A178537 -id:A178537 - cp's OEIS frontend

A233203 a(n) = floor(n^n / 2^n).

1, 0, 1, 3, 16, 97, 729, 6433, 65536, 756680, 9765625, 139312339, 2176782336, 36972058910, 678223072849, 13363461010158, 281474976710656, 6311342330065435, 150094635296999121, 3773536025353076151, 100000000000000000000, 2785962590401641140642, 81402749386839761113321

Offset: 0

Alex Ratushnyak, Dec 05 2013

nonn

			a(5) = floor(5^5 / 2^5) = floor(3125 / 32) = 97.

Cf. A000079, A000312, A178537 (n^n mod 2^n for odd n), A206344.

Bisection gives: A062206 (even part).

A233203:=n->floor((n/2)^n); seq(A233203(n), n=0..30); # Wesley Ivan Hurt, Feb 26 2014

Table[Floor[(n/2)^n], {n, 0, 30}] (* Wesley Ivan Hurt, Feb 26 2014 *)

a(n) = n^n\2^n; \\ Michel Marcus, Dec 17 2024

for n in range(33): print(n**n >> n, end=', ')

a(n) = floor((n/2)^n).

A178966 Primes of the form (2n-1)^(2n-1) mod 2^(2n-1).

Original entry on oeis.org

3, 68491, 1472837, 20933977, 115763405909, 860612330076805920786264707, 295312333859267885313149356219, 3030645565448009172324576658633624992466799, 1438089684181622246462815859549895702132195129054882453928357949021844283495791173

Offset: 1

Juri-Stepan Gerasimov, Jan 01 2011

nonn

Primes for n's: 2, 10, 11, 13, 19, 46, 50, 72, 139, 162, 344, 739, 1757, 3439, (10000)..., .

Cf. A000040, A000079, A000312, A178537.

f[n_] := Block[{a = PowerMod[2 n - 1, 2 n - 1, 2^(2 n - 1)]}, If[ PrimeQ@ a, a, 0]]; k = 1; lst = {}; While[k < 100, a = f@ k; If[a != 0, AppendTo[lst, a]]; k++]; lst
pfQ[n_]:=Module[{c=2n-1},PowerMod[c,c,2^c]]; Select[Table[pfQ[n],{n,200}],PrimeQ] (* Harvey P. Dale, Aug 31 2017 *)

A179003 Numbers k such that 2^(2k-1) mod (2k+1) is prime.

Original entry on oeis.org

1, 2, 4, 6, 7, 10, 16, 18, 19, 25, 28, 30, 31, 34, 36, 37, 38, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 78, 79, 87, 88, 91, 96, 97, 100, 106, 107, 109, 112, 115, 118, 124, 126, 133, 138, 145, 147, 148, 151, 154, 156, 157, 160, 163, 169, 180

Offset: 1

Juri-Stepan Gerasimov, Jan 03 2011

nonn

Cf. A000040, A178537, A178970.

fQ[n_] := PrimeQ@ PowerMod[2, 2 n - 1, 2 n + 1]; Select[ Range@ 180, fQ]

A172997 Numbers n such that for p=prime(n), (2p-1)^(2p-1) mod 2^(2p-1) is prime.

Original entry on oeis.org

1, 5, 6, 8, 34, 131

Offset: 1

Juri-Stepan Gerasimov, Jan 01 2011

nonn

			5 is in the sequence, since prime(5)=11 and (2*11-1)^(2*11-1) mod 2^(2*11-1) = 21^21 mod 2^21 = 1472837 is prime.

Cf. A000040, A000079, A000312, A178537, A178966.

A179002 Primes p such that 2^(2p-1) mod (2p+1) is prime.

Original entry on oeis.org

2, 7, 19, 31, 37, 43, 61, 67, 79, 97, 107, 109, 151, 157, 163, 199, 223, 241, 257, 269, 271, 307, 337, 349, 379, 409, 421, 433, 439, 523, 557, 577, 601, 613, 619, 631, 673, 691, 739, 757, 787, 811, 823, 853, 863, 877, 883, 919, 967, 991, 1009, 1051, 1061, 1063, 1117, 1153, 1181, 1213

Offset: 1

Juri-Stepan Gerasimov, Jan 03 2011

nonn

Cf. A000040, A178537, A178970.

 fQ[n_] := PrimeQ@ PowerMod[2, 2n - 1, 2n + 1]; Select[ Prime@ Range@ 200, fQ]

A178970(a(n)=A000040(i))=A000040(k).

A233398 n^n mod 3^n.

Original entry on oeis.org

0, 1, 4, 0, 13, 209, 0, 1231, 739, 0, 51850, 129587, 0, 657067, 868120, 0, 8400886, 91426058, 0, 859535812, 1483912651, 0, 30374372956, 68688648677, 0, 422259655777, 2129293909882, 0, 11623487749588, 44275834126985, 0, 329933248712302, 971441847750184, 0, 927342285792613

Offset: 0

Alex Ratushnyak, Dec 08 2013

nonn

Conjecture: a(x) equals zero for every x that's a multiple of 3. - Harvey P. Dale, Jan 10 2025

Cf. A000244, A000312, A178537, A233397.

Table[PowerMod[n,n,3^n],{n,0,40}] (* Harvey P. Dale, Jan 10 2025 *)

for n in range(55): print(f"{n**n % 3**n}, ")

cp's OEIS Frontend

A233203 a(n) = floor(n^n / 2^n).

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A178966 Primes of the form (2n-1)^(2n-1) mod 2^(2n-1).

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A179003 Numbers k such that 2^(2*k-1) mod (2*k+1) is prime.

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A172997 Numbers n such that for p=prime(n), (2p-1)^(2p-1) mod 2^(2p-1) is prime.

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A179002 Primes p such that 2^(2p-1) mod (2p+1) is prime.

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A233398 n^n mod 3^n.

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A179003 Numbers k such that 2^(2k-1) mod (2k+1) is prime.