A002997 -id:A002997 - cp's OEIS frontend

A324320 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also octagonal numbers (A000567) with index equal to their largest prime factor.

1045, 2465, 2821, 15841, 20501, 34133, 51221, 68101, 89441, 116033, 118405, 162401, 170885, 216545, 300833, 364705, 439301, 472033, 530881, 642181, 687365, 746005, 970145, 976981, 997633, 1104133, 1148245, 1193221, 1231361, 1239061, 1398101, 1654661, 1971541

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 23 2019

2465 is also a Carmichael number (A002997).
2821 is also a primary Carmichael number (A324316).
See the section on polygonal numbers in Kellner and Sondow 2019.
Subsequence of the special polygonal numbers A324973. - Jonathan Sondow, Mar 27 2019

			A324315(4) = 1045 = 5 * 11 * 19 = 19 * (3 * 19 - 2) = A000567(19), so 1045 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, #A52 Integers 21 (2021), 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A000567, A002997, A324315, A324316, A324317, A324318, A324319, A324369, A324370, A324371, A324404, A324405, A324973.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
ON[n_] := n(3n - 2);
TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
Select[ON@ Prime[Range[100]], TestS[#] &]

More terms from Amiram Eldar, Dec 05 2020

A083737 Pseudoprimes to bases 2, 3 and 5.

Original entry on oeis.org

1729, 2821, 6601, 8911, 15841, 29341, 41041, 46657, 52633, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 670033, 721801, 748657

Offset: 1

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

a(n) = n-th positive integer k(>1) such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k)
See A153580 for numbers k > 1 such that 2^k-2, 3^k-3 and 5^k-5 are all divisible by k but k is not a Carmichael number (A002997).
Note that a(1)=1729 is the Hardy-Ramanujan number. - Omar E. Pol, Jan 18 2009

			a(1)=1729 since it is the first number such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 102 from R. J. Mathar)
J. Bernheiden, Pseudoprimes (Text in German)
F. Richman, Primality testing with Fermat's little theorem
Index entries for sequences related to pseudoprimes

Proper subset of A052155. Superset of A230722. Cf. A153580, A002997, A001235, A011541.

Select[ Range[838200], !PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, 1 - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 & ]

is(n)=!isprime(n)&&Mod(2,n)^(n-1)==1&&Mod(3,n)^(n-1)==1&&Mod(5,n)^(n-1)==1 \\ Charles R Greathouse IV, Apr 12 2012

Edited by Robert G. Wilson v, May 06 2003

Edited by N. J. A. Sloane, Jan 14 2009

A132195 Number of three-prime Carmichael numbers less than 10^n.

Original entry on oeis.org

1, 7, 12, 23, 47, 84, 172, 335, 590, 1000, 1858, 3284, 6083, 10816, 19539, 35586, 65309, 120625, 224763, 420658, 790885, 1494738

Offset: 3

Jonathan Vos Post, Nov 19 2007

a(n) = C_3(n) in Table 1, p. 34 of Chick (2007-2008) = card{c such that c is in A002997 INTERSECTION A014612 and c <= 10^n}.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 220.

J. M. Chick, Carmichael number variable relations: three-prime Carmichael numbers up to 10^24, arXiv:0711.2915 [math.NT], 2007-2008, Table 1, p. 34.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.

For k-prime Carmichael numbers up to 10^n for k = 3,4,...,11, see A132195, A174612, A174613, A174614, A174615, A174616, A174617, A299710, A299711.

Cf. A002997, A006931, A055553.

