A033502 -id:A033502 - cp's OEIS frontend

A046025 Numbers k such that 6k+1, 12k+1 and 18*k+1 are all primes.

1, 6, 35, 45, 51, 55, 56, 100, 121, 195, 206, 216, 255, 276, 370, 380, 426, 506, 510, 511, 710, 741, 800, 825, 871, 930, 975, 1025, 1060, 1115, 1140, 1161, 1270, 1280, 1281, 1311, 1336, 1361, 1365, 1381, 1420, 1421, 1441, 1490, 1515, 1696, 1805, 1875, 1885

Offset: 1

Eric W. Weisstein

nonn

Main entry for this sequence is A033502.

k is a Carmichael number generator giving C(k) = (6*k+1)*(12*k+1)*(18*k+1) = A382809(k).

R. K. Guy, Unsolved Problems in Number Theory, A13.
Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, Wiley NY 1991, page 83, problem #20.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 101.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Cf. A033502, A002997, A382809.

Select[Range[2000],And@@PrimeQ[{6,12,18}#+1]&] (* Harvey P. Dale, May 26 2014 *)

is(n)=isprime(6*n+1) && isprime(12*n+1) && isprime(18*n+1) \\ Charles R Greathouse IV, Jan 04 2022

Better description from Robert G. Wilson v, Sep 27 2000

A324456 Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g.

Original entry on oeis.org

6, 10, 12, 15, 18, 20, 21, 24, 28, 33, 34, 36, 39, 40, 45, 48, 52, 57, 63, 65, 66, 68, 72, 76, 80, 85, 87, 88, 91, 93, 96, 99, 100, 105, 111, 112, 117, 120, 126, 130, 132, 133, 135, 136, 144, 145, 148, 153, 156, 160, 165, 171, 175, 176, 185, 186, 189, 190

Offset: 1

Bernd C. Kellner, Feb 28 2019

The function s_g(m) gives the sum of the base-g digits of m.
The sequence is infinite, since it contains A324460.
The sequence also contains the 3-Carmichael numbers A087788 and the primary Carmichael numbers A324316.
A term m must have at least 2 prime factors, and the divisor g satisfies the inequalities 1 < g < m^(1/(ord_g(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.
Note that the sequence contains the 3-Carmichael numbers, but not all Carmichael numbers. This is a nontrivial fact.
The subsequence A324460 mainly gives examples in which g is composite.
See Kellner 2019.
It appears that g is usually prime: compare with A324857 (g prime) and the sparser sequence A324858 (g composite). However, g is usually composite for higher values of m. - Jonathan Sondow, Mar 17 2019

			6 is a member, since 2 divides 6 and s_2(6) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..806 from Bernd C. Kellner)
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
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.

Subsequences are A033502, A087788, A324316, A324458, A324460.
Subsequence of A324455.
Cf. A324315, A324457, A324459.
Union of A324857 and A324858.

s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
f[n_] := AnyTrue[Divisors[n], s[n, #] == # &];
Select[Range[5000], f[#] &]

isok(n) = {fordiv(n, d, if ((d>1) && (sumdigits(n, d) == d), return (1)););} \\ Michel Marcus, Mar 19 2019

A174734 Prime numbers n such that 2n-1 and 3n-2 are prime.

Original entry on oeis.org

3, 7, 37, 211, 271, 307, 331, 337, 601, 727, 1171, 1237, 1297, 1531, 1657, 2221, 2281, 2557, 3037, 3061, 3067, 4261, 4447, 4801, 4951, 5227, 5581, 5851, 6151, 6361, 6691, 6841, 6967, 7621, 7681, 7687, 7867, 8017, 8167, 8191, 8287, 8521, 8527, 8647, 8941

Offset: 1

Michel Lagneau, Mar 28 2010

nonn

If n, 2n-1 and 3n-2 are prime numbers, and if n >= 5, then n*(2*n-1)*(3*n-2) is a Carmichael number (A033502).

