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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A135719 a(n) is the index of the smallest Carmichael number (A002997) with n prime divisors, or 0 if no such number exists.

Original entry on oeis.org

1, 11, 40, 403, 1224, 4886, 19096, 120137, 485941, 2974628, 25293838
Offset: 3

Views

Author

Artur Jasinski, Nov 25 2007

Keywords

Links

Crossrefs

Cf. A002997, A135717, A006931.

Formula

A002997(a(n)) = A006931(n). - M. F. Hasler, Apr 14 2015

Extensions

a(8)-a(11) from Donovan Johnson, Feb 23 2012
a(12) from Amiram Eldar, Jul 08 2019
Escape clause added by Jianing Song, Dec 12 2021
a(13) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 20 2024

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

Views

Author

Tim Johannes Ohrtmann, Oct 28 2020

Keywords

Examples

			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.

Links

Crossrefs

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).

Programs

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

Formula

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

Views

Author

Tim Johannes Ohrtmann, Oct 28 2020

Keywords

Examples

			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.

Links

Crossrefs

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).

Programs

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

Formula

Equals A002997 intersect A069272.

A306657 Least primary Carmichael number (A324316) with n prime factors, or -1 if no such number exists.

Original entry on oeis.org

1729, 10606681, 4872420815346001
Offset: 3

Views

Author

Bernd C. Kellner and Jonathan Sondow, Mar 03 2019

Keywords

Comments

Primary Carmichael numbers were introduced in Kellner and Sondow 2019. For this sequence, see Kellner 2019.
Conjecture: the sequence is infinite.
a(6) > 10^22, if it exists. - Amiram Eldar, Apr 22 2024

Examples

			1729 = 7 * 13 * 19,
10606681 = 31 * 43 * 73 * 109,
4872420815346001 = 211 * 239 * 379 * 10711 * 23801.

Links

Crossrefs

Least Carmichael number with n prime factors is A006931.
Cf. also A002997, A324316.

Extensions

Escape clause added by Amiram Eldar, Apr 22 2024

A294179 a(n) is the smallest k with n prime factors such that p^k == p (mod k) for every prime p dividing k.

Original entry on oeis.org

2, 65, 561, 41041, 825265, 321197185, 5394826801, 232250619601, 9746347772161
Offset: 1

Views

Author

Thomas Ordowski, Feb 11 2018

Keywords

Comments

All the terms are squarefree. Are all composite terms odd?
Conjecture: the sequence contains only finitely many Carmichael numbers, A006931. What is the smallest n >= 3 for which a(n) is not a Carmichael number? For n >= 3, a(n) <= A006931(n).

Crossrefs

Cf. A006931, A285549, A294169.

Programs

  • Maple
    for k from 2 to 10^6 do
  if numtheory:-issqrfree(k) then
    ps := numtheory:-factorset(k);
    n := nops(ps);
    if not assigned(A[n]) and andmap(p -> p &^ k -p mod k = 0, ps) then
      A[n] := k;
    end if
  end if;
end do:
seq(A[i],i=1..max(map(op, [indices(A)]))); # Robert Israel, Feb 11 2018
  • Mathematica
    With[{s = Select[Range[10^6], Function[k, AllTrue[FactorInteger[k][[All, 1]], PowerMod[#, k, k] == Mod[#, k] &]]]}, Select[Table[SelectFirst[s, PrimeOmega@ # == n &], {n, 5}], IntegerQ]] (* Michael De Vlieger, Feb 20 2018 *)

Extensions

a(7)-a(8) from Daniel Suteu, Feb 06 2023
a(9) from Michael S. Branicky, Aug 03 2023

A316908 a(n) is the smallest k with n prime factors such that 2^(k-1) == 1 (mod k) and p-1 does not divide k-1 for every prime p dividing k.

Original entry on oeis.org

7957, 617093, 134564501, 384266404601, 8748670222601, 6105991025919737, 901196605940857381
Offset: 2

Views

Author

Thomas Ordowski, Jul 16 2018

Keywords

Comments

Conjecture: a(n) > A006931(n) for every n > 2.
a(6)-a(8) derived from Feitsma's tables of pseudoprimes. a(9) > 2^64. - Giovanni Resta, Jul 19 2018
From Daniel Suteu, Jun 08 2020: (Start)
a(9) <= 521957994426556057126261,
a(10) <= 1315856103949347820015303981,
a(11) <= 6357507186189933506573017225316941,
a(12) <= 77822245466150976053960303855104674781. (End)

Links

Crossrefs

Cf. A001567, A121707, A316907.

Extensions

More terms from Michel Marcus, Jul 16 2018
a(6)-a(8) from Giovanni Resta, Jul 19 2018

A328938 Least imprimitive Carmichael number (A328935) with n prime factors, or -1 if no such number exists.

Original entry on oeis.org

294409, 167979421, 1152091655881, 62411762908817281, 1516087654274358001
Offset: 3

Views

Author

Amiram Eldar, Oct 31 2019

Keywords

Comments

From Daniel Suteu, Feb 17 2020: (Start)
a(8) <= 42310088783100741554666880481,
a(9) <= 21593590390253023722267234622513201,
a(10) <= 16412975107923138847512341751620644377601,
a(11) <= 325533792014488126487416882038879701391121. (End)
a(8) > 10^22. - Amiram Eldar, Apr 22 2024

Links

Crossrefs

Cf. A002997, A006931, A328935.

Extensions

Escape clause added by Amiram Eldar, Apr 22 2024

A356866 Smallest Carmichael number (A002997) with n prime factors that is also a strong pseudoprime to base 2 (A001262).

Original entry on oeis.org

15841, 5310721, 440707345, 10761055201, 5478598723585, 713808066913201, 1022751992545146865, 5993318051893040401, 120459489697022624089201, 27146803388402594456683201, 14889929431153115006659489681
Offset: 3

Views

Author

Daniel Suteu, Oct 01 2022

Keywords

Crossrefs

Cf. A001262, A002997, A006931, A063847, A180065.

Programs

  • PARI
    carmichael_strong_psp(A, B, k, base) = A=max(A, vecprod(primes(k+1))\2); (f(m, l, p, k, k_exp, congr, u=0, v=0) = my(list=List()); if(k==1, forprime(q=u, v, my(t=m*q); if((t-1)%l == 0 && (t-1)%(q-1) == 0, my(tv=valuation(q-1, 2)); if(tv > k_exp && Mod(base, q)^(((q-1)>>tv)< k_exp && Mod(base, q)^(((q-1)>>tv)<u, u=r); list=concat(list, f(t, L, r, k-1, k_exp, congr, u, v)))))))); list); my(res=f(1, 1, 3, k, 0, 1)); for(v=0, logint(B, 2), res=concat(res, f(1, 1, 3, k, v, -1))); vecsort(Vec(res));
a(n,base=2) = if(n < 3, return()); my(x=vecprod(primes(n+1))\2,y=2*x); while(1, my(v=carmichael_strong_psp(x,y,n,base)); if(#v >= 1, return(v[1])); x=y+1; y=2*x);

Formula

a(n) >= max(A180065(n), A006931(n)).
Previous Showing 21-28 of 28 results.

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