).

T. D. Noe, Table of n, a(n) for n = 0..10000
John H. Conway, Richard K. Guy, W. A. Schneeberger and N. J. A. Sloane, The Primary Pretenders, Acta Arith. 78 (1997), 307-313.
Steven Finch, The On-Line Encyclopedia of Integer Sequences, founded in 1964 by N. J. A. Sloane, A Tribute to John Horton Conway, The Mathematical Intelligencer (2021) Vol. 43, 146-147.
A. Rotkiewicz, Periodic sequences of pseudoprimes connected with Carmichael numbers and the least period of the function lx^C, Acta Arith. XCI.1 (1999), 75-83.
A. Schinzel, Sur les nombres composés n qui divisent a^n - a, Rend. Circ. Mat. Palermo (2) 7 (1958), 37-41.
W. Sierpiński, A remark on composite numbers m which are factors of a^m - a, Wiadom. Mat. 4 (1961), 183-184 (in Polish; MR 23#A87).
OEIS Index to periodic sequences.

Cf. A108574 (all values occurring in this sequence).

Cf. A002808, A090086, A295997 (it has the same set of distinct terms).

import Math.NumberTheory.Moduli (powerMod)
a000790 n = head [c | c <- a002808_list, powerMod n c c == mod n c]
-- Reinhard Zumkeller, Jul 11 2014

f:= proc(n) local c;
  for c from 4 do
    if not isprime(c) and n &^ c - n mod c = 0 then return c fi
  od
end proc:
map(f, [$0..100]); # Robert Israel, Jan 21 2018

a[n_] := For[c = 4, True, c = If[PrimeQ[c + 1], c + 2, c + 1], If[PowerMod[n, c, c] == Mod[n, c], Return[c]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 18 2013 *)

a(n)=forcomposite(c=4,554,if(Mod(n,c)^c==n,return(c))); 561 \\ Charles R Greathouse IV, Feb 23 2014

from sympy import isprime
def A000790(n):
    c = 4
    while pow(n,c,c) != (n % c) or isprime(c):
        c += 1
    return c # Chai Wah Wu, Apr 02 2021

A295997 Least composite k such that d^k == d (mod k) for every divisor d of n.

Original entry on oeis.org

4, 341, 6, 341, 4, 561, 6, 341, 6, 561, 10, 561, 4, 561, 561, 341, 4, 561, 6, 561, 6, 561, 22, 561, 4, 561, 6, 561, 4, 561, 6, 341, 561, 561, 561, 561, 4, 561, 6, 561, 4, 561, 6, 561, 561, 341, 46, 561, 6, 561, 91, 561, 4, 561, 10, 561, 6, 341, 15, 561, 4, 341

Offset: 1

Thomas Ordowski, Feb 14 2018

nonn

a(n) is the smallest weak pseudoprime k to all natural bases d|n.
For n > 1, a(n) is the smallest composite k such that p^k == p (mod k) for every prime p dividing n; so a(n) is the smallest weak pseudoprime k to all prime bases p|n (thus it is enough to check this congruence only for all prime divisors p of n, see the second program in PARI).
For n > 1, a(n) = 4 iff n has all prime divisors p == 1 (mod 4).
The sequence is bounded, namely 4 <= a(n) <= 561, see A002997.
All members of A108574 appear in the sequence. The last to appear is 538 = a(8110351). - Robert Israel, Feb 15 2018
Conjecture: all distinct terms of the sequence are A108574. - Robert Israel and Thomas Ordowski, Feb 16 2018. The conjecture is true and can be established computationally, like in Conway-Guy-Schneeberger-Sloane (1997) paper. - Max Alekseyev, Feb 27 2018
Note that a(n) >= A000790(n). - Thomas Ordowski, Feb 16 2018
The sequence is not eventually periodic: e.g., any arithmetic progression contains infinitely many terms divisible by a prime == 3 (mod 4), and thus with a(n) > 4, while on the other hand there are infinitely many terms with a(n) = 4. - Robert Israel, Feb 16 2018

Robert Israel, Table of n, a(n) for n = 1..10000
J. H. Conway, R. K. Guy, W. A. Schneeberger, and N. J. A. Sloane, The Primary Pretenders, Acta Arith. 78 (1997), 307-313.

