A002997 -id:A002997 - cp's OEIS frontend

A335584 Carmichael numbers (A002997) that are not minimal in their family.

294409, 488881, 1152271, 3057601, 3828001, 6189121, 17098369, 19384289, 53711113, 56052361, 64377991, 82929001, 115039081, 118901521, 171454321, 172947529, 214852609, 216821881, 228842209, 279377281, 288120421, 328573477, 366652201, 492559141, 542497201

Offset: 1

Jeppe Stig Nielsen, Apr 21 2021

nonn

Let a = p_1 * p_2 *...* p_k and b = q_1 * q_2 *...* q_k be two Charmichael numbers (A002997) with the same number of factors, where p_1 < p_2 <...< p_k and q_1 < q_2 <...< q_k are primes. We say that a and b are in the same family iff the vectors [p_1 - 1, ..., p_k - 1] and [q_1 - 1, ..., q_k - 1] are parallel. In other words, the ratios (p_1-1):(p_2-1):...:(p_k-1) and (q_1-1):(q_2-1):...:(q_k-1) are equal. Sequence gives Carmichael numbers that are NOT minimal in their family.

Not a subsequence of A328935 (for example 965507554621 is primitive but not minimal).

			294409 = 37*73*109 is a Carmichael number, belonging to family 36:72:108 = 1:2:3. However, 1729 = 7*13*19 is smaller Carmichael number, and the family 6:12:18 = 1:2:3 is the same. Therefore 294409 belongs to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A328935.

is(m)=!is_A002997(m)&&return(0);f=factor(m);p=f[,1]~;r=apply(x->x-1,p);g=gcd(r);a=r/g;for(i=1,g-1,t=prod(j=1,#a,i*a[j]+1);bigomega(t)==bigomega(m)&&is_A002997(t)&&return(1));0 \\ use with suitable PROG from A002997

A346569 Carmichael numbers (A002997) k such that A003961(k) is also a Carmichael number.

Original entry on oeis.org

938531360353681, 6178246534322281, 518705522457928921, 7019247908645553241, 16242056799655920481, 94812683932464811561, 94986212971063089241, 408133613144935002601, 418441276466266605481, 453648717063017803081, 556606627235843071681, 1140359076998537247001

Offset: 1

Amiram Eldar, Jul 23 2021

nonn

Each of the first 17 terms has 3 distinct prime divisors. [updated Apr 22 2024]

a(6) <= 94812683932464811561. A term with 4 prime factors is 9584146525723596902470058833132261. - Daniel Suteu, Jul 24 2021

			938531360353681 = 53881 * 107761 * 161641 is a term since it is a Carmichael number, and A003961(938531360353681) = 53887 * 107773 * 161659 = 938844932257009 is also a Carmichael number.

Amiram Eldar, Table of n, a(n) for n = 1..17 (terms below 10^22 calculated using data from Claude Goutier)
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, A003961.

Subsequence of A346568.

a(6) verified and a(7)-a(13) calculated using using data from Claude Goutier by Amiram Eldar, Apr 22 2024

A081703 Largest proper divisor of the n-th Carmichael number (A002997).

Original entry on oeis.org

187, 221, 247, 493, 403, 943, 1273, 2117, 2263, 2257, 5863, 3589, 7519, 20915, 9139, 6851, 14443, 8917, 18031, 9553, 24583, 26923, 6161, 55709, 7957, 24217, 17587, 26197, 12871, 10001, 89813, 13213, 16531, 40837, 42517, 218867, 59891, 95719

Offset: 1

Lekraj Beedassy, Apr 02 2003

nonn

			The first Carmichael number is 3*11*17, so a(1) = 11*17.

Robert Israel, Table of n, a(n) for n = 1..10000
J. Perry, Carmichael factors

Cf. A002997, A032742, A141710.

seq(A002997[n]/min(numtheory:-factorset(A002997[n])), n=1..100); # Robert Israel, Apr 13 2017

a(n) = A032742(A002997(n)) - David Wasserman, Jun 18 2004

a(n) = A002997(n)/A141710(n). - Robert Israel, Apr 13 2017

More terms from David Wasserman, Jun 18 2004

A180044 Let the n-th Carmichael number A002997(n) = p1p2...pr, where p1 < p2 < ... < pr are primes. Then a(n) = (p1-1) (p1p2...pr - 1)^(r-2) / ((p2-1)...*(pr-1)).

