A129494 -id:A129494

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A129492

Composite numbers k such that 2^k mod k is a power of 2.

+10
6

6, 9, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 34, 38, 39, 40, 44, 46, 48, 51, 52, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 72, 74, 76, 78, 80, 82, 84, 85, 86, 87, 90, 92, 93, 94, 96, 102, 106, 111, 112, 114, 116, 118, 120, 122, 123, 124, 126, 129, 132, 133, 134, 138

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Complement to composite numbers: 4, 8, 16, 18, 25, 27, 32, 35, 36, 42, 45, 49, 50, 54, 55, 64, 70, 75, 77, 81, 88, 91, 95, 98, 99, ....

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

15 is a term since 2^15 mod 15 = 8.

MAPLE

filter:= proc(n) local k;

if isprime(n) then return false fi;

k:= 2 &^ n mod n;

k > 1 and k = 2^padic:-ordp(k, 2)

end proc:

select(filter, [$4..1000]); # Robert Israel, Dec 03 2019

MATHEMATICA

Select[Range@ 141, IntegerQ@ Log[2, PowerMod[2, #, # ]] &]

PROG

(Magma) [k:k in [2..150]| not IsPrime(k) and not IsZero(a) and (PrimeDivisors(a) eq [2]) where a is 2^k mod k ]; // Marius A. Burtea, Dec 04 2019

CROSSREFS

Cf. A036236, A129493, A129494, A129495, A129496, A129497.

Contains A001567.

KEYWORD

easy,nonn

AUTHOR

Robert G. Wilson v, Apr 17 2007

STATUS

approved

A129493

Composite numbers k such that 3^k mod k is a power of 3.

+10
6

6, 10, 12, 14, 18, 22, 24, 26, 30, 33, 34, 36, 38, 39, 46, 51, 54, 56, 57, 58, 62, 63, 66, 69, 72, 74, 78, 82, 86, 87, 90, 91, 92, 93, 94, 99, 104, 106, 108, 111, 112, 116, 117, 118, 120, 121, 122, 123, 124, 129, 132, 134, 135, 141, 142, 144, 146, 148, 154, 158, 159

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Complement to composite numbers: 9, 15, 21, 25, 27, 28, 35, 42, 44, 45, 48, 49, 50, 52, 55, 60, 65, 68, 70, 75, ....

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

14 is a member of the sequence since 3^14 mod 14 = 9.

MAPLE

filter:= proc(n) local k;

if isprime(n) then return false fi;

k:= 3 &^ n mod n;

k > 1 and k = 3^padic:-ordp(k, 3)

end proc:

select(filter, [$4..1000]); # Robert Israel, Dec 03 2019

MATHEMATICA

Select[Range@ 161, IntegerQ@ Log[3, PowerMod[3, #, # ]] &]

PROG

(Magma) [k:k in [2..160]| not IsPrime(k) and not IsZero(a) and (PrimeDivisors(a) eq [3]) where a is 3^k mod k ]; // Marius A. Burtea, Dec 04 2019

CROSSREFS

Cf. A036236, A122780, A129492, A129494, A129495, A129496, A129497.

KEYWORD

easy,nonn

AUTHOR

Robert G. Wilson v, Apr 17 2007

STATUS

approved

A129495

Composite numbers k such that 5^k (mod k) is a power of 5 greater than 1.

+10
6

10, 15, 20, 26, 30, 34, 38, 40, 46, 50, 56, 58, 60, 62, 65, 74, 78, 82, 86, 94, 100, 106, 118, 120, 122, 124, 129, 130, 132, 134, 140, 141, 142, 143, 146, 150, 155, 158, 159, 166, 177, 178, 182, 183, 190, 194, 195, 200, 201, 202, 206, 213, 214, 217, 218, 219

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

EXAMPLE

26 is a member of the sequence since 5^26 (mod 26) == 25.

MATHEMATICA

Select[Range@ 225, (p = PowerMod[5, #, #]) > 1 && IntegerQ@ Log[5, p] && CompositeQ[#] &] (* corrected by Amiram Eldar, Jul 24 2021 *)

CROSSREFS

Cf. A036236, A129492, A129493, A129494, A129496, A129497.

KEYWORD

easy,nonn

AUTHOR

Robert G. Wilson v, Apr 17 2007

STATUS

approved

A129496

Composite numbers k such that 6^k (mod k) is a power of 6 greater than 1.

+10
6

10, 15, 21, 30, 35, 38, 42, 45, 46, 58, 60, 62, 70, 74, 82, 84, 86, 90, 94, 105, 106, 118, 122, 126, 132, 134, 140, 142, 146, 158, 166, 178, 180, 182, 185, 190, 194, 202, 206, 210, 214, 215, 217, 218, 219, 222, 226, 228, 231, 237, 249, 252, 254, 258, 259, 262

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

EXAMPLE

38 is a member of the sequence since 6^38 (mod 38) == 36.

MATHEMATICA

Select[Range@ 266, (p = PowerMod[6, #, #]) > 1 && IntegerQ@ Log[6, p] && CompositeQ[#] &] (* corrected by Amiram Eldar, Jul 24 2021 *)

CROSSREFS

Cf. A036236, A129492, A129493, A129494, A129495, A129497.

KEYWORD

easy,nonn

AUTHOR

Robert G. Wilson v, Apr 17 2007

STATUS

approved

A129497

Composite numbers k such that 7^k (mod k) is a power of 7 greater than 1.

+10
6

14, 21, 25, 42, 50, 56, 58, 62, 70, 74, 82, 84, 86, 94, 98, 105, 106, 112, 118, 122, 132, 133, 134, 142, 146, 147, 150, 152, 158, 166, 168, 178, 182, 194, 196, 202, 206, 210, 214, 218, 226, 231, 254, 262, 266, 274, 278, 294, 298, 301, 302, 314, 325, 326, 334

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

EXAMPLE

58 is a member of the sequence since 7^58 (mod 58) == 49.

MATHEMATICA

Select[Range@ 335, (p = PowerMod[7, #, #]) > 1 && IntegerQ@ Log[7, p] && CompositeQ[#] &] (* corrected by Amiram Eldar, Jul 24 2021 *)

CROSSREFS

Cf. A036236, A129492, A129493, A129494, A129495, A129496.

KEYWORD

easy,nonn

AUTHOR

Robert G. Wilson v, Apr 17 2007

EXTENSIONS

Corrected and edited by R. J. Mathar, May 16 2008

STATUS

approved

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