A270096 -id:A270096 - cp's OEIS frontend

A290960 Numbers k such that A276976(k) > A270096(k).

8, 32, 56, 64, 96, 128, 144, 155, 170, 176, 192, 196, 204, 215, 221, 224, 238, 248, 255, 256, 272, 288, 320, 322, 336, 341, 352, 368, 372, 374, 384, 432, 448, 465, 476, 496, 510, 512, 527, 544, 574, 576, 608, 612, 623, 635, 640, 644, 645, 658, 663, 672, 682, 697, 704, 714, 731, 736, 744

Offset: 1

Altug Alkan, Aug 15 2017, following a suggestion from N. J. A. Sloane

nonn

Odd terms are 155, 215, 221, 255, 341, 465, 527, 623, 635, 645, 663, ...

These odd terms are odd numbers k such that (k mod A002322(k)) > (k mod A002326((k-1)/2)). - Amiram Eldar and Thomas Ordowski, Nov 28 2019

			8 is a term because A276976(8) = 4 while A270096(8) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002322, A002326, A270096, A276976.

A270096:= proc(n) local d, b, t, m, c;
  d:= padic:-ordp(n, 2);
  b:= n/2^d;
  t:= 2 &^ n mod n;
  m:= numtheory:-mlog(t, 2, b, c);
  if m < d then m:= m + c*ceil((d-m)/c) fi;
  m
end proc:
A270096(1):= 0:
A276976:= proc(n) local lambda;
  lambda:= numtheory:-lambda(n);
  if n mod lambda = 0 then lambda
  elif n mod 8 = 0 and (n-2) mod lambda = 0 then lambda+2
  else n mod lambda
  fi
end proc:
A276976(1):= 0:
A276976(8):= 4:
A276976(24):= 4:
select(n -> A276976(n) > A270096(n), [$1..1000]); # Robert Israel, Aug 16 2017

With[{nn = 750}, Select[Range@ nn, Function[n, SelectFirst[Range[nn/4 + 10], Function[m, AllTrue[Range[2, n - 1], PowerMod[#, m , n] == PowerMod[#, n , n] &]]] > SelectFirst[Range[nn/4 + 10], PowerMod[2, n, n] == PowerMod[2, #, n] &]]]] (* Michael De Vlieger, Aug 15 2017 *)

A307625 Numbers k such that q = 2^k - 2^m + 1 is prime, where m = A270096(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 16, 17, 19, 22, 31, 39, 45, 61, 76, 89, 94, 100, 102, 107, 122, 127, 294, 360, 430, 460, 521, 607, 639, 694, 732, 737, 952, 1279, 1581, 1983, 2061, 2203, 2281, 2319, 2410, 2530, 3217, 4253, 4423, 5324, 6846, 7011, 9615, 9689, 9904, 9941, 10841, 11213

Offset: 1

Thomas Ordowski, Apr 19 2019

nonn

All primes in the sequence are the Mersenne exponents A000043.

It seems that the composite terms are composite numbers k <> 8 such that A307590(k) = 2.

Cf. A000043, A270096, A270427, A307590.

b[n_] := Module[{k = 0}, While[PowerMod[2, n, n] != PowerMod[2, k, n], k++]; k]; aQ[n_] := PrimeQ[2^n - 2^b[n] + 1]; Select[Range[5000], aQ] (* Amiram Eldar, Apr 19 2019 *)

f(n) = {my(m = 0); while (Mod(2, n)^m != 2^n, m++); m; } \\ A270096
isok(n) = my(m = f(n)); isprime(2^n - 2^m + 1); \\ Michel Marcus, Apr 23 2019

q == 1 (mod k).

More terms from Amiram Eldar, Apr 19 2019

A276976 Smallest m such that b^m == b^n (mod n) for every integer b.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 4, 5, 2, 9, 4, 1, 2, 1, 8, 3, 2, 11, 6, 1, 2, 3, 4, 1, 6, 1, 4, 9, 2, 1, 4, 7, 10, 3, 4, 1, 18, 15, 8, 3, 2, 1, 4, 1, 2, 3, 16, 5, 6, 1, 4, 3, 10, 1, 6, 1, 2, 15, 4, 17, 6, 1, 4, 27, 2, 1