Original entry on oeis.org

7, 23, 48, 22, 47, 45, 45, 21, 44, 163, 2105352, 162, 43, 486266, 3157729, 9859600, 5110605, 161, 6146018, 280, 8225424, 9135075, 1684, 6185169, 1363, 159, 351, 59907600, 950, 1675, 9879408, 1358, 949, 158, 95468562, 4399220, 83722500

Offset: 1

A.K. Devaraj, Aug 08 2010

nonn

a(n) is always an integer as proved at the Alekseyev link.
The conjecture referred to in A162290 was generalized as follows: Let k be an r-factor Carmichael number (p_1 < p_2 < ... < p_r). Then (p_1-1)*(k-1)^(r-2)/((p_2-1)*(p_3-1)*...*(p_r-1)) is an integer. This was proved by Max Alekseyev (see link).
Contains A162290 as a subsequence.

			Since A002997(11) = 41041 = 7*11*13*41, we have a(11) = (6*41040^2) / (10*12*40) = 2105352.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, Pomerance's proof.

Cf. A104016, A104017, A002997, A162290.

[ (d[1]-1)*(n-1)^(r-2) / &*[ d[i]-1: i in [2..r] ]: n in [3..700000 by 2] | not IsPrime(n) and IsSquarefree(n) and forall(t){x: x in d | (n-1) mod (x-1) eq 0} where r is #d where d is PrimeDivisors(n)]; // Klaus Brockhaus, Aug 10 2010

lim = 1000001; CarmichaelQ[n_] := Divisible[n - 1, CarmichaelLambda[n]] && ! PrimeQ[n]; cc = Select[Table[k, {k, 561, lim, 2}], CarmichaelQ]; lg = Length[cc]; a[n_] := (c = cc[[n]]; pp = FactorInteger[c][[All, 1]]; r = Length[pp]; (pp[[1]] - 1)*((Times @@ pp - 1)^(r - 2)/ Times @@ (Drop[pp, 1] - 1))); Table[a[n], {n, 1, lg}] (* Jean-François Alcover, Sep 28 2011 *)

Edited and extended by Max Alekseyev and Klaus Brockhaus, Aug 10 2010

A260524 Pseudoprimes to bases 2, 3, 5 and 7 that are congruent to 5 (modulo 6) but are not Carmichael numbers (A002997).

Original entry on oeis.org

468950021, 493108481, 659846021, 5936122901, 8144063621, 11408333333, 12601267541, 14252656133, 18074903681, 27223783841, 30633711701, 31093792133, 31797754721, 61426533761, 65085388961, 86610942881, 91945013333, 92380393121, 102538073177

Offset: 1

Ray Chandler, Artur Jasinski and Robert G. Wilson v, Jul 28 2015

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A153581.

fQ[n_] := !PrimeQ[n] && PowerMod[2, n - 1, n] == 1 &&
> PowerMod[3, n - 1, n] == 1 && PowerMod[5, n - 1, n] == 1 && PowerMod[7, n - 1, n] == 1 && Mod[n, CarmichaelLambda[n]] != 1; k = 1; lst = {}; While[k < 25000000001, If[ fQ@ k, AppendTo[lst, k]; Print@ k]; k += 6]; lst

Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
is(n)=n%6==5 && Mod(2,n)^n==2 && Mod(3,n)^n==3 && Mod(5,n)^(n-1)==1 && Mod(7,n)^(n-1)==1 && !isprime(n) && !Korselt(n) \\ Charles R Greathouse IV, Jul 29 2015

use ntheory ":all"; foroddcomposites { say if $%6 == 5 && is_pseudoprime($,2,3,5,7) && $ % carmichael_lambda($) != 1; } 1e9; # Dana Jacobsen, Sep 07 2015

a(9)-a(19) from Charles R Greathouse IV, Jul 29 2015

A272772 Number of prime divisors of (A002997(n) - 2) counted with multiplicity.

