A324973 -id:A324973 - cp's OEIS frontend

A324974 Rank of the n-th special polygonal number A324973(n).

3, 3, 3, 5, 3, 3, 6, 3, 6, 3, 11, 5, 3, 3, 8, 10, 5, 6, 12, 3, 15, 9, 3, 5, 3, 8, 3, 8, 19, 14, 5, 7, 3, 6, 6, 36, 21, 66, 22, 3, 10, 5, 6, 3, 3, 50, 10, 20, 5, 14, 11, 51, 3, 10, 21, 6, 13, 5, 16, 25, 3, 3, 6, 6, 12, 14, 10, 68, 5, 28, 3, 11, 29, 3, 56, 6, 19

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Mar 24 2019

nonn

While two polygonal numbers of different ranks can be equal (e.g., P(6,n) = P(3,2n-1)), that cannot occur for special polygonal numbers, since for fixed p the value of P(r,p) is strictly increasing with r. Thus the rank of a special polygonal number is well-defined.

The Carmichael numbers A002997 and primary Carmichael numbers A324316 are special polygonal numbers (see Kellner and Sondow 2019). Their ranks form the subsequences A324975 and A324976.

			If m = A324973(4) = 70 = 2*5*7, then p = 7, so a(4) = 2+2*((70/7)-1)/(7-1) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv preprint, arXiv:1902.10672 [math.NT], 2019-2021.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv preprint, arXiv:1902.11283 [math.NT], 2019-2022.
Wikipedia, Polygonal number.

A324975 and A324976 are subsequences.

Cf. A002997, A324316, A324972, A324973, A324977.

GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
T = Select[Flatten[Table[{p, (p^2*(r - 2) - p*(r - 4))/2}, {p, 3, 150}, {r, 3, 100}], 1], SquareFreeQ[Last[#]] && First[#] == GPF[Last[#]] &];
TT = Take[Union[Table[Last[T[[i]]], {i, Length[T]}]], 47];
Table[2 + 2*(t/GPF[t] - 1)/(GPF[t] - 1), {t, TT}]

a(n) = 2 + 2*((m/p)-1)/(p-1), where m = A324973(n) and p is its greatest prime factor. (Proof: solve m = P(r,p) = (p^2*(r-2) - p*(r-4))/2 for r.)

Several missing terms inserted by and more terms from Jinyuan Wang, Feb 18 2021

A002997 Carmichael numbers: composite numbers k such that a^(k-1) == 1 (mod k) for every a coprime to k.