A247074 a(n) = phi(n)/(Product_{primes p dividing n } gcd(p - 1, n - 1)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 4, 1, 4, 1, 6, 2, 8, 1, 6, 1, 8, 3, 10, 1, 8, 5, 12, 9, 4, 1, 8, 1, 16, 5, 16, 6, 12, 1, 18, 6, 16, 1, 12, 1, 20, 3, 22, 1, 16, 7, 20, 8, 8, 1, 18, 10, 24, 9, 28, 1, 16, 1, 30, 9, 32, 3, 4, 1, 32, 11, 8, 1, 24, 1, 36, 10, 12, 15, 24, 1, 32, 27, 40, 1, 24, 4, 42, 14, 40, 1, 24, 2, 44, 15, 46

Offset: 1

Eric Chen, Nov 16 2014

a(n) = A000010(n)/A063994(n). - Eric Chen, Nov 29 2014
Does every natural number appear in this sequence? If so, do they appear infinitely many times? - Eric Chen, Nov 26 2014
A063994(n) must be a factor of EulerPhi(n). - Eric Chen, Nov 26 2014
Number n is (Fermat) pseudoprimes (or prime) to one in a(n) of its coprime bases. That is, b^(n-1) = 1 (mod n) for one in a(n) of numbers b coprime to n. - Eric Chen, Nov 26 2014
a(n) = 1 if and only if n is 1, prime (A000040), or Carmichael number (A002997). - Eric Chen, Nov 26 2014
a(A191311(n)) = 2. - Eric Chen, Nov 26 2014
a(p^n) = p^(n-1), where p is a prime. - Eric Chen, Nov 26 2014
a(A209211(n)) = EulerPhi(A209211(n)), because A063994(A209211(n)) = 1. - Eric Chen, Nov 26 2014

			EulerPhi(15) = 8, and that 15 is a Fermat pseudoprime in base 1, 4, 11, and 14, the total is 4 bases, so a(15) = 8/4 = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Gérard P. Michon, Pseudoprimes

Cf. A063994, A182816, A002997, A191311, A064234, A000010, A063994, A003557, A209211.

a063994[n_] := Times @@ GCD[n - 1, First /@ FactorInteger@ n - 1]; a063994[1] = 1; a247074[n_] := EulerPhi[n]/a063994[n]; Array[a247074, 150]

a(n)=my(f=factor(n));eulerphi(f)/prod(i=1,#f~,gcd(f[i,1]-1,n-1)) \\ Charles R Greathouse IV, Nov 17 2014

A003557(n) <= a(n) <= n, and a(n) is a multiple of A003557(n). - Charles R Greathouse IV, Nov 17 2014

A299710 Number of ten-prime Carmichael numbers less than 10^n.

Original entry on oeis.org

23, 340, 3058, 20738, 114232, 547528, 2347828

Offset: 16

Tim Johannes Ohrtmann, Feb 17 2018

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 220.

Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22.
Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22. [Local copy, with permission]
Richard Pinch, Tables relating to Carmichael numbers.

For k-prime Carmichael numbers up to 10^n for k = 3,4,...,11, see A132195, A174612, A174613, A174614, A174615, A174616, A174617, A299710, A299711.

Cf. A002997, A006931, A055553.

a(22) from Claude Goutier added by Amiram Eldar, Apr 19 2024

A324404 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 2 (mod p-1), where s_p(m) is the sum of the base p digits of m.

Original entry on oeis.org

1122, 3458, 5642, 6734, 11102, 13202, 17390, 17822, 21170, 22610, 27962, 31682, 46002, 58682, 61778, 79730, 82082, 93314, 105266, 106262, 125490, 127946, 136202, 150722, 153254, 177122, 182002, 202202, 203870, 214370, 231842, 252434, 274298, 278462, 305102, 315282

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 26 2019

For d >= 1 define S_d = (terms m in A324315 such that s_p(m) == d (mod p-1) if prime p divides m). Then S_1 is precisely the Carmichael numbers (A002997), S_2 is A324404, S_3 is A324405, and the union of all S_d for d >= 1 is A324315.
Subsequence of the 2-Knödel numbers (A050990). Generally, for d > 1 the terms of S_d that are greater than d form a subsequence of the d-Knödel numbers.
See Kellner and Sondow 2019.

			1122 = 2*3*11*17 is squarefree and equals 10001100010_2, 1112120_3, 930_11, and 3f0_17 in base p = 2, 3, 11, and 17. Then s_2(1122) = 1+1+1+1 = 4 >= 2, s_3(1122) = 1+1+1+2+1+2 = 8 >= 3, s_11(1122) = 9+3 = 12 >= 11, and s_17(1122) = 3+f = 3+15 = 18 >= 17. Also, s_2(1122) = 4 == 2 (mod 1), s_3(1122) = 8 == 2 (mod 2), s_11(1122) = 12 == 2 (mod 10), and s_17(1122) = 18 == 2 (mod 16), so 1122 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..2200
Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019-2021.