Proof: there exist numbers m such that n=6m+1 is prime (if n=6m+5, then 2n-1 = 12m+9 is composite). Let p=(6m+1)(12m+1)(18m+1) = a*b*c. Then p-1 = 6*12*18*m^3 + (6*12 + 6*18 + 12*18)*m^2 + (6 + 12 + 19)*m, so p-1 is divisible by a-1=6m, by b-1=12m, and by c-1=18m; thus p is a Carmichael number.

			For n=3, 2n-1 = 5, 3n-2 = 7.
For n=7, 2n-1 = 13, 3n-2 = 19 and 7*13*19 = 1729 (a Carmichael number).
For n=37, 2n-1 = 73, 3n-2 = 109 and 37*73*109 = 294409 (a Carmichael number).

R. K. Guy, Unsolved Problems in Number Theory, A13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
W. R. Alford, Andrew Granville, and Carl Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703-722.
Richard Pinch, Carmichael numbers up to 10^18, April 2006.
Richard Pinch, Carmichael numbers up to 10^18, arXiv:math/0604376 [math.NT], 2006.

Cf. A002476, A002997, A033502.

[ n: n in PrimesUpTo(10000) | IsPrime(2*n-1) and IsPrime(3*n-2) ];

with(numtheory): for n from 2 to 15000 do: if type(n,prime)=true and type(2*n-1,prime)=true and type(3*n-2,prime)=true then print (n):else fi:od:

Select[Prime[Range[1000]], PrimeQ[2*#-1] && PrimeQ[3*#-2]&] (* Vladimir Joseph Stephan Orlovsky, Jan 13 2011 *)

forprime(p=3,10^3, isprime(2*p-1) && isprime(3*p-2) && print1(p,", ")); \\ Joerg Arndt, Nov 29 2014

Typo in term corrected by D. S. McNeil, Nov 20 2010

A318646 The least Chernick's "universal form" Carmichael number with n prime factors.

Original entry on oeis.org

1729, 63973, 26641259752490421121, 1457836374916028334162241, 24541683183872873851606952966798288052977151461406721, 53487697914261966820654105730041031613370337776541835775672321, 58571442634534443082821160508299574798027946748324125518533225605795841

Offset: 3

Amiram Eldar, Aug 31 2018

nonn

Chernick proved that U(k, m) = (6m + 1)*(12m + 1)*Product_{i = 1..k-2} (9*(2^i)m + 1), for k >= 3 and m >= 1 is a Carmichael number, if all the factors are primes and, for k >= 4, 2^(k-4) divides m. He called U(k, m) "universal forms". This sequence gives a(k) = U(k, m) with the least value of m. The least values of m for k = 3, 4, ... are 1, 1, 380, 380, 780320, 950560, 950560, 3208386195840, 31023586121600, ...

			For k=3, m = 1, a(3) = U(3, 1) = (6*1 + 1)*(12*1 + 1)*(18*1 + 1) = 1729.
For k=4, m = 1, a(4) = U(4, 1) = (6*1 + 1)*(12*1 + 1)*(18*1 + 1)*(36*1 + 1) = 63973.
For k=5, m = 380, a(5) = U(5, 1) = (6*380 + 1)*(12*380 + 1)*(18*380 + 1)*(36*380 + 1)*(72*380 + 1) =  26641259752490421121.

Amiram Eldar, Table of n, a(n) for n = 3..11
Jack Chernick, On Fermat's simple theorem, Bulletin of the American Mathematical Society, Vol. 45, No. 4 (1939), pp. 269-274.
Daniel Suteu, C++ program
Samuel S. Wagstaff, Jr., Large Carmichael numbers, Mathematical Journal of Okayama University, Vol. 22, (1980), pp. 33-41.

