A054715 -id:A054715 - cp's OEIS frontend

A054703 Number of distinct powers of 2 modulo n.

1, 2, 2, 3, 4, 3, 3, 4, 6, 5, 10, 4, 12, 4, 4, 5, 8, 7, 18, 6, 6, 11, 11, 5, 20, 13, 18, 5, 28, 5, 5, 6, 10, 9, 12, 8, 36, 19, 12, 7, 20, 7, 14, 12, 12, 12, 23, 6, 21, 21, 8, 14, 52, 19, 20, 6, 18, 29, 58, 6, 60, 6, 6, 7, 12, 11, 66, 10, 22, 13, 35, 9, 9, 37, 20, 20, 30, 13, 39, 8, 54

Offset: 1

Henry Bottomley, Apr 20 2000

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A007733, A007814.

Cf. A054704 (base 3), A054705 (4), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054717 (9), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := Module[{e = IntegerExponent[n, 2]}, e + MultiplicativeOrder[2, n/2^e]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007814(n) + A007733(n). - Max Alekseyev, May 13 2016

Definition improved by T. D. Noe, Jul 21 2008

A054710 Number of powers of 10 mod n.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 6, 4, 1, 2, 2, 3, 6, 7, 2, 5, 16, 2, 18, 3, 6, 3, 22, 4, 3, 7, 3, 8, 28, 2, 15, 6, 2, 17, 7, 3, 3, 19, 6, 4, 5, 7, 21, 4, 2, 23, 46, 5, 42, 3, 16, 8, 13, 4, 3, 9, 18, 29, 58, 3, 60, 16, 6, 7, 7, 3, 33, 18, 22, 7, 35, 4, 8, 4, 3, 20, 6, 7, 13, 5, 9, 6, 41, 8, 17, 22, 28, 5, 44, 2

Offset: 1

Henry Bottomley, Apr 20 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A007732, A051628, A121341.

Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054717 (9), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

Table[Length[Union[PowerMod[10, Range[0,n], n]]], {n,100}] (* T. D. Noe, Aug 30 2006 *)
a[n_] := Module[{e = IntegerExponent[n, {2, 5}]}, Max[e] + MultiplicativeOrder[10, n/Times @@ ({2, 5}^e)]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007732(n) + A051628(n). - Amiram Eldar, Aug 25 2024

A054717 Number of powers of 9 modulo n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 5, 2, 3, 3, 3, 2, 8, 2, 9, 2, 4, 5, 11, 2, 10, 3, 3, 3, 14, 3, 15, 4, 6, 8, 6, 2, 9, 9, 4, 2, 4, 4, 21, 5, 3, 11, 23, 3, 21, 10, 9, 3, 26, 3, 10, 3, 10, 14, 29, 3, 5, 15, 4, 8, 6, 6, 11, 8, 12, 6, 35, 2, 6, 9, 11, 9, 15, 4, 39, 2, 3, 4, 41, 4, 8, 21, 15, 5, 44, 3, 3

Offset: 1

Henry Bottomley, Apr 20 2000

			Take the sequence 1, 9, 81, 729, ... and reduce mod n; count distinct terms. For n = 5 we get 1, 4, 1, 4, ... so a(5) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from David W. Wilson)

Cf. A007740.

Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

With[{p9=9^Range[0,50]},Table[Length[Union[Mod[#,n]&/@p9]],{n,100}]] (* Harvey P. Dale, Apr 22 2012 *)
a[n_] := IntegerExponent[3*n, 9] + MultiplicativeOrder[9, n/3^IntegerExponent[n, 3]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = valuation(3*n, 9) + A007740(n). - Amiram Eldar, Aug 25 2024

A351524 Number of powers of 11 modulo n.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 2, 6, 1, 2, 2, 12, 3, 2, 4, 16, 6, 3, 2, 6, 2, 22, 2, 5, 12, 18, 6, 28, 2, 30, 8, 3, 16, 3, 6, 6, 3, 12, 2, 40, 6, 7, 3, 6, 22, 46, 4, 21, 5, 16, 12, 26, 18, 2, 6, 6, 28, 58, 2, 4, 30, 6, 16, 12, 3, 66, 16, 22, 3, 70, 6, 72, 6, 10, 6, 4, 12, 39, 4, 54, 40, 41, 6, 16, 7, 28

Offset: 1

Georg Fischer, Feb 13 2022

This is the original version of A054711 rev. #1 as defined by Henry Bottomley, Apr 20 2000.

A054711 is now different from the sequence here.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A054711.

Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054717 (9), A054710 (10), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := Module[{e = IntegerExponent[n, 11]}, e + MultiplicativeOrder[11, n/11^e]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

A054704 Number of powers of 3 modulo n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 2, 3, 4, 5, 3, 3, 6, 5, 4, 16, 3, 18, 4, 7, 5, 11, 3, 20, 3, 4, 6, 28, 5, 30, 8, 6, 16, 12, 4, 18, 18, 4, 4, 8, 7, 42, 10, 6, 11, 23, 5, 42, 20, 17, 6, 52, 4, 20, 6, 19, 28, 29, 5, 10, 30, 8, 16, 12, 6, 22, 16, 12, 12, 35, 4, 12, 18, 21, 18, 30, 4, 78, 4, 5, 8, 41, 7, 16

Offset: 1

Henry Bottomley, Apr 20 2000

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from David W. Wilson)

Cf. A007734, A007949.