Offset: 1

Thomas Ordowski, Sep 23 2016

It suffices to check all bases 0 < b < n for n > 2.
The congruence n == a(n) (mod A002322(n)) is always true.
a(n) = 1 iff n is a prime or a Carmichael number.
We have a(n) > 0 for n > 1, and a(n) <= n/2.
If n > 2 then a(n) is odd iff n is odd.
Conjecture: a(n) <= n/3 for every n >= 9.
Professor Andrzej Schinzel proved this conjecture (in a letter to the author). - Thomas Ordowski, Sep 30 2016
Note: a(n) = n/3 for n = A038754 >= 3.
Numbers n such that a(n) > A270096(n) are A290960.
Information from Carl Pomerance: a(n) > A002322(n) if and only if 8|n and n is in A050990; such n = 8, 24, 56, ... - Thomas Ordowski, Jun 21 2017
Number of integers k < n such that b^k == b^n (mod n) for every integer b is f(n) = (n - a(n))/lambda(n). For n > 1, f(n) = floor((n-1)/lambda(n)) if and only if a(n) <= lambda(n), where lambda(n) = A002322(n). - Thomas Ordowski, Jun 21 2017
a(n) >= A051903(n); numbers n such that a(n) = A051903(n) are 1, primes, Carmichael numbers, and A327295. - Thomas Ordowski, Dec 06 2019

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002322, A002997, A038754, A050990, A051903, A124240, A219175, A270096, A290960, A327295.

With[{nn = 83}, Table[SelectFirst[Range[nn/4 + 10], Function[m, AllTrue[Range[2, n - 1], PowerMod[#, m , n] == PowerMod[#, n , n] &]]] - Boole[n == 1], {n, nn}]] (* Michael De Vlieger, Aug 15 2017 *)
a[1] = 0; a[8] = a[24] = 4; a[n_] := If[(rem = Mod[n, (lam = CarmichaelLambda[n])]) >= Max @@ Last /@ FactorInteger[n], rem, rem + lam]; Array[a, 100] (* Amiram Eldar, Nov 30 2019 *)

a(n)=if(n<3, return(n-1)); forstep(m=1,n,n%2+1, for(b=0,n-1, if(Mod(b,n)^m-Mod(b,n)^n, next(2))); return(m)) \\ Charles R Greathouse IV, Sep 23 2016

def a(n): return next(m for m in range(0, n+1) if all(pow(b,m,n)==pow(b,n,n) for b in range(1, 2*n+1))) # Nicholas Stefan Georgescu, Jun 03 2022

a(p) = 1 for prime p.
a(2*p) = 2 for prime p.
a(3*p) = 3 for odd prime p.
a(p^k) = p^(k-1) for odd prime p and k >= 1.
a(2*p^k) = 2*p^(k-1) for odd prime p and k >= 1.
a(2^k) = 2^(k-2) for k >= 4.
From Thomas Ordowski, Jul 09 2017: (Start)
Full description of the function:
a(n) = lambda(n) if lambda(n)|n except n = 1, 8, and 24;
a(n) = lambda(n)+2 if lambda(n)|(n-2) and 8|n;
a(n) = n mod lambda(n) otherwise;
where lambda(n) = A002322(n). (End)
For n <> 8 and 24, a(n) = A(n) if A(n) >= A051903(n) or a(n) = A002322(n) + A(n) otherwise, where A(n) = A219175(n). - Thomas Ordowski, Nov 30 2019

More terms from Altug Alkan, Sep 23 2016

A327295 Numbers k such that e(k) > 1 and k == e(k) (mod lambda(k)), where e(k) = A051903(k) is the maximal exponent in prime factorization of k.

Original entry on oeis.org

4, 12, 16, 48, 80, 112, 132, 208, 240, 1104, 1456, 1892, 2128, 4144, 5852, 12208, 17292, 18544, 21424, 25456, 30160, 45904, 78736, 97552, 106384, 138864, 153596, 154960, 160528, 289772, 311920, 321904, 399212, 430652, 545584, 750064, 770704, 979916, 1037040, 1058512

Offset: 1

Thomas Ordowski, Dec 05 2019

nonn

The condition e(k) > 1 excludes primes and Carmichael numbers.
Numbers n such that e(k) > 1 and b^k == b^e(k) (mod k) for all b.
These are numbers k such that A276976(k) = e(k) > 1.
Are there infinitely many such numbers? Are all such numbers even?
A number k is a term if and only if k is e(k)-Knödel number with e(k) > 1. So they may have the name nonsquarefree e(k)-Knodel numbers k.
It seems that if k is in this sequence, then e(k) = A007814(k) and k/2^e(k) is squarefree.
Conjecture: there are no composite numbers m > 4 such that m == e(m) (mod phi(m)). By Lehmer's totient conjecture, there are no such squarefree numbers.
Problem: are there odd numbers n such that e(n) > 1 and n == e(n) (mod ord_{n}(2)), where ord_{n}(2) = A002326((n-1)/2)? These are odd numbers n such that 2^n == 2^e(n) (mod n) with e(n) > 1.
Numbers k for which A051903(k) > 1 and A219175(k) = A329885(k). - Antti Karttunen, Dec 11 2019

