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.

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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

Views

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 ))}

A141704 Indices of records in A141703.

Original entry on oeis.org

1, 2, 3, 4, 11, 14, 47, 193, 331, 2967, 5557
Offset: 1

Views

Author

M. F. Hasler, Jul 01 2008

Keywords

Comments

Based on an idea by Jack Brennen, who pointed out that 2221=prime(331) yields the value A141703(331)=29.
No more terms below 50000. The corresponding records are 0, 1, 3, 6, 7, 11, 17, 22, 29, 32, 40, ... - Amiram Eldar, Sep 25 2019

Examples

			The sequence starts 1,2,3,4,... since the first 4 terms in A141703 are not preceded by a value larger or equal to them.
a(5) = 11 since A141703(11)=6 is the next record value.
a(9) = 331 since A141703(331)=29 is the 9th value establishing a (quite long-lasting) record.

Crossrefs

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

Programs

  • PARI
    m=-1;for(n=1,10^4, m < A141703(n) & print1(n," /*",m=A141703(n),"*/, "));

Extensions

a(10)-a(11) from Amiram Eldar, Sep 25 2019
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