A087788 -id:A087788 - cp's OEIS frontend

A290481 The number of 3-Carmichael numbers that are divisible by the n-th odd prime.

1, 3, 6, 1, 8, 5, 4, 2, 4, 9, 8, 9, 12, 3, 3, 1, 16, 4, 7, 11, 2, 2, 5, 8, 4, 6, 3, 12, 6, 8, 11, 5, 6, 2, 11, 14, 8, 2, 3, 4, 15, 6, 11, 1, 9, 22, 5, 4, 7, 2, 5, 15, 8, 6, 4, 4, 21, 9, 10, 2, 5, 12, 9, 20, 2, 20, 19, 2, 6, 8, 2, 9, 8, 12, 3, 1, 10, 14, 10, 3

Offset: 1

Amiram Eldar, Aug 03 2017

nonn

Beeger proved in 1950 that if p < q < r are primes such that p*q*r is a 3-Carmichael number, then q < 2p^2 and r < p^3. Therefore the number of 3-Carmichael numbers that are divisible by a fixed prime is finite.

The terms were calculated using Pinch's tables of Carmichael numbers (see link below).

			There is only one 3-Carmichael number that is divisible by 3 (561); there are three that are divisible by 5 (1105, 2465 and 10585) and six that are divisible by 7 (1729, 2821, 6601, 8911, 15841 and 52633). Thus a(1)=1, a(2)=3 and a(3)=6.

N. G. W. H. Beeger, On composite numbers n for which a^n == 1 (mod n) for every a prime to n, Scripta Mathematica, Vol. 16 (1950), pp. 133-135.

Max Alekseyev, Table of n, a(n) for n = 1..1000
R. G. E. Pinch, Tables relating to Carmichael numbers.
Carlos Rivera, Conjecture 19, A bound to the largest prime factor of certain Carmichael numbers, The Prime Puzzles and Problems Connection.

Cf. A065091 (Odd primes), A087788 (3-Carmichael numbers).

A324857 Numbers m > 1 such that there exists a prime divisor p of m with s_p(m) = p.

Original entry on oeis.org

6, 10, 12, 15, 18, 20, 21, 24, 33, 34, 36, 39, 40, 45, 48, 57, 63, 65, 66, 68, 72, 80, 85, 87, 91, 93, 96, 99, 105, 111, 117, 130, 132, 133, 135, 136, 144, 145, 160, 165, 171, 175, 185, 189, 192, 205, 217, 225, 231, 249, 255, 258, 259, 260, 261, 264, 265, 272, 273, 279, 285, 288, 297, 301, 305, 320, 325, 327, 333, 341, 351, 384, 385

Offset: 1

Jonathan Sondow, Mar 17 2019

The function s_p(m) gives the sum of the base-p digits of m.
m must have at least 2 prime factors, since s_p(p^k) = 1 < p.
The sequence contains the primary Carmichael numbers A324316.
The main entry for this sequence is A324456 = numbers m > 1 such that there exists a divisor d > 1 of m with s_d(m) = d. It appears that d is usually prime: compare the sparser sequence A324858 = numbers m > 1 such that there exists a composite divisor c of m with s_c(m) = c. However, d is usually composite for higher values of m.
The sequence contains the 3-Carmichael numbers A087788, but not all Carmichael numbers A002997. This is a nontrivial fact. The smallest Carmichael number that is not a member is 173085121 = 11*31*53*61*157. For further properties of the terms see A324456 and Kellner 2019. - Bernd C. Kellner, Apr 02 2019

			s_p(m) = 1 < p for m = 2, 3, 4, 5 with prime p dividing m, but if m = 6 and p = 2 then s_p(m) = s_2(2 + 2^2) = 1 + 1 = 2 = p, so a(1) = 6.

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

A324456 is the union of A324857 and A324858.

Includes A083558.

S:= (p,m) -> convert(convert(m,base,p),`+`):
filter:= proc(m) ormap(p -> S(p,m) = p, numtheory:-factorset(m)) end proc:
select(filter, [$2..500]); # Robert Israel, Mar 20 2019

s[n_, b_] := If[n < 1 || b < 2, 0, Plus @@ IntegerDigits[n, b]];
f[n_] := AnyTrue[Divisors[n], PrimeQ[#] && s[n, #] == # &];
Select[Range[400], f[#] &]n (* simplified by Bernd C. Kellner, Apr 02 2019 *)

isok(n) = {if (n>1, my(vp=factor(n)[,1]); for (k=1, #vp, if (sumdigits(n, vp[k]) == vp[k], return (1)))); } \\ Michel Marcus, Mar 19 2019

A141703 a(n) is the number of Carmichael numbers of the form prime(n)prime(n')prime(n") with n < n' < n".