Original entry on oeis.org

561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461, 530881, 552721

Offset: 1

N. J. A. Sloane

V. Šimerka found the first 7 terms of this sequence 25 years before Carmichael (see the link and also the remark of K. Conrad). - Peter Luschny, Apr 01 2019
k is composite and squarefree and for p prime, p|k => p-1|k-1.
An odd composite number k is a pseudoprime to base a iff a^(k-1) == 1 (mod k). A Carmichael number is an odd composite number k which is a pseudoprime to base a for every number a prime to k.
A composite odd number k is a Carmichael number if and only if k is squarefree and p-1 divides k-1 for every prime p dividing k. (Korselt, 1899)
Ghatage and Scott prove using Fermat's little theorem that (a+b)^k == a^k + b^k (mod k) (the freshman's dream) exactly when k is a prime (A000040) or a Carmichael number. - Jonathan Vos Post, Aug 31 2005
Alford et al. have constructed a Carmichael number with 10333229505 prime factors, and have also constructed Carmichael numbers with m prime factors for every m between 3 and 19565220. - Jonathan Vos Post, Apr 01 2012
Thomas Wright proved that for any numbers b and M in N with gcd(b,M) = 1, there are infinitely many Carmichael numbers k such that k == b (mod M). - Jonathan Vos Post, Dec 27 2012
Composite numbers k relatively prime to 1^(k-1) + 2^(k-1) + ... + (k-1)^(k-1). - Thomas Ordowski, Oct 09 2013
Composite numbers k such that A063994(k) = A000010(k). - Thomas Ordowski, Dec 17 2013
Odd composite numbers k such that k divides A002445((k-1)/2). - Robert Israel, Oct 02 2015
If k is a Carmichael number and gcd(b-1,k)=1, then (b^k-1)/(b-1) is a pseudoprime to base b by Steuerwald's theorem; see the reference in A005935. - Thomas Ordowski, Apr 17 2016
Composite numbers k such that p^k == p (mod k) for every prime p <= A285512(k). - Max Alekseyev and Thomas Ordowski, Apr 20 2017
If a composite m < A285549(n) and p^m == p (mod m) for every prime p <= prime(n), then m is a Carmichael number. - Thomas Ordowski, Apr 23 2017
The sequence of all Carmichael numbers can be defined as follows: a(1) = 561, a(n+1) = smallest composite k > a(n) such that p^k == p (mod k) for every prime p <= n+2. - Thomas Ordowski, Apr 24 2017
An integer m > 1 is a Carmichael number if and only if m is squarefree and each of its prime divisors p satisfies both s_p(m) >= p and s_p(m) == 1 (mod p-1), where s_p(m) is the sum of the base-p digits of m. Then m is odd and has at least three prime factors. For each prime factor p, the sharp bound p <= a*sqrt(m) holds with a = sqrt(17/33) = 0.7177.... See Kellner and Sondow 2019. - Bernd C. Kellner and Jonathan Sondow, Mar 03 2019
Carmichael numbers are special polygonal numbers A324973. The rank of the n-th Carmichael number is A324975(n). See Kellner and Sondow 2019. - Jonathan Sondow, Mar 26 2019
An odd composite number m is a Carmichael number iff m divides denominator(Bernoulli(m-1)). The quotient is A324977. See Pomerance, Selfridge, & Wagstaff, p. 1006, and Kellner & Sondow, section on Bernoulli numbers. - Jonathan Sondow, Mar 28 2019
This is setwise difference A324050 \ A008578. Many of the same identities apply also to A324050. - Antti Karttunen, Apr 22 2019
If k is a Carmichael number, then A309132(k) = A326690(k). The proof generalizes that of Theorem in A309132. - Jonathan Sondow, Jul 19 2019
Composite numbers k such that A111076(k)^(k-1) == 1 (mod k). Proof: the multiplicative order of A111076(k) mod k is equal to lambda(k), where lambda(k) = A002322(k), so lambda(k) divides k-1, qed. - Thomas Ordowski, Nov 14 2019
For all positive integers m, m^k - m is divisible by k, for all k > 1, iff k is either a Carmichael number or a prime, as is used in the proof by induction for Fermat's Little Theorem. Also related are A182816 and A121707. - Richard R. Forberg, Jul 18 2020
From Amiram Eldar, Dec 04 2020, Apr 21 2024: (Start)
Ore (1948) called these numbers "Numbers with the Fermat property", or, for short, "F numbers".
Also called "absolute pseudoprimes". According to Erdős (1949) this term was coined by D. H. Lehmer.
Named by Beeger (1950) after the American mathematician Robert Daniel Carmichael (1879 - 1967). (End)
For ending digit 1,3,5,7,9 through the first 10000 terms, we see 80.3, 4.1, 7.4, 3.8 and 4.3% apportionment respectively. Why the bias towards ending digit "1"? - Bill McEachen, Jul 16 2021
It seems that for any m > 1, the remainders of Carmichael numbers modulo m are biased towards 1. The number of terms congruent to 1 modulo 4, 6, 8, ..., 24 among the first 10000 terms: 9827, 9854, 8652, 8034, 9682, 5685, 6798, 7820, 7880, 3378 and 8518. - Jianing Song, Nov 08 2021
Alford, Granville and Pomerance conjectured in their 1994 paper that a statement analogous to Bertrand's Postulate could be applied to Carmichael numbers. This has now been proved by Daniel Larsen, see link below. - David James Sycamore, Jan 17 2023

