A122780 -id:A122780 - cp's OEIS frontend

A153519 Nonprime numbers k such that 10^k == 10 (mod k) and are not Carmichael numbers.

1, 6, 9, 10, 15, 18, 30, 33, 45, 55, 90, 91, 99, 165, 246, 259, 370, 385, 451, 481, 495, 505, 657, 703, 715, 909, 1035, 1045, 1233, 1626, 2035, 2409, 2981, 3333, 3367, 3585, 4005, 4141, 4187, 4521, 4545, 5005, 5461, 6533, 6541, 6565, 7107, 7471, 7777, 8149

Offset: 1

Artur Jasinski, Dec 28 2008

nonn

Old name: Members of A121014 which are not Carmichael numbers A002997.

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

Cf. A002997, A122780, A153508, A153513, A153514, A153515.

Select[Range[8000], !PrimeQ[#] && PowerMod[10, #, #] == Mod[10, #] && !(# > 1 && Divisible[# - 1, CarmichaelLambda[#]]) &] (* Amiram Eldar, Mar 19 2020 *)

isok(n) = !isprime(n) && !is_A002997(n) && (Mod(10^n, n) == Mod(10, n)); \\ Michel Marcus, Nov 06 2013

New name from Michel Marcus, Nov 06 2013

A290543 Composite numbers n such that A290542(n) >= 2.

Original entry on oeis.org

28, 65, 66, 85, 91, 105, 117, 121, 124, 133, 145, 153, 154, 165, 185, 186, 190, 205, 217, 221, 231, 244, 246, 247, 259, 273, 276, 280, 286, 292, 301, 305, 310, 325, 341, 343, 344, 357, 364, 366, 369, 370, 377, 385, 396, 418, 425, 427, 429, 430, 435, 451

Offset: 1

Arkadiusz Wesolowski, Aug 05 2017

nonn

Is a(n) ~ n * log n as n -> infinity?

Cf. A001567, A006935, A122780, A122781, A122782, A122783, A122784, A122785, A122786, A290542.

lst:=[]; for n in [4..451] do if not IsPrime(n) then r:=Floor(Sqrt(n)); for k in [2..r] do if Modexp(k, n, n) eq k then Append(~lst, n); break; end if; end for; end if; end for; lst;

Select[Flatten@ Position[#, k_ /; k >= 2], CompositeQ] &@ Table[SelectFirst[Range[2, Sqrt@ n], PowerMod[#, n , n] == Mod[#, n] &], {n, 451}] (* Michael De Vlieger, Aug 09 2017 *)

A306451 Non-coprime pseudoprimes or primes to base 3: numbers k that are multiples of 3 and are such that k divides 3^k - 3.

Original entry on oeis.org

3, 6, 66, 561, 726, 7107, 8205, 8646, 62745, 100101, 140097, 166521, 237381, 237945, 566805, 656601, 876129, 1053426, 1095186, 1194285, 1234806, 1590513, 1598871, 1938021, 2381259, 2518041, 3426081, 4125441, 5398401, 5454681, 5489121, 5720331, 5961441

Offset: 1

Jianing Song, Feb 17 2019

nonn

Union of {3} and (A122780 - {1} - A005935).
Numbers of the form 3*m such that 3^(3*m-1) == 1 (mod m).
The squarefree terms are listed in A306450.

Jianing Song, Table of n, a(n) for n = 1..163 (all terms below 10^9)

Cf. A005935, A006935, A122780, A306450.

A258801 is a subsequence.

forstep(n=3, 1e7, 3, if(Mod(3, n)^n==3, print1(n, ", ")))

66 is a term because 66 divides 3^66 - 3 = 3*(3^65 - 1) = 3*(3^5 - 1)*(3^60 + 3^55 + ... + 3^5 + 1) and 66 is divisible by 3.

A306452 Pseudoprimes to base 3 that are not squarefree, including the non-coprime pseudoprimes.

Original entry on oeis.org

121, 726, 3751, 4961, 7381, 11011, 29161, 32791, 142901, 228811, 239701, 341341, 551881, 566401, 595441, 671671, 784201, 856801, 1016521, 1053426, 1237951, 1335961, 1433971, 1804231, 1916761, 2000251, 2254351, 2446741, 2817001, 2983981, 3078361, 3307051, 3562361

Offset: 1

Jianing Song, Feb 17 2019

nonn

Numbers k such that 3^k == 3 (mod k) and k is divisible by the square of a Mirimanoff prime (or base-3 Wieferich prime), A014127.
A non-coprime pseudoprime in base b is a number k such that b^k == b (mod k) and that gcd(b, k) > 1, and the non-coprime pseudoprime in base 3 (726, 1053426, 6498426, ...) that are not squarefree are listed in A306450 while the others terms in this sequence (121, 3751, 4961, ...) are listed in A244065. So this sequence is the union of A244065 and A306450.
Intersection of A122780 and A013929.

			121 is a term because 3^120 == (3^5)^24 == 1 (mod 121) and 121 = 11^2.
Although 3^725 = 243 rather than 1 mod 726, we see that nevertheless 3^726 = 3 mod 726, and since 726 = 2 * 3 * 11^2, 726 is in the sequence. - _Alonso del Arte_, Mar 16 2019

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A013929, A014127, A122780, A158358, A244065, A306450.

Select[Range[5000], PowerMod[3, #, #] == 3 && MoebiusMu[#] == 0 &] (* Alonso del Arte, Mar 16 2019 *)

forcomposite(n=1, 10^7, if(Mod(3, n)^n==3 && !issquarefree(n), print1(n, ", ")))

A247033 Numbers of the form (3^k - 3)/k.

Original entry on oeis.org

0, 3, 8, 48, 121, 312, 16104, 122640, 7596480, 61171656, 4093181688, 2366564736720, 19924948267224, 12169835294351280, 889585277491970400, 7633882962663652968, 565719454445904325272, 365721616371321130128240, 239498069351503974657030696, 2084811062715550992506283600

Offset: 1

Juri-Stepan Gerasimov, Sep 09 2014

nonn

Generated by k: 1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 66, 67, ...

Subsequence of A246445.

			121 is in this sequence because (3^k - 3)/k = (3^6 - 3)/6 = 121.

Cf. A000040, A005935, A064535 (with form (2^k - 2)/k), A122780 (nonprimes k in a(n)), A246445, A247307 (with form (4^k - 4)/k).

cp's OEIS Frontend

A153519 Nonprime numbers k such that 10^k == 10 (mod k) and are not Carmichael numbers.

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A290543 Composite numbers n such that A290542(n) >= 2.

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A306451 Non-coprime pseudoprimes or primes to base 3: numbers k that are multiples of 3 and are such that k divides 3^k - 3.

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A306452 Pseudoprimes to base 3 that are not squarefree, including the non-coprime pseudoprimes.

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A247033 Numbers of the form (3^k - 3)/k.

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