cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A141705 a(n) is the least Carmichael number of the form prime(n)*prime(n')*prime(n") with n < n' < n", or 0 if no such number exists.

Original entry on oeis.org

0, 561, 1105, 1729, 0, 29341, 162401, 334153, 1615681, 3581761, 399001, 294409, 252601, 1152271, 104569501, 2508013, 178837201, 6189121, 10267951, 10024561, 14469841, 4461725581, 985052881, 19384289, 23382529, 3828001, 90698401
Offset: 1

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Author

M. F. Hasler, Jul 03 2008

Keywords

Comments

Primes for which there are no such numbers (i.e. prime(n) such that a(n)=0) are given in A051663. Sequence A135720 is similar, but without restriction to 3-factor Carmichael numbers.

Examples

			a(1)=0 since there is no Carmichael number having prime(1)=2 as factor.
a(2)=561 since this is the smallest Carmichael number of the form pqr with prime r>q>p=prime(2)=3.
a(5)=0 since there is no Carmichael number of the form pqr with prime r>q>p=prime(5)=11.

Links

Crossrefs

Cf. A002997, A051663, A135720, A141702-A141706.

Programs

  • PARI
    A141705(n) = { /* 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++ )); if( V, vecmin( V )); }

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

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Author

M. F. Hasler, Jul 01 2008

Keywords

Comments

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

Examples

			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}.

Links

Crossrefs

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

Programs

  • PARI
    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 }

Formula

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

A141706 a(n) is the largest Carmichael number of the form prime(n)*prime(n')*prime(n") with n < n' < n", or 0 if no such number exists.

Original entry on oeis.org

0, 561, 10585, 52633, 0, 530881, 7207201, 1024651, 1615681, 5444489, 471905281, 36765901, 2489462641, 564651361, 958762729, 17316001, 178837201, 1574601601, 7991602081, 597717121, 962442001, 4461725581, 167385219121, 43286923681
Offset: 1

Views

Author

M. F. Hasler, Jul 03 2008

Keywords

Comments

Primes for which there are no such numbers (i.e. prime(n) such that a(n)=0) are given in A051663.

Examples

			a(1)=0 since there is no Carmichael number having prime(1)=2 as factor.
a(2)=561 since this is the largest (since only) Carmichael number of the form pqr with prime r>q>p=prime(2)=3.
a(5)=0 since there is no Carmichael number of the form pqr with prime r>q>p=prime(5)=11.

Links

Crossrefs

Cf. A002997, A051663, A135720, A141702-A141705.

Programs

  • PARI
    A141706(n) = { /* 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++ )); if( V, vecmax( V ))}

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

Views

Author

Max Alekseyev, Jan 31 2024

Keywords

Comments

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

Links

Crossrefs

Subsequence of A051663.
Cf. A087788, A290481, A290484

A290484 Odd prime numbers that are factors of only one 3-Carmichael number.

Original entry on oeis.org

3, 11, 59, 197, 389, 467, 479, 503, 563, 719, 839, 887, 1523, 1907, 2087, 2339, 2837, 3167, 3989, 4229, 4259, 4643, 4679, 4787, 4903, 4919, 5417, 5849, 5879, 6299, 7307, 7331, 7577, 7583, 8117
Offset: 1

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Author

Amiram Eldar, Aug 03 2017

Keywords

Comments

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 there is a finite number of 3-Carmichael numbers which divisible by a given prime.
An odd prime p is a term if and only if A290481(A033270(p)) = 1. - Max Alekseyev, Jan 31 2024

Examples

			59 is in the sequence since it is a prime factor of only one 3-Carmichael number: 178837201 = 59 * 1451 * 2089.

References

  • 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.

Links

Crossrefs

Cf. A065091 (Odd primes), A087788 (3-Carmichael numbers), A051663, A290481, A369777.

Extensions

a(1)-a(12) were calculated using Pinch's tables of Carmichael numbers (see links).
a(13)-a(35) from Max Alekseyev, Jan 31 2024
Showing 1-5 of 5 results.

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