Cf. A000790, A002808, A002997, A007947, A108574.

f := n -> g(map(t -> t[1], ifactors(n)[2])):
g:= proc (P) local k; option remember;
  for k from 4 do
    if not isprime(k) and andmap(p -> (p &^ k - p mod k = 0), P)
    then return k
    end if
  end do
end proc:
map(f, [$1..100]); # Robert Israel, Feb 14 2018

With[{c = Table[FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1], {n, 561}]}, Table[With[{d = Divisors@ n}, SelectFirst[c, Function[k, AllTrue[d, PowerMod[#, k, k] == Mod[#, k] &]]]], {n, 62}]] (* Michael De Vlieger, Feb 17 2018, after Robert G. Wilson v at A066277 *)

a(n) = forcomposite(k=1,, my (ok=1); fordiv (n, d, if (Mod(d,k)!=Mod(d,k)^k, ok=0; break)); if (ok, return (k))); \\ Rémy Sigrist, Feb 14 2018

a(n)=my(f=factor(n)[,1],p); forcomposite(k=4,561, for(i=1,#f, p=f[i]; if(Mod(p,k)^k!=p, next(2))); return(k)); \\ Charles R Greathouse IV, Feb 14 2018

a(n) = a(rad(n)), where rad(n) = A007947(n).

For prime p, a(p) = A000790(p). - Max Alekseyev, Feb 27 2018

More terms from Rémy Sigrist, Feb 14 2018

A309316 Euler primary pretenders: a(n) is the smallest odd composite k such that n^((k+1)/2) == +-n (mod k).

Original entry on oeis.org

9, 9, 341, 121, 341, 15, 15, 21, 9, 9, 9, 33, 33, 21, 15, 15, 15, 9, 9, 9, 15, 15, 21, 33, 15, 15, 9, 9, 9, 15, 15, 15, 25, 33, 21, 9, 9, 9, 39, 15, 15, 21, 21, 21, 9, 9, 9, 65, 21, 21, 15, 15, 39, 9, 9, 9, 21, 21, 57, 15, 15, 15, 9, 9, 9, 15, 15, 33, 25, 15, 15, 9, 9, 9, 15, 15, 15, 21, 21, 39, 9, 9, 9

Offset: 0

Thomas Ordowski, Jul 23 2019

nonn

Note that if p is an odd prime, then n^((p+1)/2) == +-n (mod p) for all n.
a(n) is the least Euler weak pseudoprime to base n, as A000790(n) is the least Fermat weak pseudoprime to base n.
This sequence is bounded, namely a(n) <= 1729, the smallest absolute Euler pseudoprime, because n^((1729+1)/2) == n (mod 1729) for every n, see A033181.
The set A = {a(n)} of terms contains all odd semiprimes pq < 1729 such that p-1 | q-1. Other numbers in the set are 561, 645, 1065, 1247, and 1729. Probably complete. Cf. A108574.
This sequence is periodic with a very long period P = LCM(A) = p#*q#/2^2, where p and q are the largest primes such that p^2 < 1729 and 3q < 1729. Such primes are p = 41 and q = 571, so the period P = 41#*571#/4 (248 digits) is much longer than period of the (Fermat) primary pretenders.
Problem: is P the fundamental period of this sequence?
Records of a(n) are 9, 341, 561, 703, 793, 1145, 1263, 1477, and 1729; for n = 0, 2, 383, 1822, 3308, 4702, 10103, 12252, and 21821. - Amiram Eldar, Jul 24 2019

			a(2) = 341 since 2^((341+1)/2) = 2^171 == 2 (mod 341) and 341 is the smallest odd composite number satisfying this congruence.
a(5) = 15 since 5^((15+1)/2) = 5^8 == -5 (mod 15) and 15 is the smallest odd composite number with this property.
a(8) = 9 since 8^((9+1)/2) = 8^5 == 8 (mod 9) and 9 is the smallest odd composite number.