N. G. W. H. Beeger, On composite numbers n for which a^n == 1 (mod n) for every a prime to n, Scripta Mathematica, Vol. 16 (1950), pp. 133-135.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover Publications, Inc. New York, 1966, Table 18, Page 44.
David M. Burton, Elementary Number Theory, 5th ed., McGraw-Hill, 2002.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 142.
CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 87.
Richard K. Guy, Unsolved Problems in Number Theory, A13.
Øystein Ore, Number Theory and Its History, McGraw-Hill, 1948, Reprinted by Dover Publications, 1988, Chapter 14.
Paul Poulet, Tables des nombres composés vérifiant le théorème du Fermat pour le module 2 jusqu'à 100.000.000, Sphinx (Brussels), 8 (1938), 42-45.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 22, 100-103.
Wacław Sierpiński, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 51.
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, pages 145-146.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 561 at p. 157.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (from the Pinch web site mentioned below)
W. R. Alford, Jon Grantham, Steven Hayman, and Andrew Shallue, Constructing Carmichael numbers through improved subset-product algorithms, Mathematics of Computation, Vol. 83, No. 286 (2014), pp. 899-915, arXiv preprint, arXiv:1203.6664v1 [math.NT], Mar 29 2012.
W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703-722.
W. R. Alford, A. Granville, and C. Pomerance (1994). "On the difficulty of finding reliable witnesses". Lecture Notes in Computer Science 877, 1994, pp. 1-16.
François Arnault, Thesis.
François Arnault, Constructing Carmichael numbers which are strong pseudoprimes to several bases, Journal of Symbolic Computation, vol. 20, no 2, Aug. 1995, pp. 151-161.
François Arnault, Rabin-Miller primality test: Composite numbers which pass it, Mathematics of Computation, vol. 64, no 209, 1995, pp. 355-361.
François Arnault, The Rabin-Monier theorem for Lucas pseudoprimes, Mathematics of Computation, vol. 66, no 218, April 1997, pp. 869-881.
Joerg Arndt, Matters Computational (The Fxtbook), p. 786.
Eric Bach and Rex Fernando, Infinitely Many Carmichael Numbers for a Modified Miller-Rabin Prime Test, ISSAC '16: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, July 2016, pp. 47-54, arXiv preprint, arXiv:1512.00444 [math.NT], 2015.
Sunghan Bae, Su Hu, and Min Sha, On the Carmichael rings, Carmichael ideals and Carmichael polynomials, arXiv:1809.05432 [math.NT], 2018.
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, Carmichael numbers (Text in German).
John D. Brillhart, N. J. A. Sloane, and J. D. Swift, Correspondence, 1972.
Ronald Joseph Burthe, Jr. Upper bounds for least witnesses and generating sets, Acta Arith. 80 (1997), no. 4, 311-326.
C. K. Caldwell, The Prime Glossary, Carmichael number.
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1910), 232-238.
Keith Conrad, Carmichael numbers and Korselt's criterion, expository paper (2016), 1-3.
K. A. Draziotis, V. Martidis, and S. Tiganourias, Product Subset Problem: Applications to number theory and cryptography, arXiv:2002.07095 [math.NT], 2020. See also Chapter 5, Analysis, Cryptography and Information Science, World Scientific (2023), p. 108.
Harvey Dubner, Carmichael Numbers of the form (6m+1)(12m+1)(18m+1), Journal of Integer Sequences, Vol. 5 (2002) Article 02.2.1.
James Emery, Number Theory, 2013. [Broken link]
Paul Erdős, On the converse of Fermat's theorem, The American Mathematical Monthly, Vol. 56, No. 9 (1949), pp. 623-624; alternative link.
Jan Feitsma and William Galway, Tables of pseudoprimes and related data.
Pratibha Ghatage and Brian Scott, Exactly When is (a+b)^n == a^n + b^n (mod n)?, College Mathematics Journal, Vol. 36, No. 4 (Sep 2005), p. 322.
Claude Goutier, Compressed text file carm10e22.7z containing all the Carmichael numbers up to 10^22. [Local copy, with permission. This is a very large file.]
Andrew Granville, Papers on Carmichael numbers.
Andrew Granville, Primality testing and Carmichael numbers, Notices Amer. Math. Soc., 39 (No. 7, 1992), 696-700.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comp. 71 (2002), no. 238, 883-908.
Gerhard Jaeschke, The Carmichael numbers to 10^12, Math. Comp., Vol. 55, No. 191 (1990), pp. 383-389.
Eugene Karolinsky and Dmytro Seliutin, Carmichael numbers for GL(m), arXiv:2001.10315 [math.NT], 2020.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Paul Kinlaw, The reciprocal sums of pseudoprimes and Carmichael numbers, Mathematics of Computation (2023).
A. Korselt, G. Tarry, I. Franel, and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), 142-144.
Daniel Larsen, Bertrand's Postulate for Carmichael Numbers, arXiv:2111.06963 [math.NT], 2021.
D. H. Lehmer, Errata for Poulet's table, Math. Comp., 25 (1971), 944-945. 25 944 1971.
Max Lewis and Victor Scharaschkin, k-Lehmer and k-Carmichael Numbers, Integers, 16 (2016), #A80.
Renaud Lifchitz, A generalization of the Korselt's criterion - Nested Carmichael numbers.
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.
Romeo Meštrović, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Yoshio Mimura, Carmichael Numbers up to 10^12. [broken link, WayBackMachine]
Math Reference Project, Carmichael Numbers.
R. G. E. Pinch, The Carmichael numbers up to 10^18, 2008.
R. G. E. Pinch, Carmichael numbers up to 10^15, 10^16, 10^16 to 10^17, 10^17 to 10^18, 10^19, and 10^21.
Carl Pomerance, J. L. Selfridge, and Samuel S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Math. Comp., Vol. 35, No. 151 (1980), pp. 1003-1026.
Carl Pomerance and N. J. A. Sloane, Correspondence, 1991.
Fred Richman, Primality testing with Fermat's little theorem.
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. - From _N. J. A. Sloane_, Feb 07 2013
Václav Šimerka, Zbytky z arithmetické posloupnosti, (On the remainders of an arithmetic progression), Časopis pro pěstování matematiky a fysiky. 14 (1885), 221-225.
Eric Weisstein's World of Mathematics, Carmichael Number, Knoedel Numbers, and Pseudoprime.
Wikipedia, Carmichael number.
Thomas Wright, Infinitely many Carmichael numbers in arithmetic progressions, Bulletin of the London Mathematical Society, 45 (2013) 943-952, arXiv preprint, arXiv:1212.5850 [math.NT], Dec 2012.
Index entries for sequences related to Carmichael numbers.