Original entry on oeis.org

0, 1, 3, 6, 0, 5, 2, 2, 1, 2, 7, 5, 7, 11, 3, 3, 1, 10, 3, 7, 4, 1, 2, 5, 6, 2, 5, 3, 10, 5, 5, 11, 4, 6, 2, 9, 11, 7, 2, 3, 4, 11, 6, 10, 0, 7, 17, 5, 4, 6, 1, 5, 10, 7, 5, 4, 4, 14, 8, 9, 2, 5, 12, 9, 16, 2, 16, 15, 2, 6, 5, 2, 9, 8, 8, 3, 1, 7, 13, 7, 3, 13, 5, 14, 6, 8, 4, 9, 6, 4, 1, 1, 9, 7, 3, 1

Offset: 1

M. F. Hasler, Jul 01 2008

nonn

It is known that there is a finite number of Carmichael numbers with k prime factors if k-2 of the factors are fixed. Here we consider the case k=3 and impose the additional condition that prime(n) be the smallest of the 3 factors.

The primes related to the zeros in this sequence are in A051663. - Jack Brennen, Jul 01 2008

			a(1)=0 since prime(1)=2 and there is no even Carmichael number.
a(2)=1 since prime(2)=3 and 561 is the only Carmichael number of the form 3pq with p,q prime.
a(3)=3 since prime(3)=5 and the only Carmichael numbers of the form 5pq are {1105, 2465, 10585}.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from M. F. Hasler)
Index entries related to Carmichael numbers.

Cf. A002997 and references therein ; A087788 ; A141702 ff.

A141703(n,verbose=0) = { /* based on code by J.Brennen (jb AT brennen.net) */ local( V=[], B, p=prime(n), q, r); for( A=1, p-1, B=ceil((p^2+1)/A); while( 1, r=(p*B-p+A*B-B)/(A*B-p*p); q=(A*r-A+1)/p; q<=p && break; denominator(q)==1 && denominator(r)==1 && r>q && isprime(q) && isprime(r) && (p*q*r)%(p-1)==1 && V=concat(V,[p*q*r]); B++ )); verbose && print1(V); #V }

a(n) = # { pqr | p=prime(n) < q=prime(n') < r=prime(n") ; p-1 | pqr-1 ; q-1 | pqr-1 ; r-1 | pqr-1 }

A324858 Numbers m > 1 such that there exists a composite divisor c of m with s_c(m) = c.

Original entry on oeis.org

28, 40, 52, 66, 76, 88, 96, 100, 112, 120, 126, 136, 148, 153, 156, 160, 176, 186, 190, 196, 208, 225, 232, 246, 268, 276, 280, 288, 292, 297, 304, 306, 328, 336, 340, 344, 352, 366, 369, 370, 378, 388, 396, 400, 408, 435, 441, 448, 456, 460, 486, 496, 513, 516, 520, 532, 540, 544, 546, 550, 560, 568, 576, 580, 585, 592

Offset: 1

Jonathan Sondow, Mar 17 2019

The function s_c(m) gives the sum of the base-c digits of m.
The main entry for this sequence is A324456 = numbers m > 1 such that there exists a divisor d > 1 of m with s_d(m) = d. It appears that d is usually prime: compare the subsequence A324857 = numbers m > 1 such that there exists a prime divisor p of m with s_p(m) = p. However, d is usually composite for higher values of m.
For any composite c, 0 < b < c, and 0 < i < j, b*c^i + (c-b)*c^j is in the sequence. - Robert Israel, Mar 19 2019
The sequence does not contain the 3-Carmichael numbers A087788, but intersects the Carmichael numbers A002997 that have at least four factors. This is a nontrivial fact. Examples for such Carmichael numbers below one million: 41041 = 7*11*13*41, 172081 = 7*13*31*61, 188461 = 7*13*19*109, 278545 = 5*17*29*113, 340561 = 13*17*23*67, 825265 = 5*7*17*19*73. For further properties of the terms see A324456 and Kellner 2019. - Bernd C. Kellner, Apr 02 2019

			s_4(28) = 4 as 28 = 3 * 4 + 1 * 4^2, so 28 is a member.

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

A324456 is the union of A324857 and A324858.