Original entry on oeis.org

2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 4, 1, 1, 3, 2, 3, 2, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 2, 1, 3, 2, 2, 1, 3, 1, 2, 3, 3, 4, 3, 2, 1, 2, 3, 2, 2, 2, 3, 3, 2, 2, 4, 1, 3, 3, 2, 4, 3, 2, 2, 2, 1, 2, 2, 3, 3, 2, 2, 2, 2, 4, 2, 1, 2, 2, 4, 2, 2, 2, 1, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 3, 2, 1, 2, 2, 2, 4, 2, 2, 2, 4, 2

Offset: 1

Altug Alkan, May 06 2016

nonn

62756641 is the first Carmichael number k such that k-2 has 5 prime divisors (counted with multiplicity).
What is the average value function of a(n) when n goes to infinity?
If these number act like typical numbers of their size, then standard heuristics suggest an average value of log log n since there are between x^(1/3) and x Carmichael numbers up to x for large enough x. - Charles R Greathouse IV, May 09 2016

			a(1) = 2 because 561 - 2 = 559 has 2 prime divisors that are 13 and 43.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A002997, A135717.

isA002997(n)=my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1
for(n=561, 1e7, if(isA002997(n), print1(bigomega(n-2), ", ")));

a(n) = A001222(A002997(n)-2).

A303791 Carmichael numbers (A002997) that are central polygonal numbers (A002061).

Original entry on oeis.org

5310721, 2278677961, 9593125081, 29859667201, 467593730289953281, 98538479002618905601, 146842414757227736821

Offset: 1

Max Alekseyev, Apr 30 2018

Also, Carmichael numbers of the form k^2 + k + 1.
Also, Carmichael numbers of the form k^2 - k + 1.
There are no other terms below 10^22.
Carmichael numbers m such that 4m - 3 is square. - Thomas Ordowski, Apr 30 2018

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

Intersection of A002997 and A002061.

Cf. A265285, A265628.

a(6)-a(7) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 20 2024

A352987 Carmichael numbers (A002997) that are overpseudoprimes to base 2 (A141232).

Original entry on oeis.org

65700513721, 168003672409, 459814831561, 13685652197857, 34477679139751, 74031531351121, 92327722290241, 206175669172201, 704077371354601, 1882982959757929, 2901482064497017, 3715607011189609, 5516564718607489, 5636724028491697, 6137426373439681, 14987802403246609

Offset: 1

Daniel Suteu, May 06 2022

nonn

If we define f(n) to be the smallest number in the sequence with n prime factors, then we have:
f(3) = 65700513721,
f(4) <= 84286331493236478328609,
f(5) <= 3848515708338676403444146123852434164444641.

Amiram Eldar, Table of n, a(n) for n = 1..450 (calculated using data from Claude Goutier; terms 1..102 from Daniel Suteu)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Intersection of A002997 and A141232.

Intersection of A291637 and A141232.

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

Daniel Suteu, Oct 01 2022

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

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);

a(n) >= max(A180065(n), A006931(n)).

A001567 Fermat pseudoprimes to base 2, also called Sarrus numbers or Poulet numbers.

Original entry on oeis.org

341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911, 10261, 10585, 11305, 12801, 13741, 13747, 13981, 14491, 15709, 15841, 16705, 18705, 18721, 19951, 23001, 23377, 25761, 29341