Cf. A002997, A050990, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324405.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestSd[n_, d_] := (n > 1) && (d > 0) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # && Mod[SD[n, #] - d, # - 1] == 0 &];
Select[Range[200000], TestSd[#, 2] &]

More terms from Amiram Eldar, Dec 05 2020

A324405 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 3 (mod p-1), where s_p(m) is the sum of the base p digits of m.

Original entry on oeis.org

3003, 3315, 5187, 7395, 8463, 14763, 19803, 26733, 31755, 47523, 50963, 58035, 62403, 88023, 105339, 106113, 123123, 139971, 152643, 157899, 166611, 178923, 183183, 191919

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 26 2019

For d >= 1 define S_d = (terms m in A324315 such that s_p(m) == d (mod p-1) if prime p divides m). Then S_1 is precisely the Carmichael numbers (A002997), S_2 is A324404, S_3 is A324405, and the union of all S_d for d >= 1 is A324315.
Subsequence of the 3-Knödel numbers (A033553). Generally, for d > 1 the terms of S_d that are greater than d form a subsequence of the d-Knödel numbers.
See Kellner and Sondow 2019.

			3003 = 3*7*11*13 is squarefree and equals 11010020_3, 11520_7, 2290_11, and 14a0_13 in base p = 3, 7, 11, and 13. Then s_3(3003) = 1+1+1+2 = 5 >= 3, s_7(3003) = 1+1+5+2 = 9 >= 7, s_11(3003) = 2+2+9 = 13 >= 11, and s_13(3003) = 1+4+a = 1+4+10 = 15 >= 13. Also, s_3(3003) = 5 == 3 (mod 2), s_7(3003) = 9 == 3 (mod 6), s_11(3003) = 13 == 3 (mod 10), and s_13(3003) = 15 == 3 (mod 12), so 3003 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..2000
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019-2021.

Cf. A002997, A033553, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestSd[n_, d_] := (n > 1) && (d > 0) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # && Mod[SD[n, #] - d, # - 1] == 0 &];
Select[Range[200000], TestSd[#, 3] &]

A339869 Carmichael numbers k for which A053575(k) [the odd part of phi] divides k-1.

Original entry on oeis.org

561, 1105, 2465, 6601, 8911, 10585, 46657, 62745, 162401, 410041, 449065, 5148001, 5632705, 6313681, 6840001, 7207201, 11119105, 11921001, 19683001, 21584305, 26719701, 41298985, 55462177, 64774081, 67371265, 79411201, 83966401, 87318001, 99861985, 100427041, 172290241, 189941761, 484662529, 790623289, 809883361

Offset: 1

Antti Karttunen, Dec 22 2020

nonn

Lehmer conjectured that the equation k * phi(n) = n - 1 (with k integer) has no solutions for any composite n (i.e., when k > 1). If this sequence has no common terms with A339818, then the conjecture certainly holds.

Amiram Eldar, Table of n, a(n) for n = 1..1150 (terms below 10^22 calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
Wikipedia, Lehmer's totient problem.
Index entries for sequences related to Carmichael numbers.

Intersection of A002997 and A339880.
Complement of A340092 in A002997.
Cf. A000010, A000265, A002322, A053575.
Cf. also A339818, A339878, A339909.

carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]]; oddPart[n_] := n/2^IntegerExponent[n, 2]; q[n_] := Divisible[n - 1, oddPart[EulerPhi[n]]]; Select[carmichaels, q] (* Amiram Eldar, Dec 26 2020 *)

A000265(n) = (n>>valuation(n, 2));
A002322(n) = lcm(znstar(n)[2]);
isA339869(n) = ((n>1)&&!isprime(n)&&(!((n-1)%A002322(n)))&&!((n-1)%A000265(eulerphi(n))));

A033181 Absolute Euler pseudoprimes: odd composite numbers n such that a^((n-1)/2) == +-1 (mod n) for every a coprime to n.

