A015910 -id:A015910 - cp's OEIS frontend

A294390 a(n) = 2^(n-4) mod n, for n >= 4.

1, 2, 4, 1, 0, 5, 4, 7, 4, 5, 2, 8, 0, 15, 4, 12, 16, 11, 14, 3, 16, 2, 10, 5, 8, 11, 4, 4, 0, 17, 30, 23, 4, 14, 24, 20, 16, 36, 4, 27, 12, 32, 6, 6, 16, 8, 14, 26, 40, 20, 22, 13, 16, 29, 22, 37, 16, 23, 8, 32, 0, 2, 4, 42, 52, 35, 64, 9, 40, 64, 28, 23, 20, 30, 4

Offset: 4

Enrique Navarrete, Oct 29 2017

Every nonnegative integer seems to appear in the sequence, and every integer seems to appear in the sequence of first differences (see link).
From Robert Israel, Dec 04 2017: (Start)
a(n) = 0 iff n>=8 is a power of 2.
a(n) = 1 iff n=4 or n is in A033984.
a(n) = 2 iff n>=4 is in A015925 and is not divisible by 4. (End)

			For n=9, 2^5 = 32 == 5 mod 9.

Robert Israel, Table of n, a(n) for n = 4..10000
Enrique Navarrete, Sequences derived from residues of 2^n (mod n)

Cf. A015910, A015925, A033984, A294389.

[Modexp(2, n-4, n): n in [4..120]]; // G. C. Greubel, Dec 27 2024

A294390:=n->2&^(n-4) mod n: seq(A294390(n), n=4..150); # Wesley Ivan Hurt, Nov 30 2017

Array[Mod[2^(# - 4), #] &, 75, 4] (* Michael De Vlieger, Dec 02 2017 *)
Array[PowerMod[2,#-4,#]&,80,4] (* Harvey P. Dale, Dec 01 2018 *)

a(n) = lift(Mod(2, n)^(n-4)); \\ Michel Marcus, Oct 30 2017

print([power_mod(2, n-4, n) for n in range(4, 101)]) # G. C. Greubel, Dec 27 2024

More terms from Michel Marcus, Oct 30 2017

A357531 Final value obtained by traveling clockwise around a circular array with positions numbered clockwise from 1 to n. Each move consists of traveling clockwise k places, where k is the position at the beginning of the move. The first move begins at position 1. a(n) is the position at the end of the n-th move.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 8, 8, 4, 2, 4, 2, 4, 8, 16, 2, 10, 2, 16, 8, 4, 2, 16, 7, 4, 26, 16, 2, 4, 2, 32, 8, 4, 18, 28, 2, 4, 8, 16, 2, 22, 2, 16, 17, 4, 2, 16, 30, 24, 8, 16, 2, 28, 43, 32, 8, 4, 2, 16, 2, 4, 8, 64, 32, 64, 2, 16, 8, 44, 2, 64, 2, 4, 68, 16, 18, 64, 2, 16, 80, 4, 2, 64, 32, 4, 8, 80

Offset: 1

Moosa Nasir, Nov 19 2022

This is only an empirical observation, but when we graph this sequence, a point always exists at the intersection of y = 2^b and y = -x + 2^(b+1), where b is any integer greater than or equal to 1. This means that a(2^b) = 2^b. This is shown in a link.

Many of the terms seem to be of the form 2^b.

			For n = 5, with a circular array of positions numbered clockwise from 1 to 5, start at position 1.
On move 1, travel 1 unit clockwise, reaching position 2.
On move 2, travel 2 units clockwise, reaching position 4.
On move 3, travel 4 units clockwise (almost a full circle), reaching position 3.
On move 4, travel 3 units clockwise, reaching position 1.
On move 5, travel 1 unit clockwise, reaching position 2.
Since the final position at the end of the 5th move is 2, a(5) = 2. (See the illustration in the links.)