Cf. A000790, A033181, A108574.

f:= proc(n) local k,r;
  for k from 9 by 2 do
    if isprime(k) then next fi;
    r:= n &^ ((k+1)/2) mod k;
    if r = (n mod k) or r = (-n mod k) then return k fi
  od
end proc:
map(f, [$0..100]); # Robert Israel, Jul 23 2019

a[n_] := Module[{k=9}, While[!CompositeQ[k] || ((m = PowerMod[n, (k+1)/2, k]) != Mod[n, k] && m != Mod[-n, k]), k+=2]; k]; Array[a, 100, 0] (* Amiram Eldar, Jul 23 2019 *)

a(n) = 9 if and only if n == {-1,0,1} (mod 9).

a(n + P) = a(n), where the period P = 41#*571#/4.

More terms from Amiram Eldar, Jul 23 2019

A321575 Nexus primary pretenders: a(n) is the smallest composite k such that n^k - (n-1)^k == 1 (mod k).

Original entry on oeis.org

9, 4, 341, 4, 6, 4, 9, 4, 14, 4, 6, 4, 9, 4, 21, 4, 6, 4, 9, 4, 15, 4, 6, 4, 9, 4, 10, 4, 6, 4, 9, 4, 62, 4, 6, 4, 9, 4, 49, 4, 6, 4, 9, 4, 33, 4, 6, 4, 9, 4, 14, 4, 6, 4, 9, 4, 10, 4, 6, 4, 9, 4, 65, 4, 6, 4, 9, 4, 49, 4, 6, 4, 9, 4, 111, 4, 6, 4, 9, 4, 15

Offset: 0

Thomas Ordowski, Nov 13 2018

nonn

The sequence is bounded, namely a(n) <= 561 (the smallest Carmichael number), since if n^k == n (mod k) and (n-1)^k == n-1 (mod k), then n^k - (n-1)^k == 1 (mod k).
Problem: find all distinct terms of the sequence. Is this sequence periodic like the primary pretenders?
Note that a(n) > 9 if and only if n == 2 (mod 6). We have a(6m+2) = 341, 14, 21, 15, 10, 62, 49, 33, 14, 10, 65, 49, 111, 15, 10, ... for m >= 0. Found a(n) = 561 for the smallest n = 6*70+2 = 422.
From Robert Israel, Nov 27 2018: (Start)
Since a(n) depends only on the residues of n mod k for composites k <= 561, it must be periodic with period at most the lcm of those composites.
Up to n=2*10^6, the last term to appear for the first time is 478 = a(184748).
Conjecture: the only terms of the sequence that are not squarefree are 4, 9 and 49. (End)

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000790, A108574.

Comps:= remove(isprime, [$4..561]):
f:= proc(n) local k;
  for k in Comps do if n&^k - (n-1)&^k - 1 mod k = 0 then return k fi od
end proc:
map(f, [$0..100]); # Robert Israel, Nov 27 2018

a[n_]:=Module[{k=4}, While[PrimeQ[k] || Mod[n^k-(n-1)^k,k]!=1, k++]; k]; Array[a, 100, 0] (* Amiram Eldar, Nov 13 2018 *)

a(n)=forcomposite(k=4,,Mod(n,k)^k-Mod(n-1,k)^k==1&&return(k)) \\ M. F. Hasler, Nov 13 2018

a(n) = 4 iff n == 1,3,5 (mod 6), thus n is odd.
a(n) = 6 iff n == 4 (mod 6).
a(n) = 9 iff n == 0 (mod 6).

More terms from Amiram Eldar, Nov 13 2018

cp's OEIS Frontend

A326610 Least k such that A000790(k) = A108574(n).

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A000790 Primary pretenders: least composite c such that n^c == n (mod c).

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A295997 Least composite k such that d^k == d (mod k) for every divisor d of n.

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A309316 Euler primary pretenders: a(n) is the smallest odd composite k such that n^((k+1)/2) == +-n (mod k).

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A321575 Nexus primary pretenders: a(n) is the smallest composite k such that n^k - (n-1)^k == 1 (mod k).

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