Cf. A001567, A002445, A002322, A006931, A024556, A027748, A055553, A064238-A064262, A083737, A087441, A087442, A135717, A141711, A153581, A225498, A285512, A285549, A309132, A324290, A324315, A324316, A324973, A324975, A324977, A326690.

Subsequence of A324050.

a002997 n = a002997_list !! (n-1)
a002997_list = [x | x <- a024556_list,
all (== 0) $ map ((mod (x - 1)) . (subtract 1)) $ a027748_row x]
-- Reinhard Zumkeller, Apr 12 2012

[n: n in [3..53*10^4 by 2] | not IsPrime(n) and n mod CarmichaelLambda(n) eq 1]; // Bruno Berselli, Apr 23 2012

filter:= proc(n)
  local q;
  if isprime(n) then return false fi;
  if 2 &^ (n-1) mod n <> 1 then return false fi;
  if not numtheory:-issqrfree(n) then return false fi;
  for q in numtheory:-factorset(n) do
    if (n-1) mod (q-1) <> 0 then return false fi
  od:
  true;
end proc:
select(filter, [seq(2*k+1,k=1..10^6)]); # Robert Israel, Dec 29 2014
isA002997 := n -> 0 = modp(n-1, numtheory:-lambda(n)) and not isprime(n) and n <> 1:
select(isA002997, [$1..10000]); # Peter Luschny, Jul 21 2019

