A141705 -id:A141705 - cp's OEIS frontend

A135720 a(n) is the smallest Carmichael number (A002997) with the n-th prime as its smallest prime divisor, or 0 if no such number exists.

561, 1105, 1729, 75361, 29341, 162401, 334153, 1615681, 3581761, 399001, 294409, 252601, 1152271, 104569501, 2508013, 178837201, 6189121, 10267951, 10024561, 14469841, 4461725581, 985052881, 19384289, 23382529, 3828001, 90698401, 84350561, 6733693, 17098369

Offset: 2

Artur Jasinski, Nov 25 2007

nonn

			a(2) = 561 because the smallest prime divisor of 561 is 3 which is the second prime.

Amiram Eldar, Table of n, a(n) for n = 2..1383 (calculated using data from Claude Goutier; terms 2..447 from Donovan Johnson, terms 448..615 from Max Alekseyev)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A135717, A006931, A135719, A135721, A141705.

Two missing terms and terms up to a(447) added by Donovan Johnson, Dec 25 2013
a(448)-a(615) in b-file from Max Alekseyev, Mar 11 2018
Escape clause added by Jianing Song, Dec 12 2021

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

M. F. Hasler, Jul 03 2008

nonn

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

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
OEIS index entries for Carmichael numbers

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

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 ))}

A290486 The least 3-Carmichael number that is divisible by the n-th odd prime, or 0 if no such number exists.

Original entry on oeis.org

561, 1105, 1729, 561, 1105, 561, 1729, 6601, 2465, 2821, 29341, 6601, 334153, 104569501, 2508013, 178837201, 29341, 8911, 10024561, 10585, 2508013, 985052881, 19384289, 46657, 252601, 52633, 84350561, 294409, 3581761, 1152271, 139952671, 410041, 79624621

Offset: 1

Amiram Eldar, Aug 03 2017

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..198
R. G. E. Pinch, Tables relating to Carmichael numbers.

Cf. A065091 (Odd primes), A087788 (3-Carmichael numbers), A135721, A141705.

Escape clause added by Jianing Song, Dec 12 2021

cp's OEIS Frontend

A135720 a(n) is the smallest Carmichael number (A002997) with the n-th prime as its smallest prime divisor, or 0 if no such number exists.

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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.

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A290486 The least 3-Carmichael number that is divisible by the n-th odd prime, or 0 if no such number exists.

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cp's OEIS Frontend

A135720 a(n) is the smallest Carmichael number (A002997) with the n-th prime as its smallest prime divisor, or 0 if no such number exists.

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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.

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PARI

A290486 The least 3-Carmichael number that is divisible by the n-th odd prime, or 0 if no such number exists.

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Author

Keywords

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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.