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 1-2 of 2 results.

A083737 Pseudoprimes to bases 2, 3 and 5.

Original entry on oeis.org

1729, 2821, 6601, 8911, 15841, 29341, 41041, 46657, 52633, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 670033, 721801, 748657
Offset: 1

Views

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

Keywords

Comments

a(n) = n-th positive integer k(>1) such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k)
See A153580 for numbers k > 1 such that 2^k-2, 3^k-3 and 5^k-5 are all divisible by k but k is not a Carmichael number (A002997).
Note that a(1)=1729 is the Hardy-Ramanujan number. - Omar E. Pol, Jan 18 2009

Examples

			a(1)=1729 since it is the first number such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k).

Links

Crossrefs

Proper subset of A052155. Superset of A230722. Cf. A153580, A002997, A001235, A011541.

Programs

  • Mathematica
    Select[ Range[838200], !PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, 1 - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 & ]
  • PARI
    is(n)=!isprime(n)&&Mod(2,n)^(n-1)==1&&Mod(3,n)^(n-1)==1&&Mod(5,n)^(n-1)==1 \\ Charles R Greathouse IV, Apr 12 2012

Extensions

Edited by Robert G. Wilson v, May 06 2003
Edited by N. J. A. Sloane, Jan 14 2009

A230746 Carmichael numbers of the form (30*k + 1)*(120*k + 1)*(150*k + 1), where 30*k + 1, 120*k + 1 and 150*k + 1 are all primes.

Original entry on oeis.org

68154001, 3713287801, 63593140801, 122666876401, 193403531401, 227959335001, 246682590001, 910355497801, 4790779641001, 5367929037001, 6486222838801, 24572944746001, 25408177226401, 27134994772801, 55003376283001, 63926508701401, 108117809748001, 112614220996801
Offset: 1

Views

Author

Arkadiusz Wesolowski, Oct 29 2013

Keywords

Links

Crossrefs

Subsequence of A083739 and of A230722.
Cf. A002997, A007304, A157956.

Programs

  • Magma
    [n : k in [1..593 by 2] | IsPrime(a) and IsPrime(b) and IsPrime(c) and IsOne(n mod CarmichaelLambda(n)) where n is a*b*c where a is 30*k+1 where b is 120*k+1 where c is 150*k+1]
  • Mathematica
    carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; v = {30, 120, 150}; Times @@ (v*# + 1) & /@ Select[Range[1000], AllTrue[(w = v*# + 1), PrimeQ] && carmQ[Times @@ w] &] (* Amiram Eldar, Nov 11 2019 *)

Formula

(A007304 INTERSECT A157956) INTERSECT A230722.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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