A153513 -id:A153513 - cp's OEIS frontend

A153514 Terms of A122780 which are not Carmichael numbers A002997.

1, 6, 66, 91, 121, 286, 671, 703, 726, 949, 1541, 1891, 2665, 2701, 3281, 3367, 3751, 4961, 5551, 7107, 7381, 8205, 8401, 8646, 11011, 12403, 14383, 15203, 15457, 16471, 16531, 18721, 19345, 23521, 24046, 24661, 24727, 28009, 29161, 30857, 31621

Offset: 1

Artur Jasinski, Dec 28 2008

nonn

For the intersection of this sequence and A153508, see A153513.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A002997, A122780, A153508, A153513.

filter:= proc(n) local p;
  if isprime(n) or (3 &^n - 3 mod n <> 0) then return false fi;
  if n::even then return true fi;
  if not numtheory:-issqrfree(n) then return true fi;
  for p in numtheory:-factorset(n) do
    if n-1 mod (p-1) <> 0 then return true fi
  od;
false
end proc:
filter(1):= true:
select(filter, [$1..10^5]); # Robert Israel, Jan 29 2017

okQ[n_] := !PrimeQ[n] && PowerMod[3, n, n] == Mod[3, n] && Mod[n, CarmichaelLambda[n]] != 1;
Select[Range[10^5], okQ] (* Jean-François Alcover, Mar 27 2019 *)

A153515 Terms of A122782 which are not Carmichael numbers A002997.

Original entry on oeis.org

1, 4, 10, 15, 20, 65, 124, 190, 217, 310, 435, 781, 1541, 1891, 3565, 3820, 4123, 4495, 5461, 5611, 5662, 5731, 6735, 7449, 7813, 8029, 8290, 9881, 11041, 11476, 12801, 13021, 13333, 13981, 14981, 15751, 16297, 17767, 20345, 20710, 21361, 22791

Offset: 1

Artur Jasinski, Dec 28 2008

nonn

Are there entries in this sequence which are also in A153513 ?

Yes. This subsequence starts 721801, 873181, 4504501, 8646121, 9006401, 9863461, 10403641, 10680265,... (similar to A153580). - R. J. Mathar, Mar 30 2011

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

Cf. A002997, A122780, A153508, A153513.

Select[Range[10^4], !PrimeQ[#] && PowerMod[5, #, # ] == Mod[5, #] && Mod[#, CarmichaelLambda[#]] != 1 &] (* Amiram Eldar, Sep 19 2019 *)

A153580 Terms of A083737 which are not Carmichael numbers (A002997).

Original entry on oeis.org

721801, 873181, 4504501, 8646121, 9006401, 9863461, 10403641, 12322133, 14609401, 15913261, 18595801, 18736381, 20234341, 21397381, 22066201, 22369621, 22885129, 25326001, 25696133, 28312921, 36307981, 42702661

Offset: 1

Ray Chandler & Artur Jasinski, Dec 28 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2871 (calculated from the b-file at A083737)

Cf. A002997, A083737, A122780, A153508, A153513, A153514, A153515, A153519, A153581.

Select[Range[5*10^7], ! PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, # - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 && Mod[ #, CarmichaelLambda[ # ]] != 1 &] (* Ray Chandler, Dec 28 2008 *)

A153581 Pseudoprimes to bases 2,3,5 and 7 which are not Carmichael numbers (A002997).

Original entry on oeis.org

721801, 8646121, 10403641, 22885129, 36307981, 42702661, 46094401, 48064021, 52204237, 79398901, 80918281, 81954133, 114329881, 116151661, 143168581, 170782921, 188985961, 217145881, 220531501, 282707461, 299671921, 303373801, 326695141, 353815801, 361307521

Offset: 1

Ray Chandler & Artur Jasinski, Dec 28 2008

nonn

Terms congruent to 5 (mod 6): 468950021, 493108481, 659846021, 5936122901, 8144063621, ... - Robert G. Wilson v, Sep 03 2014

Terms not congruent to 1 (mod 12): 468950021, 493108481, 643767931, 659846021, 773131927, 5779230451, 5936122901, 7294056727, 8144063621, 9671001451, ... - Robert G. Wilson v, Sep 03 2014