			The number 4 = 2^2 is a term, because e(4) = A051903(4) = 2 > 1 and 4 == 2 (mod lambda(4)), where lambda(4) = A002322(4) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000010, A002322, A002326, A002997, A050990, A050992, A051903, A068494, A219175, A270096, A276976, A324050, A327979, A329885.

A proper subset of A013929.

Select[Range[10^5], (e = Max @@ Last /@ FactorInteger[#]) > 1 && Divisible[# -e, CarmichaelLambda[#]] &] (* Amiram Eldar, Dec 05 2019 *)

isok(n) = ! issquarefree(n) && (Mod(n, lcm(znstar(n)[2])) == vecmax(factor(n)[, 2])); \\ Michel Marcus, Dec 05 2019

More terms from Amiram Eldar, Dec 05 2019

A385989 a(n) is the least m > n such that 2^n and 2^m are congruent modulo n.

Original entry on oeis.org

2, 3, 5, 5, 9, 8, 10, 9, 15, 14, 21, 14, 25, 17, 19, 17, 25, 24, 37, 24, 27, 32, 34, 26, 45, 38, 45, 31, 57, 34, 36, 33, 43, 42, 47, 42, 73, 56, 51, 44, 61, 48, 57, 54, 57, 57, 70, 50, 70, 70, 59, 64, 105, 72, 75, 59, 75, 86, 117, 64, 121, 67, 69, 65, 77, 76

Offset: 1

Rémy Sigrist, Jul 14 2025

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000

See A270096 for a similar sequence.

Cf. A007733.

a[n_]:=Module[{m=n+1},While[PowerMod[2,n,n]!=PowerMod[2,m,n], m++]; m]; Array[a,66] (* Stefano Spezia, Jul 16 2025 *)

a(n) = { my (u = Mod(2, n)^n, v = u); for (m = n+1, oo, if (u==v*=2, return (m));); }

a(2^k) = 2^k + 1 for any k >= 0.

a(n) <= n + A007733(n).

A277129 Largest m < n such that 2^m == 2^n (mod n).

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 4, 7, 3, 6, 1, 10, 1, 11, 11, 15, 9, 12, 1, 16, 15, 12, 12, 22, 5, 14, 9, 25, 1, 26, 26, 31, 23, 26, 23, 30, 1, 20, 27, 36, 21, 36, 29, 34, 33, 35, 24, 46, 28, 30, 43, 40, 1, 36, 35, 53, 39, 30, 1, 56, 1, 57, 57, 63, 53, 56, 1, 60, 47, 58, 36, 66, 64, 38, 55, 58, 47, 66, 40, 76, 27, 62, 1

Offset: 1

Thomas Ordowski, Oct 01 2016

nonn

If n is odd, then a(n) = n - A002326((n-1)/2).

Cf. A002326, A015910, A270096.

Table[m = n - 1; While[Mod[2^m, n] != Mod[2^n, n], m--]; m, {n, 83}] (* Michael De Vlieger, Oct 02 2016 *)

a(n) = {if(n==0,return(0));my(pt = valuation(n, 2), odd = n/2^pt, ul = odd-A002326(odd\2)); forstep(i = n-1, ul, -1, if(Mod(2,n)^i==Mod(2,n)^n,return(i)))} \\ David A. Corneth, Oct 01 2016
A002326(n)=if(n<0, 0, znorder(Mod(2, 2*n+1)))

More terms from Altug Alkan, Oct 01 2016

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A290960 Numbers k such that A276976(k) > A270096(k).

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A307625 Numbers k such that q = 2^k - 2^m + 1 is prime, where m = A270096(k).

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A276976 Smallest m such that b^m == b^n (mod n) for every integer b.

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A327295 Numbers k such that e(k) > 1 and k == e(k) (mod lambda(k)), where e(k) = A051903(k) is the maximal exponent in prime factorization of k.

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A385989 a(n) is the least m > n such that 2^n and 2^m are congruent modulo n.

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A277129 Largest m < n such that 2^m == 2^n (mod n).

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