Offset: 1

N. J. A. Sloane

A composite number n is a Fermat pseudoprime to base b if and only if b^(n-1) == 1 (mod n). Fermat pseudoprimes to base 2 are often simply called pseudoprimes.
Theorem: If both numbers q and 2q - 1 are primes (q is in the sequence A005382) and n = q*(2q-1) then 2^(n-1) == 1 (mod n) (n is in the sequence) if and only if q is of the form 12k + 1. The sequence 2701, 18721, 49141, 104653, 226801, 665281, 721801, ... is related. This subsequence is also a subsequence of the sequences A005937 and A020137. - Farideh Firoozbakht, Sep 15 2006
Also, composite odd numbers n such that n divides 2^n - 2 (cf. A006935). It is known that all primes p divide 2^(p-1) - 1. There are only two known numbers n such that n^2 divides 2^(n-1) - 1, A001220(n) = {1093, 3511} Wieferich primes p: p^2 divides 2^(p-1) - 1. 1093^2 and 3511^2 are the terms of a(n). - Alexander Adamchuk, Nov 06 2006
An odd composite number 2n + 1 is in the sequence if and only if multiplicative order of 2 (mod 2n+1) divides 2n. - Ray Chandler, May 26 2008
The Carmichael numbers A002997 are a subset of this sequence. For the Sarrus numbers which are not Carmichael numbers, see A153508. - Artur Jasinski, Dec 28 2008
An odd number n greater than 1 is a Fermat pseudoprime to base b if and only if ((n-1)! - 1)*b^(n-1) == -1 (mod n). - Arkadiusz Wesolowski, Aug 20 2012
The name "Sarrus numbers" is after Frédéric Sarrus, who, in 1819, discovered that 341 is a counterexample to the "Chinese hypothesis" that n is prime if and only if 2^n is congruent to 2 (mod n). - Alonso del Arte, Apr 28 2013
The name "Poulet numbers" appears in Monografie Matematyczne 42 from 1932, apparently because Poulet in 1928 produced a list of these numbers (cf. Miller, 1975). - Felix Fröhlich, Aug 18 2014
Numbers n > 2 such that (n-1)! + 2^(n-1) == 1 (mod n). Composite numbers n such that (n-2)^(n-1) == 1 (mod n). - Thomas Ordowski, Feb 20 2016
The only twin pseudoprimes up to 10^13 are 4369, 4371. - Thomas Ordowski, Feb 12 2016
Theorem (A. Rotkiewicz, 1965): (2^p-1)*(2^q-1) is a pseudoprime if and only if p*q is a pseudoprime, where p,q are different primes. - Thomas Ordowski, Mar 31 2016
Theorem (W. Sierpiński, 1947): if n is a pseudoprime (odd or even), then 2^n-1 is a pseudoprime. - Thomas Ordowski, Apr 01 2016
If 2^n-1 is a pseudoprime, then n is a prime or a pseudoprime (odd or even). - Thomas Ordowski, Sep 05 2016
From Amiram Eldar, Jun 19 2021, Apr 21 2024: (Start)
Erdős (1950) called these numbers "almost primes".
According to Erdős (1949) and Piza (1954), the term "pseudoprime" was coined by D. H. Lehmer.
Named after the French mathematician Pierre de Fermat (1607-1665), or, alternatively, after the Belgian mathematician Paul Poulet (1887-1946), or, the French mathematician Pierre Frédéric Sarrus (1798-1861). (End)
If m is a term of this sequence, then (m-1)/ord(2,m) >= 5, where ord(a,m) is the multiplicative order of a modulo m; see my link below. Actually, it seems that we always have (m-1)/ord(2,m) >= 9. - Jianing Song, Nov 04 2024

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 80.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A12, pp. 44-50.
George P. Loweke, The Lore of Prime Numbers. New York: Vantage Press (1982), p. 22.
Øystein Ore, Number Theory and Its History, McGraw-Hill, 1948.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 88-92.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 145.

N. J. A. Sloane, Table of n, a(n) for n = 1..101629 [The pseudoprimes up to 10^12, from Richard Pinch's web site - see links below]
Jonathan Bayless and Paul Kinlaw, Explicit Bounds for the Sum of Reciprocals of Pseudoprimes and Carmichael Numbers, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.4.
Jens Bernheiden, Pseudoprimes (Text in German).
John Brillhart, N. J. A. Sloane, J. D. Swift, Correspondence, 1972.
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
Paul Erdős, On the converse of Fermat's theorem, The American Mathematical Monthly, Vol. 56, No. 9 (1949), pp. 623-624; alternative link.
Paul Erdős, On almost primes, Amer. Math. Monthly, Vol. 57, No. 6 (1950), pp. 404-407; alternative link.
Jan Feitsma, The pseudoprimes below 2^64.
William Galway, Tables of pseudoprimes and related data [Includes a file with pseudoprimes up to 2^64.]
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly, Vol. 95, No. 8 (1988), pp. 697-712. [Annotated scanned copy]
Paul Kinlaw, The reciprocal sums of pseudoprimes and Carmichael numbers, Mathematics of Computation (2023).
D. H. Lehmer, Guide to Tables in the Theory of Numbers, Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 48.
D. H. Lehmer, Errata for Poulet's table, Math. Comp., Vol. 25, No. 116 (1971), pp. 944-945.
D. H. Lehmer, Errata for Poulet's table. [annotated scanned copy]
Gérard P. Michon, Pseudoprimes.
J. C. P. Miller, On factorization, with a suggested new approach, Math. Comp., Vol. 29, No. 129 (1975), pp. 155-172. - _Felix Fröhlich_, Aug 18 2014
Robert Morris, Some observations on the converse of Fermat's theorem, unpublished memorandum, Oct 03 1973.
Richard Pinch, Pseudoprimes.
P. A. Piza, Fermat Coefficients, Mathematics Magazine, Vol. 27, No. 3 (1954), pp. 141-146.
Carl Pomerance & N. J. A. Sloane, Correspondence, 1991.
Paul Poulet, Tables des nombres composés vérifiant le théorème de Fermat pour le module 2 jusqu'à 100.000.000, Sphinx (Brussels), Vol. 8 (1938), pp. 42-45. [annotated scanned copy]
Fred Richman, Primality testing with Fermat's little theorem.
Andrzej Rotkiewicz, Sur les nombres pseudopremiers de la forme MpMq, Elemente der Mathematik, Vol. 20 (1965), pp. 108-109.
Waclaw Sierpiński, Remarque sur une hypothèse des Chinois concernant les nombres (2^n-2)/n, Colloquium Mathematicum, Vol. 1 (1947), p. 9.
Waclaw Sierpiński, Elementary Theory of Numbers, Państ. Wydaw. Nauk., Warszawa, 1964, p. 215.
Jianing Song, Notes on Fermat Pseudoprimes.
Eric Weisstein's World of Mathematics, Chinese Hypothesis, Fermat Pseudoprime, Poulet Number, and Pseudoprime.
Wikipedia, Chinese hypothesis and Pseudoprime.
Index entries for sequences related to pseudoprimes