Moosa Nasir, Example of a(5) = 2
Moosa Nasir, a(2^b) = 2^b

Cf. A015910, A082495.
Cf. A358647 (stepping in digits of n).
Equals {A082495} + 1. - Hugo Pfoertner, Nov 30 2022

int a(int n)
{
    int current = 1;
    for (int j = 0; j < n; j++) {
        current += current;
        if (current > n) {
            current = current - n;
        }
    }
    return current;
}

a(n) = lift(Mod(2,n)^n - 1) + 1; \\ Kevin Ryde, Nov 20 2022

def A357531(n): return m if (m:=pow(2,n,n)) else n # Chai Wah Wu, Dec 01 2022

a(n) = ((2^n - 1) mod n) + 1 = A082495(n) + 1. - Jon E. Schoenfield, Nov 20 2022

A065891 The a(n)-th composite number is 2^n.

Original entry on oeis.org

1, 3, 9, 20, 45, 96, 201, 414, 851, 1738, 3531, 7163, 14483, 29255, 58993, 118820, 239143, 480897, 966550, 1941540, 3898356, 7824444, 15699344, 31490742, 63151054, 126614174, 253804612, 508678161, 1019341795, 2042386082, 4091687074, 8196318785, 16416930072

Offset: 2

Labos Elemer, Nov 28 2001

nonn

Index of n-th power of 2 in A002808.
Remainder of division 2^n/c(n) equals zero, where c(n) = A002808(n), the n-th composite number.
Exponential increase with a factor > 2 and approaching two.

			For n = 4, 2^4 = 16 is the 9th composite number: 4,6,8,9,10,12,14,15,16, so a(4) = 9.

Cf. A000079, A000720, A002808, A015910, A065855, A073800.

seq(2^k - numtheory:-pi(2^k)-1, k=2..28); # Robert Israel, Dec 10 2024

Do[s=Mod[2^n, c[n]]; If[s==0, Print[n]], {n, 2, 1000000}]
Table[2^n-(PrimePi[2^n])-1, {n, 2, 31}]

lista(kmax) = {my(c = 0); forcomposite(k = 1, kmax, c++; if(k >> valuation(k, 2) == 1, print1(c, ", ")));} \\ Amiram Eldar, Jun 04 2024

a(n) = 2^n - A065855(2^n) - 1. - Robert Israel, Dec 10 2024

Edited by Robert G. Wilson v, Jun 18 2002

a(32)-a(34) from Amiram Eldar, Jun 04 2024

A073816 a(n) = 2^sigma(n) mod n.

Original entry on oeis.org

0, 0, 1, 0, 4, 4, 4, 0, 2, 4, 4, 4, 4, 8, 1, 0, 4, 8, 4, 4, 4, 20, 4, 16, 23, 12, 16, 4, 4, 16, 4, 0, 25, 30, 1, 20, 4, 26, 22, 24, 4, 22, 4, 16, 19, 18, 4, 16, 36, 42, 1, 4, 4, 46, 26, 8, 28, 6, 4, 16, 4, 2, 4, 0, 1, 16, 4, 64, 49, 36, 4, 8, 4, 64, 16, 44, 64, 40, 4, 64, 11, 64, 4, 4, 16

Offset: 1

Labos Elemer, Aug 16 2002

nonn

			n=17: sigma(17) = 18, 2^18 = 262144, 262144 mod 17 = 4 = a(17).

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

Cf. A000203, A015910.

Table[PowerMod[2,DivisorSigma[1,n],n],{n,90}] (* Harvey P. Dale, Jun 23 2014 *)

A096385 a(n) = smallest prime p with p^n mod n = 1.