Cf. A002997, A033502 (3 prime factors), A206024 (4 prime factors), A206349 (5 prime factors), A126797.

fc[k_] := If[k < 4, 1, 2^(k - 4)]; a={};Do[v = Join[{6, 12}, 2^Range[k-2]*9];
w = fc[k]; x = v*w; m = 1; While[! AllTrue[x*m + 1, PrimeQ], m++]; c=Times @@ (x*m + 1);AppendTo[a,c], {k, 3, 9}]; a

A036060 Number of 3-component Carmichael numbers C = (6M + 1)(12M + 1)(18M + 1) < 10^n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 7, 10, 16, 25, 50, 86, 150, 256, 436, 783, 1435, 2631, 4765, 8766, 16320, 30601, 57719, 109504, 208822, 400643, 771735, 1494772, 2903761, 5658670, 11059937, 21696205, 42670184, 84144873, 66369603, 329733896, 655014986, 1303918824, 2601139051

Offset: 3

N. J. A. Sloane

nonn

Note that this is different from the count of 3-Carmichael numbers, A132195. The numbers counted here are neither those listed in A087788 (3 arbitrary prime factors) nor those listed in A033502 (where 6m + 1, 12m + 1 and 18m + 1 are all prime). - M. F. Hasler, Apr 14 2015

Posting by Harvey Dubner (harvey(AT)dubner.com) to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Nov 23 1998.

Amiram Eldar, Table of n, a(n) for n = 3..42
H. Dubner, 3-Component Carmichael Numbers-correction, Post to Number Theory List, Nov 23 1998.
Harvey Dubner, Carmichael Numbers of the form (6m+1)(12m+1)(18m+1), Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.1.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A033502, A055553, A087788, A132195.

Terms updated (from Dubner's paper) by Amiram Eldar, Aug 11 2017

A242980 Carmichael numbers of the form (6k + 1)(12k + 1)(18k + 1), where only two of the three numbers 6k + 1, 12k + 1, 18k + 1 are prime.

Original entry on oeis.org

172081, 1773289, 4463641, 47006785, 295643089, 798770161, 1150270849, 1420379065, 1976295241, 18390744505, 122160500281, 134642101321, 215741809801, 228944441089, 263610459505, 321140603665, 374464040689, 444722065201, 676328168881, 1009514855521

Offset: 1

Arkadiusz Wesolowski, May 28 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..2665 (terms < 4*10^30)
Carlos Rivera, Puzzle 739
Eric Weisstein's World of Mathematics, Carmichael Number
Index entries for sequences related to Carmichael numbers

Cf. A002997, A033502, A242981.

lst:=[]; for k in [1..920] do t:={n: n in [1..3] | IsPrime(6*k*n+1)}; if #Set(t) eq 2 then c:=&*[6*k*n+1: n in [1..3]]; if IsOne(c mod CarmichaelLambda(c)) then lst:=Append(lst, c); end if; end if; end for; lst;

A242981 Carmichael numbers of the form (6k + 1)(12k + 1)(18k + 1), where only one of the three numbers 6k + 1, 12k + 1, 18k + 1 is prime.

Original entry on oeis.org

13992265, 1504651681, 14782305601, 292869912385, 2387706608354185, 4484354372982001, 70895950685971489, 807481759780458361, 1659264916949696161, 118023300545190612481, 140695625117781970705, 11710597360056333492601, 19818019625768288167321

Offset: 1

Arkadiusz Wesolowski, May 28 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..26 (terms < 4*10^33)
Carlos Rivera, Puzzle 739
Eric Weisstein's World of Mathematics, Carmichael Number
Index entries for sequences related to Carmichael numbers

Cf. A002997, A033502, A242980.

lst:=[]; for k in [1..2482095] do t:={n: n in [1..3] | IsPrime(6*k*n+1)}; if #Set(t) eq 1 then c:=&*[6*k*n+1: n in [1..3]]; if IsOne(c mod CarmichaelLambda(c)) then lst:=Append(lst, c); end if; end if; end for; lst;

A182087 Carmichael numbers of the form C = (30n-p)(60n-(2p+1))(90n-(3p+2)), where n is a natural number and p, 2p+1, 3p+2 are all three prime numbers.