Cf. A002997, A005382, A020137, A052155, A083737, A084653, A153508.
Cf. A001220 = Wieferich primes p: p^2 divides 2^(p-1) - 1.
Cf. A005935, A005936, A005937, A005938, A005939, A020136-A020228 (pseudoprimes to bases 3 through 100).

[n: n in [3..3*10^4 by 2] | IsOne(Modexp(2,n-1,n)) and not IsPrime(n)]; // Bruno Berselli, Jan 17 2013

select(t -> not isprime(t) and 2 &^(t-1) mod t = 1, [seq(i,i=3..10^5,2)]); # Robert Israel, Feb 18 2016

Select[Range[3,30000,2], ! PrimeQ[ # ] && PowerMod[2, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)

q=1;vector(50,i,until( !isprime(q) & (1<<(q-1)-1)%q == 0, q+=2);q) \\ M. F. Hasler, May 04 2007

is_A001567(n)={Mod(2,n)^(n-1)==1 && !isprime(n) && n>1}  \\ M. F. Hasler, Oct 07 2012, updated to current PARI syntax and to exclude even pseudoprimes on Mar 01 2019

Sum_{n>=1} 1/a(n) is in the interval (0.015260, 33) (Bayless and Kinlaw, 2017). The upper bound was reduced to 0.0911 by Kinlaw (2023). - Amiram Eldar, Oct 15 2020, Feb 24 2024

More terms from David W. Wilson, Aug 15 1996
Replacement of broken geocities link by Jason G. Wurtzel, Sep 05 2010
"Poulet numbers" added to name by Joerg Arndt, Aug 18 2014

cp's OEIS Frontend

A335584 Carmichael numbers (A002997) that are not minimal in their family.

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A346569 Carmichael numbers (A002997) k such that A003961(k) is also a Carmichael number.

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A081703 Largest proper divisor of the n-th Carmichael number (A002997).

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A180044 Let the n-th Carmichael number A002997(n) = p1*p2*...*pr, where p1 < p2 < ... < pr are primes. Then a(n) = (p1-1) * (p1*p2*...*pr - 1)^(r-2) / ((p2-1)*...*(pr-1)).

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Magma

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A260524 Pseudoprimes to bases 2, 3, 5 and 7 that are congruent to 5 (modulo 6) but are not Carmichael numbers (A002997).

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A272772 Number of prime divisors of (A002997(n) - 2) counted with multiplicity.

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A303791 Carmichael numbers (A002997) that are central polygonal numbers (A002061).

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A352987 Carmichael numbers (A002997) that are overpseudoprimes to base 2 (A141232).

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A180044 Let the n-th Carmichael number A002997(n) = p1p2...pr, where p1 < p2 < ... < pr are primes. Then a(n) = (p1-1) (p1p2...pr - 1)^(r-2) / ((p2-1)...*(pr-1)).