Original entry on oeis.org

3, 7, 3, 11, 5, 29, 3, 7, 11, 23, 5, 53, 13, 31, 3, 103, 5, 191, 3, 37, 23, 47, 5, 11, 53, 7, 13, 59, 11, 311, 3, 67, 67, 71, 5, 149, 37, 61, 3, 83, 5, 173, 23, 31, 47, 283, 5, 29, 11, 103, 5, 107, 5, 31, 13, 7, 59, 709, 7, 367, 61, 37, 3, 131, 23, 269, 13, 139, 29, 569

Offset: 2

Reinhard Zumkeller, Aug 05 2004

nonn

			n=5: 2^5=32=5*6+2, 3^5=243=5*48+3, 5^5 mod 5 = 0, 7^5=16807=5*3361+2, 11^5=161051=5*32210+1: a(5)=11.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A015910, A066601, A066603, A066438, A066441.

With[{prs=Prime[Range[200]]},Table[SelectFirst[prs,PowerMod[#,n,n]==1&],{n,2,80}]] (* The program uses the SelectFirst function from Mathematica version 10 *) (* Harvey P. Dale, Aug 31 2015 *)

a(n) = my(p=2); while (Mod(p,n)^n !=1, p=nextprime(p+1)); p; \\ Michel Marcus, Feb 07 2021

A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 0, 2, 0, 3, 4, 1, 0, 1, 0, 1, 4, 3, 2, 1, 0, 0, 1, 0, 0, 4, 3, 0, 1, 0, 1, 2, 1, 1, 1, 4, 1, 8, 1, 0, 0, 0, 0, 2, 0, 5, 0, 0, 4, 1, 0, 1, 1, 1, 3, 1, 6, 1, 1, 9, 2, 1, 0, 0, 2, 0, 4, 4, 0, 0, 8, 6, 3, 4, 1, 0, 1, 0, 1, 0, 3, 1, 1, 0, 5, 4, 9, 2, 1

Offset: 1

Leroy Quet, Feb 14 2006

Alternate description: triangular array a(n, k) = n^k (mod k) read by rows (n > 1, 0 < k < n). This is equivalent because a(n, k) = a(n-k, k). - David Wasserman, Jan 25 2007

			2^6 = 64 and 64 (mod 6) is 4. So a(2,6) = 4.

Cf. A051127, A051128, A094263. Transpose of A060154. First 13 rows are A057427(n-1), A015910, A066601, A066602, A066603, A066604, A066438, A066439, A066440, A056969, A066441, A066442, A116609. First 12 columns are A000004, A000035, A010872, A000035, A010874, A070431, A010876, A000035, A021559(n+1), A008959, A010880, A070435.

a[n_, k_] := Mod[n^k, k]; Table[a[n - k + 1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

More terms from David Wasserman, Jan 25 2007

A155886 Least k such that 2^(2^k) = n (mod k).

Original entry on oeis.org

1, 3, 14, 11, 6, 1941491, 10, 83, 31, 13, 123, 35, 71, 27, 34913, 241, 18, 8059, 34, 349, 44, 25, 39, 100867561, 76, 231, 253, 66203, 57, 227, 139, 45, 184, 37, 111, 97, 55, 41, 103, 1099, 81, 66791, 53

Offset: 0

T. D. Noe, Jan 29 2009

First occurrence of n in sequence A155836.

a(43) > 12500000. - Tyler Busby, Mar 15 2024

Cf. A015910 (2^n mod n), A036236, A155836.

A180074 Squarefree semiprimes s=p*q, p

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 133, 134, 141, 142, 145, 146, 158, 159, 166, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 217, 218, 219, 226, 237

Offset: 1

Juri-Stepan Gerasimov, Jan 14 2011

nonn

It may seem that this is a subsequence of A162730, but it is not so, 131801 being the first counterexample. - Michel Marcus, Sep 19 2018

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

Cf. A006881, A015910, A162730, A179976.

f[n_]:=With[{f=FactorInteger[n][[All,1]]},PowerMod[ 2,Times@@f,Times@@f] == 2^f[[1]]]; Select[Range[250],PrimeOmega[#]==2&&SquareFreeQ[#]&&f[#]&] (* Harvey P. Dale, Jun 06 2017 *)

isok(n) = {if ((bigomega(n) == 2) && (omega(n) == 2), my(p = factor(n)[1, 1]); lift(Mod(2, n)^n) == 2^p);} \\ Michel Marcus, Sep 19 2018