Original entry on oeis.org

1729, 172081, 294409, 1773289, 4463641, 56052361, 118901521, 172947529, 216821881, 228842209, 295643089, 798770161, 1150270849, 1299963601, 1504651681, 1976295241, 2301745249, 9624742921, 11346205609, 13079177569

Offset: 1

Marius Coman, Apr 11 2012

nonn

These numbers can be reduced to only two possible forms: C =(30n-23)*(60n-47)*(90n-71) or C = (30n-29)*(60n-59)*(90n-89). In the first form, for the particular case when 30n-23,60n-47 and 90n-71 are all three prime numbers, we obtain the Chernick numbers of the form 10m+1 (for k = 5n-4 we have C = (6k+1)*(12k+1)*(18k+1)). In the second form, for the particular case when 30n-29,60n-59 and 90n-89 are all three prime numbers, we obtain the Chernick numbers of the form 10m+9 (for k = 5n-5 we have C = (6k+1)*(12k+1)*(18k+1)).

So the Chernick numbers can be divided into two categories: Chernick numbers of the form (30n+7)*(60n+13)*(90n+19) and Chernick numbers of the form (30n+1)*(60n+1)*(90n+1).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Carmichael Number.

Cf. A033502, A206347.

list(lim)={
    my(v=List(),f);
    for(k=1,round(solve(x=(lim/162000)^(1/3),lim^(1/3),(30*x-23)*(60*x-47)*(90*x-71)-lim)),
        n=(30*k-23)*(60*k-47)*(90*k-71)-1;
        f=factor(30*k-23);
        for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
        f=factor(60*k-47);
        for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
        f=factor(90*k-71);
        for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
        listput(v,n+1)
    );
    for(k=2,round(solve(x=(lim/162000)^(1/3),lim^(1/3),(30*x-29)*(60*x-59)*(90*x-89)-lim)),
        n=(30*k-29)*(60*k-59)*(90*k-89)-1;
        f=factor(30*k-29);
        for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
        f=factor(60*k-59);
        for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
        f=factor(90*k-89);
        for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
        listput(v,n+1)
    );
    vecsort(Vec(v))
}; \\ Charles R Greathouse IV, Oct 02 2012

A221742 Carmichael numbers of the form (6k+1)(12k+1)(18*k+1) which are the product of four prime numbers.

Original entry on oeis.org

172081, 1773289, 4463641, 295643089, 798770161, 1976295241, 122160500281, 374464040689, 444722065201, 676328168881, 1009514855521, 2382986541601, 3022286597929, 9031805532361, 33648448111489, 155773422536761, 206932492972801, 366715617643441, 708083570971801

Offset: 1

Bruno Berselli, Jan 23 2013, based on the Cerruti paper

nonn

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..87 from Vincenzo Librandi)
Umberto Cerruti, Pseudoprimi di Fermat e numeri di Carmichael (in Italian), 2013. The sequence is on page 11.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A033502, A221743 (associated k).

Subsequence of A182087.

[c: n in [1..10^4] | #PrimeDivisors(c) eq 4 and IsOne(c mod CarmichaelLambda(c)) where c is (6*n+1)*(12*n+1)*(18*n+1)];

with(numtheory);P:=proc(q)local a,b,k,ok,n;
for n from 0 to q do a:=(6*n+1)*(12*n+1)*(18*n+1); b:=ifactors(a)[2];
if issqrfree(a) and nops(b)=4 then ok:=1;
for k from 1 to 4 do if not type((a-1)/(b[k][1]-1),integer) then ok:=0;
break; fi; od; if ok=1 then print(a); fi;
fi; od; end: P(10^6); # Paolo P. Lava, Oct 11 2013

g[n_] := (6*n+1)*(12*n+1)*(18*n+1); testQ[n_] := Block[{p,e}, {p, e} = Transpose@ FactorInteger@ n; e == {1,1,1,1} && Max[Mod[n-1, p-1]] == 0]; Select[g /@ Range[10^4], testQ] (* Giovanni Resta, May 21 2013 *)

A221743 Numbers k such that (6k+1)(12k+1)(18*k+1) is a Carmichael number which is the product of four prime numbers.