S:= proc(c,m) convert(convert(m,base,c),`+`) end proc:
filter:= proc(m) ormap(c -> (S(c,m)=c), remove(isprime,numtheory:-divisors(m) minus {1})) end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 19 2019

s[n_, b_] := If[n < 1 || b < 2, 0, Plus @@ IntegerDigits[n, b]];
f[n_] := AnyTrue[Divisors[n], CompositeQ[#] && s[n, #] == # &];
Select[Range[600], f[#] &] (* simplified by Bernd C. Kellner, Apr 02 2019 *)

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

A202562 Carmichael numbers whose prime factors do not divide any smaller Carmichael number.

Original entry on oeis.org

561, 84350561, 851703301, 2436691321, 34138047673, 60246018673, 63280622521, 83946864769, 110296864801, 114919915021, 155999871721, 225593397919, 342267565249, 534919693681, 660950414671, 733547013841, 1079942171239, 1301203515361, 1333189866793

Offset: 1

Arkadiusz Wesolowski, Dec 21 2011

nonn

Note that all terms so far have only three prime factors.

			a(2) = 84350561 because 84350561 = 107*743*1061 and the smaller Carmichael numbers do not have the factors 107, 743 or 1061.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..1377, terms below 10^18, from Donovan Johnson)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A087788.

A290810 Numbers k such that 6k-1, 12k-1 and 18k-1 are all primes.

Original entry on oeis.org

1, 4, 5, 14, 15, 29, 39, 40, 49, 70, 110, 159, 169, 204, 235, 260, 264, 315, 334, 355, 390, 425, 449, 490, 560, 565, 599, 634, 725, 729, 735, 820, 824, 889, 1019, 1029, 1349, 1379, 1419, 1510, 1580, 1590, 1694, 1719, 1765, 1925, 1930, 1950, 1985, 2044, 2150

Offset: 1

Amiram Eldar, Aug 11 2017

nonn

If k is in the sequence then (6k-1)(12k-1)(18k-1) = 36k * (36k^2 - 11k + 1) - 1 is a Lucas-Carmichael number (A006972).

Analogous to A046025 as A006972 (Lucas-Carmichael numbers) is analogous to A002997 (Carmichael numbers).

			1 is in the sequence since 6*1 - 1 = 5, 12*1 - 1 = 11 and 18*1 - 1 = 17 are all primes, and 5*11*17 = 935 is a Lucas-Carmichael number.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A002997, A006972, A046025, A067256, A087788, A216925.

Intersection of A024898, A138620 and A138918.

seq = {}; Do[ If[ AllTrue[{6 m - 1, 12 m - 1, 18 m - 1}, PrimeQ ], AppendTo[seq, m] ], {m, 1, 10^5} ]; seq

isok(n) = isprime(6*n-1) && isprime(12*n-1) && isprime(18*n-1); \\ Michel Marcus, Aug 11 2017

6*a(n) - 1 = A067256(n+1).

A290947 Primes p1 > 3, such that p2 = 3p1-2 and p3 = (p1p2+1)/2 are also primes, so p1p2*p3 is a triangular 3-Carmichael number.

Original entry on oeis.org

7, 13, 37, 43, 61, 193, 211, 271, 307, 331, 601, 673, 727, 757, 823, 1063, 1297, 1447, 1597, 1621, 1657, 1693, 2113, 2281, 2347, 2437, 2503, 3001, 3067, 3271, 3733, 4093, 4201, 4957, 5581, 6073, 6607, 7321, 7333, 7723, 7867, 8287, 8581, 8647, 9643, 10243

Offset: 1

Amiram Eldar, Aug 14 2017

nonn

The primes are of the form p1=(6k+1), p2=(18k+1), and p3=(54k^2+12k+1), with k = 1, 2, 6, 7, 10, 32, 35, 45, 51, 55, 100, ...

The generated triangular 3-Carmichael numbers are: 8911, 115921, 8134561, 14913991, 60957361, 6200691841, 8863329511, 24151953871, 39799655911, 53799052231, 585796503601, ...

			p1 = 7 is in the sequence since with p2 = 3*7-2 = 19 and p3 = (7*19+1)/2 = 67 they are all primes. 7*19*67 = 8911 is a triangular 3-Carmichael number.

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

Cf. A000217, A002997, A087788, A290945.

seq = {}; Do[p1 = 6 k + 1; p2 = 3 p1 - 2; p3 = (p1*p2 + 1)/2;
If[AllTrue[{p1, p2, p3}, PrimeQ], AppendTo[seq, p1]], {k, 1,
  2000}]; seq

list(lim)=my(v=List()); forprime(p=7,lim, if(isprime(3*p-2) && isprime((p*(3*p-2)+1)/2), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 14 2017

A338442 Carmichael numbers with 10 prime factors.