Definition and terms corrected by R. J. Mathar, Jan 14 2011

A374911 a(n) = a(2^n mod n) + a(3^n mod n), with a(0) = 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 7, 7, 7, 3, 4, 7, 7, 7, 7, 7, 10, 3, 7, 11, 7, 5, 10, 7, 7, 7, 18, 7, 8, 21, 7, 7, 7, 3, 11, 7, 18, 25, 7, 7, 11, 5, 7, 17, 7, 10, 18, 7, 7, 14, 14, 21, 11, 10, 7, 29, 14, 7, 11, 7, 7, 13, 7, 7, 11, 3, 17, 7, 7, 10, 11, 21, 7, 7, 7, 7, 21, 10, 32, 11, 7, 5, 6, 7, 7, 14, 10, 7, 11, 19

Offset: 0

Bryle Morga, Jul 23 2024

Conjectured to contain all positive integers. Here are the indexes where each of the first few positive integers appear:
0
1
2, 4, 8, 16, 32, ... (2^k, k > 0)
3, 9, ...
20, 40, 80, 272, 320, 328, ...
81, 66469, 144937, ...
5, 6, 7, 10, 11, 12, 13,... (all primes appear except 2 and 3)
27, 301, 729, 1099, 2107, 2187, 85085, 1594323, ...
Most solutions to a(n) = 5 seem to be divisible by 5 and all of them seem to be even. Why?
Are 3 and 9 the only solutions to a(n) = 4?

John Tyler Rascoe, Table of n, a(n) for n = 0..10000

Cf. A000079, A015910, A066601.

a[0]=1; a[n_]:=a[PowerMod[2,n,n]]+a[PowerMod[3,n,n]]; Array[a,89,0] (* Stefano Spezia, Jul 23 2024 *)

a(n) = if (n==0, 1, a(lift(Mod(2,n)^n)) + a(lift(Mod(3,n)^n))); \\ Michel Marcus, Jul 25 2024

def a(n):
  return 1 if n == 0 else a(pow(2, n, n)) + a(pow(3, n, n))

a(p) = 7 for primes p except 2 and 3.

a(2^n) = 3 for n > 0.

A069049 Numbers k such that 2^k mod k = 2^phi(k) mod phi(k).

Original entry on oeis.org

1, 2, 4, 8, 14, 16, 22, 26, 32, 44, 46, 52, 62, 64, 92, 94, 108, 112, 118, 124, 128, 154, 164, 166, 188, 214, 222, 234, 236, 244, 252, 256, 258, 264, 288, 332, 334, 336, 358, 390, 412, 428, 438, 454, 456, 504, 512, 526, 534, 546, 576, 582, 630, 664, 668, 672

Offset: 1

Benoit Cloitre, Apr 03 2002

Numbers k such that A015910(k) = A015910(A000010(k)). - Michel Marcus, Feb 11 2021

Cf. A000010, A015910.

Select[Range[1000], PowerMod[2, #, #] == PowerMod[2, (e = EulerPhi[#]), e] &] (* Amiram Eldar, Feb 11 2021 *)

f(n) = lift(Mod(2, n)^n); \\ A015910
isok(k) = f(k) == f(eulerphi(k)); \\ Michel Marcus, Feb 11 2021

cp's OEIS Frontend

A294390 a(n) = 2^(n-4) mod n, for n >= 4.

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A065891 The a(n)-th composite number is 2^n.

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A073816 a(n) = 2^sigma(n) mod n.

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Mathematica

A096385 a(n) = smallest prime p with p^n mod n = 1.

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A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

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Mathematica

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A155886 Least k such that 2^(2^k) = n (mod k).

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A180074 Squarefree semiprimes s=p*q, p

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