Original entry on oeis.org

1729, 2465, 15841, 41041, 46657, 75361, 162401, 172081, 399001, 449065, 488881, 530881, 656601, 670033, 838201, 997633, 1050985, 1615681, 1773289, 1857241, 2113921, 2433601, 2455921, 2704801, 3057601, 3224065, 3581761, 3664585, 3828001, 4463641, 4903921

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

These numbers n have the property that, for each prime divisor p, p-1 divides (n-1)/2. E.g., 2465 = 5*17*29; 1232/4 = 308; 1232/16 = 77; 1232/28 = 44. - Karsten Meyer, Nov 08 2005
All these numbers are Carmichael numbers (A002997). - Daniel Lignon, Sep 12 2015
These are odd composite numbers n such that b^((n-1)/2) == 1 (mod n) for every base b that is a quadratic residue modulo n and coprime to n. There are no odd composite numbers n such that b^((n-1)/2) == -1 (mod n) for every base b that is a quadratic non-residue modulo n and coprime to n. Note: the absolute Euler-Jacobi pseudoprimes do not exist. Theorem: if an absolute Euler pseudoprime n is a Proth number, then b^((n-1)/2) == 1 for every b coprime to n; by Proth's theorem. Such numbers are 1729, 8355841, 40280065, 53282340865, ...; for example, 1729 = 27*2^6 + 1 with 27 < 2^6. However, it seems that all absolute Euler pseudoprimes n satisfy the stronger congruence b^((n-1)/2) == 1 (mod n) for every b coprime to n, as evidenced by the modified Korselt's criterion (see the first comment). It should be noted that these are odd numbers n such that Carmichael's lambda(n) divides (n-1)/2. Also, these are odd numbers n > 1 coprime to Sum_{k=1..n-1} k^{(n-1)/2}. - Amiram Eldar and Thomas Ordowski, Apr 29 2019
Carmichael numbers k such that (p-1)|(k-1)/2 for each prime p|k. These are odd composite numbers k with half (the maximal possible fraction) of the numbers 1 <= b < k coprime to k that are bases in which k is an Euler-Jacobi pseudoprime, i.e. A329726((k-1)/2)/phi(k) = 1/2. - Amiram Eldar, Nov 20 2019
By Karsten Meyer's and Amiram Eldar's comment, this sequence is numbers k > 1 such that 2*psi(k) | (k-1), where psi = A002322. This means that if k is a term in this sequence, then we actually have a^((k-1)/2) == 1 (mod k) for every a coprime to k. - Jianing Song, Sep 03 2024

Daniel Lignon and Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 124 terms from Daniel Lignon)
Lorenzo Di Biagio, Euler Pseudoprimes for Half of the Bases, Integers, Vol. 12, No. 6 (2012), pp. 1231-1237, arXiv preprint, arXiv:1109.3596 [math.NT] (2011).
Math Help Forum, how many absolute euler pseudoprimes less than a million, Sep 2009.
Louis Monier, Evaluation and comparison of two efficient primality testing algorithms, Theoretical Computer Science, Vol. 11 (1980), pp. 97-108.
Index entries for sequences related to pseudoprimes

Cf. A002997, A006970, A047713, A080075, A329726.

filter:=  proc(n)
  local q;
  if isprime(n) then return false fi;
  if 2 &^ (n-1) mod n <> 1 then return false fi;
  if not numtheory:-issqrfree(n) then return false fi;
  for q in numtheory:-factorset(n) do
    if (n-1)/2 mod (q-1) <> 0 then return false fi
  od:
  true;
end proc:
select(filter, [seq(i,i=3..10^7,2)]); # Robert Israel, Nov 24 2015

absEulerpspQ[n_Integer?PrimeQ]:=False;
absEulerpspQ[n_Integer?EvenQ]:=False;
absEulerpspQ[n_Integer?OddQ]:=Module[{a=2},
While[aDaniel Lignon, Sep 09 2015 *)
aQ[n_] := Module[{f = FactorInteger[n], p},p=f[[;;,1]]; Length[p] > 1 && Max[f[[;;,2]]]==1 && AllTrue[p, Divisible[(n-1)/2, #-1] &]];Select[Range[3, 2*10^5], aQ] (* Amiram Eldar, Nov 20 2019 *)