Cf. A054703 (base 2), A054705 (4), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054717 (9), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := Module[{e = IntegerExponent[n, 3]}, e + MultiplicativeOrder[3, n/3^e]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007734(n) + A007949(n). - Amiram Eldar, Aug 25 2024

A054705 Number of powers of 4 modulo n.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 3, 3, 3, 3, 5, 2, 6, 4, 2, 3, 4, 4, 9, 3, 3, 6, 11, 3, 10, 7, 9, 4, 14, 3, 5, 4, 5, 5, 6, 4, 18, 10, 6, 4, 10, 4, 7, 6, 6, 12, 23, 3, 21, 11, 4, 7, 26, 10, 10, 5, 9, 15, 29, 3, 30, 6, 3, 4, 6, 6, 33, 5, 11, 7, 35, 5, 9, 19, 10, 10, 15, 7, 39, 4, 27, 11, 41, 4, 4, 8, 14, 7, 11

Offset: 1

Henry Bottomley, Apr 20 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from David W. Wilson)

Cf. A007735, A244415.

Cf. A054703 (base 2), A054704 (3), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054717 (9), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := IntegerExponent[2*n, 4] + MultiplicativeOrder[4, n/2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007735(n) + A244415(n). - Amiram Eldar, Aug 25 2024

A054706 Number of powers of 5 modulo n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 6, 2, 6, 2, 5, 2, 4, 6, 3, 4, 16, 6, 9, 2, 6, 5, 22, 2, 3, 4, 18, 6, 14, 3, 3, 8, 10, 16, 7, 6, 36, 9, 4, 3, 20, 6, 42, 5, 7, 22, 46, 4, 42, 3, 16, 4, 52, 18, 6, 6, 18, 14, 29, 3, 30, 3, 6, 16, 5, 10, 22, 16, 22, 7, 5, 6, 72, 36, 4, 9, 30, 4, 39, 5, 54, 20, 82, 6, 17, 42, 14

Offset: 1

Henry Bottomley, Apr 20 2000

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from David W. Wilson)

Cf. A007736, A112765.

Cf. A054703 (base 2), A054704 (3), A054705 (4), A054707 (6), A054708 (7), A054709 (8), A054717 (9), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := Module[{e = IntegerExponent[n, 5]}, e + MultiplicativeOrder[5, n/5^e]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007736(n) + A112765(n). - Amiram Eldar, Aug 25 2024

A054707 Number of powers of 6 modulo n.

Original entry on oeis.org

1, 2, 2, 3, 1, 2, 2, 4, 3, 2, 10, 3, 12, 3, 2, 5, 16, 3, 9, 3, 3, 11, 11, 4, 5, 13, 4, 4, 14, 2, 6, 6, 11, 17, 2, 3, 4, 10, 13, 4, 40, 3, 3, 12, 3, 12, 23, 5, 14, 6, 17, 14, 26, 4, 10, 5, 10, 15, 58, 3, 60, 7, 4, 7, 12, 11, 33, 18, 12, 3, 35, 4, 36, 5, 6, 11, 10, 13, 78, 5, 5, 41, 82, 4, 16

Offset: 1

Henry Bottomley, Apr 20 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from David W. Wilson)

Cf. A007737, A244417.

Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054708 (7), A054709 (8), A054717 (9), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := Module[{e = IntegerExponent[n, {2, 3}]}, Max[e] + MultiplicativeOrder[6, n/Times @@ ({2, 3}^e)]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007737(n) + A244417(n). - Amiram Eldar, Aug 25 2024

A054708 Number of powers of 7 modulo n.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 2, 2, 3, 4, 10, 2, 12, 2, 4, 2, 16, 3, 3, 4, 2, 10, 22, 2, 4, 12, 9, 3, 7, 4, 15, 4, 10, 16, 5, 6, 9, 3, 12, 4, 40, 2, 6, 10, 12, 22, 23, 2, 3, 4, 16, 12, 26, 9, 20, 3, 3, 7, 29, 4, 60, 15, 4, 8, 12, 10, 66, 16, 22, 5, 70, 6, 24, 9, 4, 6, 11, 12, 78, 4, 27, 40, 41, 3, 16, 6

Offset: 1

Henry Bottomley, Apr 20 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from David W. Wilson)

Cf. A007738, A214411.

Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054707 (6), A054709 (8), A054717 (9), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := Module[{e = IntegerExponent[n, 7]}, e + MultiplicativeOrder[7, n/7^e]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007738(n) + A214411(n). - Amiram Eldar, Aug 25 2024

A054709 Number of powers of 8 modulo n.

Original entry on oeis.org

1, 2, 2, 2, 4, 3, 1, 2, 2, 5, 10, 3, 4, 2, 4, 3, 8, 3, 6, 5, 2, 11, 11, 3, 20, 5, 6, 2, 28, 5, 5, 3, 10, 9, 4, 3, 12, 7, 4, 5, 20, 3, 14, 11, 4, 12, 23, 4, 7, 21, 8, 5, 52, 7, 20, 2, 6, 29, 58, 5, 20, 6, 2, 3, 4, 11, 22, 9, 22, 5, 35, 3, 3, 13, 20, 7, 10, 5, 13, 6, 18, 21, 82, 3, 8, 15, 28

Offset: 1

Henry Bottomley, Apr 20 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from David W. Wilson)

Cf. A007739.

Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054707 (6), A054708 (7), A054717 (9), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := IntegerExponent[4*n, 8] + MultiplicativeOrder[8, n/2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = valuation(4*n, 8) + A007739(n). - Amiram Eldar, Aug 25 2024

cp's OEIS Frontend

A054703 Number of distinct powers of 2 modulo n.

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A054710 Number of powers of 10 mod n.

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A054717 Number of powers of 9 modulo n.

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A351524 Number of powers of 11 modulo n.

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A054704 Number of powers of 3 modulo n.

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A054705 Number of powers of 4 modulo n.

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A054706 Number of powers of 5 modulo n.

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A054707 Number of powers of 6 modulo n.

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A054708 Number of powers of 7 modulo n.

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