Original entry on oeis.org

1436697831295441, 1493812621027441, 2094319836529921, 2349991949342401, 2842648863161185, 2859959706040801, 3455134500424321, 3871703982953521, 4177950872896801, 4289150794129201, 4937378437571041, 5071419883911745, 5778659093725441, 6665161459969441, 6682056104892961

Offset: 1

Tim Johannes Ohrtmann, Oct 28 2020

nonn

			1436697831295441 = 11*13*19*29*31*37*41*43*71*127 and 10, 12, 18, 28, 30, 36, 40, 42, 70, 126 all divide 1436697831295440.

Daniel Suteu, Table of n, a(n) for n = 1..10000
Richard Pinch, Tables relating to Carmichael numbers.

Cf. A002997 (Carmichael numbers).
Cf. A006931 (Least Carmichael number with n prime factors).
Cf. A299710 (Number of terms less than 10^n).
Cf. A087788, A074379, A112428, A112429, A112430, A112431, A112432, A338443 (Carmichael numbers with 3-9 and 11 prime factors).

PARI
```
is(n)={omega(n)==10&&is_A002997(n)}
```

Equals A002997 intersect A046314.

A338443 Carmichael numbers with 11 prime factors.

Original entry on oeis.org

60977817398996785, 105083995864811041, 107473646345582881, 132819104923908481, 145671955835893201, 161802381510126721, 165167398073764801, 206063729626916161, 263076030916096321, 292433912163313921, 292561243007134465, 337365329710615921, 388219799621120545

Offset: 1

Tim Johannes Ohrtmann, Oct 28 2020

nonn

			60977817398996785 = 5*7*17*19*23*37*53*73*79*89*233 and 4, 6, 16, 18, 22, 36, 52, 72, 78, 88, 232 all divide 60977817398996784.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..1142 from Daniel Suteu)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Richard Pinch, Tables relating to Carmichael numbers.

Cf. A002997 (Carmichael numbers).
Cf. A006931 (Least Carmichael number with n prime factors).
Cf. A299710 (Number of terms less than 10^n).
Cf. A087788, A074379, A112428, A112429, A112430, A112431, A112432, A338442 (Carmichael numbers with 3-10 prime factors).

PARI
```
is(n)={omega(n)==11&&is_A002997(n)}
```

Equals A002997 intersect A069272.

A369777 Primes that do not divide any 3-Carmichael numbers.

Original entry on oeis.org

2, 1223, 1487, 4007, 4547, 7823, 9839, 10259, 11483, 11807, 11909, 13259, 13967, 14207, 15629, 15803, 16139, 16889, 18287, 19583, 23039, 23879, 24359, 25349, 29339, 30707, 32027, 34883, 36929, 38747, 39113, 39119, 42787, 43223, 44207, 46829, 47189, 49003, 49019, 49157, 53093, 56267, 56909

Offset: 1

Max Alekseyev, Jan 31 2024

nonn

An odd prime p is a term if and only if A290481(A033270(p)) = 0.

Max Alekseyev, Table of n, a(n) for n = 1..496
Index entries for sequences related to Carmichael numbers

Subsequence of A051663.

Cf. A087788, A290481, A290484

cp's OEIS Frontend

A290481 The number of 3-Carmichael numbers that are divisible by the n-th odd prime.

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A324857 Numbers m > 1 such that there exists a prime divisor p of m with s_p(m) = p.

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A141703 a(n) is the number of Carmichael numbers of the form prime(n)*prime(n')*prime(n") with n < n' < n".

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A324858 Numbers m > 1 such that there exists a composite divisor c of m with s_c(m) = c.

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A202562 Carmichael numbers whose prime factors do not divide any smaller Carmichael number.

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A290810 Numbers k such that 6k-1, 12k-1 and 18k-1 are all primes.

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A290947 Primes p1 > 3, such that p2 = 3p1-2 and p3 = (p1*p2+1)/2 are also primes, so p1*p2*p3 is a triangular 3-Carmichael number.

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A338442 Carmichael numbers with 10 prime factors.

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A141703 a(n) is the number of Carmichael numbers of the form prime(n)prime(n')prime(n") with n < n' < n".

A290947 Primes p1 > 3, such that p2 = 3p1-2 and p3 = (p1p2+1)/2 are also primes, so p1p2*p3 is a triangular 3-Carmichael number.