Original entry on oeis.org

5, 11, 15, 61, 85, 115, 455, 661, 700, 805, 920, 1225, 1326, 1910, 2961, 4935, 5425, 6565, 8175, 10885, 11375, 12155, 13230, 18315, 37800, 39325, 45325, 59726, 69440, 99645, 113120, 121365, 129850, 144685, 211945, 353465, 378940, 389896, 392625

Offset: 1

Bruno Berselli, Jan 23 2013, based on the Cerruti paper

nonn

Amiram Eldar, Table of n, a(n) for n = 1..500
Umberto Cerruti, Pseudoprimi di Fermat e numeri di Carmichael (in Italian), 2013. The sequence is on page 11.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A033502, A221742 (associated Carmichael numbers).

Subsequence of A101187.

[n: n in [1..4*10^5] | #PrimeDivisors(c) eq 4 and IsOne(c mod CarmichaelLambda(c)) where c is (6*n+1)*(12*n+1)*(18*n+1)];

with(numtheory);P:=proc(q)local a,b,k,ok,n;
for n from 0 to q do a:=(6*n+1)*(12*n+1)*(18*n+1); b:=ifactors(a)[2];
if issqrfree(a) and nops(b)=4 then ok:=1;
for k from 1 to 4 do if not type((a-1)/(b[k][1]-1),integer) then ok:=0;
break; fi; od; if ok=1 then print(n); fi;
fi; od; end: P(10^6); # Paolo P. Lava, Oct 11 2013

IsCarmichaelQ[n_] := Module[{f}, If[EvenQ[n] || PrimeQ[n], False, f = Transpose[FactorInteger[n]][[1]]; Union[Mod[n-1, f-1]] == {0}]]; n = 0; t = {}; While[Length[t] < 39, n++; c = (6*n + 1)*(12*n + 1)*(18*n + 1); If[SquareFreeQ[c] && Length[FactorInteger[c]] == 4 && IsCarmichaelQ[c], AppendTo[t, n]]]; t (* T. D. Noe, Jan 23 2013 *)

cp's OEIS Frontend

A046025 Numbers k such that 6*k+1, 12*k+1 and 18*k+1 are all primes.

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A324456 Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g.

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A174734 Prime numbers n such that 2n-1 and 3n-2 are prime.

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A318646 The least Chernick's "universal form" Carmichael number with n prime factors.

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A036060 Number of 3-component Carmichael numbers C = (6M + 1)(12M + 1)(18M + 1) < 10^n.

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A242980 Carmichael numbers of the form (6*k + 1)*(12*k + 1)*(18*k + 1), where only two of the three numbers 6*k + 1, 12*k + 1, 18*k + 1 are prime.

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A242981 Carmichael numbers of the form (6*k + 1)*(12*k + 1)*(18*k + 1), where only one of the three numbers 6*k + 1, 12*k + 1, 18*k + 1 is prime.

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A182087 Carmichael numbers of the form C = (30n-p)*(60n-(2p+1))*(90n-(3p+2)), where n is a natural number and p, 2p+1, 3p+2 are all three prime numbers.

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A046025 Numbers k such that 6k+1, 12k+1 and 18*k+1 are all primes.

A242980 Carmichael numbers of the form (6k + 1)(12k + 1)(18k + 1), where only two of the three numbers 6k + 1, 12k + 1, 18k + 1 are prime.

A242981 Carmichael numbers of the form (6k + 1)(12k + 1)(18k + 1), where only one of the three numbers 6k + 1, 12k + 1, 18k + 1 is prime.

A182087 Carmichael numbers of the form C = (30n-p)(60n-(2p+1))(90n-(3p+2)), where n is a natural number and p, 2p+1, 3p+2 are all three prime numbers.

A221742 Carmichael numbers of the form (6k+1)(12k+1)(18*k+1) which are the product of four prime numbers.

A221743 Numbers k such that (6k+1)(12k+1)(18*k+1) is a Carmichael number which is the product of four prime numbers.