Amiram Eldar, Table of n, a(n) for n = 1..1077 (terms 1..125 from Robert G. Wilson v)

Cf. A002997, A083737, A122780, A153508, A153513, A153514, A153515, A153519, A153580.

fQ[n_] := ! PrimeQ[n] && PowerMod[2, n - 1, n] == 1 && PowerMod[3, n - 1, n] == 1 && PowerMod[5, n - 1, n] == 1 && PowerMod[7, n - 1, n] == 1 && Mod[n, CarmichaelLambda[n]] != 1; Select[ Range[ 365000000], fQ] (* Ray Chandler, Dec 28 2008; corrected by Robert G. Wilson v, Sep 01 2014 *)

Terms a(8) and onward from Robert G. Wilson v, Sep 01 2014

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

Original entry on oeis.org

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

A333130 Numbers that are super pseudoprimes to both bases 2 and 3.

Original entry on oeis.org

2701, 18721, 31621, 49141, 83333, 90751, 104653, 226801, 282133, 653333, 665281, 721801, 873181, 1373653, 1530787, 1537381, 1584133, 1690501, 1755001, 1987021, 2008597, 2035153, 2284453, 2746589, 2944261, 3059101, 3116107, 3363121, 3375041, 3375487, 4082653, 4314967

Offset: 1

Amiram Eldar, Mar 08 2020

nonn

The first term that has more than 2 prime factors is a(1067) = A333131(1) = 11500521553.

The first term that is also a Carmichael number is a(1131) = 13079177569.

			2701 is a term since it is a Fermat pseudoprime to both bases 2 and 3, and its proper divisors that are larger than 1 are all primes: 37 and 73.

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

Intersection of A050217 and A328662.

Subsequence of A001567, A005935, A052155 and A153513.

pspQ[n_] := CompositeQ[n] && AllTrue[Rest @ Divisors[n], PowerMod[2, # - 1, #] == 1 && PowerMod[3, # - 1, #] == 1 &]; Select[Range[10^6], pspQ]

A333131 Super pseudoprimes to both bases 2 and 3 (A333130) with more than two prime factors (counted with multiplicity).

Original entry on oeis.org

11500521553, 13079177569, 52474339009, 168003672409, 229352039821, 280792563977, 318289021201, 428178002569, 918660756421, 2015841188197, 2367478228501, 2544457029601, 2639665216117, 3023595814801, 3457449931321, 3712164285421, 4348114583017, 6046196043229

Offset: 1

Amiram Eldar, Mar 08 2020

nonn

Up to 2^64 all the 1085 terms are nonsquarefree, 2 terms have 4 prime factors: a(163) = 18362297383286473 = 3037 * 6073 * 9109 * 109297 and a(651) = 2587580959818925201 = 18121 * 36241 * 54361 * 72481, and no term have more than 4 prime factors.

			11500521553 is a term since it is a Fermat pseudoprime to both bases 2 and 3, and its proper divisors that are larger than 1 are either primes (937, 1873, 6553) or Fermat pseudoprimes to both bases 2 and 3 (1755001, 6140161, 12273769, 11500521553).

Amiram Eldar, Table of n, a(n) for n = 1..1085 (terms below 2^64)

Subsequence of A001567, A005935, A050217, A052155, A153513, A178997, A328662, A328663, A333130.

pspQ[n_] := PrimeOmega[n] > 2 && AllTrue[Rest @ Divisors[n], PowerMod[2, # - 1, #] == 1 && PowerMod[3, # - 1, #] == 1 &]; seq = {}; Do[If[pspQ[n], AppendTo[seq, n]], {n, 1, 6*10^10}]; seq

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A153514 Terms of A122780 which are not Carmichael numbers A002997.

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A153515 Terms of A122782 which are not Carmichael numbers A002997.

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A153580 Terms of A083737 which are not Carmichael numbers (A002997).

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A153581 Pseudoprimes to bases 2,3,5 and 7 which are not Carmichael numbers (A002997).

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

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A333130 Numbers that are super pseudoprimes to both bases 2 and 3.

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A333131 Super pseudoprimes to both bases 2 and 3 (A333130) with more than two prime factors (counted with multiplicity).

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