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.

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

Author

Thomas Ordowski, Dec 05 2019

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

A proper subset of A013929.

Programs

Mathematica

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

PARI

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

Extensions

More terms from Amiram Eldar, Dec 05 2019