use ntheory ":all"; my $n; foroddcomposites { say if is_carmichael($) && vecall { (($n-1)>>1) % ($-1) == 0 } factor($n=$); } 1e6; # _Dana Jacobsen, Dec 27 2015

a(n) == 1 (mod 4). - Thomas Ordowski, May 02 2019

"Absolute Euler pseudoprimes" added to name by Daniel Lignon, Sep 09 2015

Definition edited by Thomas Ordowski, Apr 29 2019

A074379 Carmichael numbers with exactly 4 prime factors.

Original entry on oeis.org

41041, 62745, 63973, 75361, 101101, 126217, 172081, 188461, 278545, 340561, 449065, 552721, 656601, 658801, 670033, 748657, 838201, 852841, 997633, 1033669, 1082809, 1569457, 1773289, 2100901, 2113921, 2433601, 2455921

Offset: 1

Jani Melik, Sep 24 2002

nonn

Original name was: "Super-Carmichael numbers with exactly 4 factors", and a comment explained that the prefix "super" means that the Moebius function (A008683) equals mu(N) = +1 for these. But for squarefree numbers such as Carmichael numbers (A002997), this just means that they have an even number of prime factors, which is trivial if that number is 4.

In the literature there are other definitions of "super-Carmichael numbers", see the McIntosh and Meštrović references, so we prefer not to use this terminology at all.

			41041 = 7 * 11 * 13 * 41.
62745 = 3 * 5 * 47 * 89.

R. J. Mathar and Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..6042 from R. J. Mathar)
Richard J. McIntosh Carmichael numbers with (p + 1) | (n - 1), Integers 14 (2014) #A59.
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867, May 04 2013

Cf. A002997 (Carmichael numbers), A006931 (least Carmichael with n prime factors), A046386 (products of four distinct primes).

p = Table[ Prime[i], {i, 1, 10}]; f[n_] := Union[ PowerMod[ Select[p, GCD[ #, n] == 1 & ], n - 1, n]]; Select[ Range[2500000], !PrimeQ[ # ] && OddQ[ # ] && Length[ FactorInteger[ # ]] == 4 && MoebiusMu[ # ] == 1 && f[ # ] == {1} & ]

is_A074379(n)=is_A002997(n) && is_A046386(n) \\ M. F. Hasler, Mar 24 2022

list(lim)=my(v=List()); forprime(p=3,sqrtnint(lim\=1,4), forprime(q=p+2,sqrtnint(lim\p,3), if(q%p==1, next); forprime(r=q+2,sqrtint(lim\p\q), if(r%p==1 || r%q==1, next); my(m=lcm([p-1,q-1,r-1]),pqr=p*q*r,t=Mod(1,m)/pqr,L=lim\pqr); fordiv(pqr-1,d, my(s=d+1); if(s>L, break); if(s==t && s>r && isprime(s), listput(v,pqr*s)))))); Set(v) \\ Charles R Greathouse IV, Apr 23 2022

Intersection of A002997 (Carmichael numbers) and A046386 (product of four distinct primes). - M. F. Hasler, Mar 24 2022

Edited and extended by Robert G. Wilson v, Oct 03 2002

Edited by M. F. Hasler, Mar 24 2022

cp's OEIS Frontend

A324320 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also octagonal numbers (A000567) with index equal to their largest prime factor.

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A083737 Pseudoprimes to bases 2, 3 and 5.

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A132195 Number of three-prime Carmichael numbers less than 10^n.

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A247074 a(n) = phi(n)/(Product_{primes p dividing n } gcd(p - 1, n - 1)).

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A299710 Number of ten-prime Carmichael numbers less than 10^n.

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A324404 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 2 (mod p-1), where s_p(m) is the sum of the base p digits of m.

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A324405 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 3 (mod p-1), where s_p(m) is the sum of the base p digits of m.

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A339869 Carmichael numbers k for which A053575(k) [the odd part of phi] divides k-1.

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