A320465 - OEIS

A320465

a(n) = 2^n - (2^(n-1) mod n), where "mod" is the nonnegative remainder operator.

2, 4, 7, 16, 31, 62, 127, 256, 508, 1022, 2047, 4088, 8191, 16382, 32764, 65536, 131071, 262130, 524287, 1048568, 2097148, 4194302, 8388607, 16777208, 33554416, 67108862, 134217715, 268435448, 536870911, 1073741822, 2147483647, 4294967296, 8589934588

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OFFSET

1,1

COMMENTS

Thomas Ordowski (private communication) observes that a(n) appears to be composite whenever n is composite. Is there any prime a(n) for composite n ?

Conjecture: for n > 2, a(n) is prime if and only if n is in A000043. Note that a(n) = 2^n - 1 if and only if n is an odd prime or pseudoprime to base 2 (A001567), so a counterexample cannot be a Fermat pseudoprime to base 2. - Thomas Ordowski, Oct 14 2018

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = 2^(n-1) + n*floor(2^(n-1)/n). (Due to Thomas Ordowski).

For k >= 0, a(2^k) = 2^(2^k) = A001146(k). For n > 1, a(prime(n)) = 2^prime(n) - 1 = A001348(n). If p is an odd prime, then a(2p) = 4^p - 2. - Thomas Ordowski, Oct 14 2018

a(n) = 2^n - 1 = A000225(n) for n in A065091 U A001567 (odd prime or pseudoprime to base 2 = Sarrus or Poulet number).

MATHEMATICA

Table[2^n-PowerMod[2, n-1, n], {n, 40}] (* Harvey P. Dale, Jun 08 2024 *)

PROG

(PARI) A320465(n)=2^n-2^(n-1)%n

CROSSREFS

A001146 is a subsequence.

Cf. A000043, A001348 (the Mersenne numbers > 3 are a subsequence).

Cf. A000225, A065091, A001567.

Sequence in context: A026764 A027230 A217929 * A345333 A259544 A294376

Adjacent sequences: A320462 A320463 A320464 * A320466 A320467 A320468

KEYWORD

nonn

AUTHOR

M. F. Hasler, Oct 13 2018

STATUS

approved