Cases[Range[1,100000,2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]] (* Artur Jasinski, Apr 05 2008; minor edit from Zak Seidov, Feb 16 2011 *)
Select[Range[1,600001,2],CompositeQ[#]&&Mod[#,CarmichaelLambda[#]]==1&] (* Harvey P. Dale, Jul 08 2023 *)

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
isA002997(n)=n%2 && !isprime(n) && Korselt(n) && n>1 \\ Charles R Greathouse IV, Jun 10 2011

is_A002997(n, F=factor(n)~)={ #F>2 && !foreach(F,f,(n%(f[1]-1)==1 && f[2]==1) || return)} \\ No need to check parity: if efficiency is needed, scan only odd numbers. - M. F. Hasler, Aug 24 2012, edited Mar 24 2022

from itertools import islice
from sympy import nextprime, factorint
def A002997_gen(): # generator of terms
    p, q = 3, 5
    while True:
        for n in range(p+2,q,2):
            f = factorint(n)
            if max(f.values()) == 1 and not any((n-1) % (p-1) for p in f):
                yield n
        p, q = q, nextprime(q)
A002997_list = list(islice(A002997_gen(),20)) # Chai Wah Wu, May 11 2022

def isCarmichael(n):
    if n == 1 or is_even(n) or is_prime(n):
        return False
    factors = factor(n)
    for f in factors:
        if f[1] > 1: return False
        if (n - 1) % (f[0] - 1) != 0:
            return False
    return True
print([n for n in (1..20000) if isCarmichael(n)]) # Peter Luschny, Apr 02 2019

Sum_{n>=1} 1/a(n) is in the interval (0.004706, 27.8724) (Bayless and Kinlaw, 2017). The upper bound was reduced to 0.0058 by Kinlaw (2023). - Amiram Eldar, Oct 26 2020, Feb 24 2024

Links for lists of Carmichael numbers updated by Jan Kristian Haugland, Mar 25 2009 and Danny Rorabaugh, May 05 2017

A324316 Primary Carmichael numbers.

Original entry on oeis.org

1729, 2821, 29341, 46657, 252601, 294409, 399001, 488881, 512461, 1152271, 1193221, 1857241, 3828001, 4335241, 5968873, 6189121, 6733693, 6868261, 7519441, 10024561, 10267951, 10606681, 14469841, 14676481, 15247621, 15829633, 17098369, 17236801, 17316001, 19384289, 23382529, 29111881, 31405501, 34657141, 35703361, 37964809

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 21 2019

Squarefree integers m > 1 such that if prime p divides m, then the sum of the base-p digits of m equals p. It follows that m is then a Carmichael number (A002997).
Dickson's conjecture implies that the sequence is infinite, see Kellner 2019.
If m is a term and p is a prime factor of m, then p <= a*sqrt(m) with a = sqrt(66337/132673) = 0.7071..., where the bound is sharp.
The distribution of primary Carmichael numbers is A324317.
See Kellner and Sondow 2019 and Kellner 2019.
Primary Carmichael numbers are special polygonal numbers A324973. The rank of the n-th primary Carmichael number is A324976(n). See Kellner and Sondow 2019. - Jonathan Sondow, Mar 26 2019
The first term is the Hardy-Ramanujan number. - Omar E. Pol, Jan 09 2020

			1729 = 7 * 13 * 19 is squarefree, and 1729 in base 7 is 5020_7 = 5 * 7^3 + 0 * 7^2 + 2 * 7 + 0 with 5+0+2+0 = 7, and 1729 in base 13 is a30_13 with a+3+0 = 10+3+0 = 13, and 1729 in base 19 is 4f0_19 with 4+f+0 = 4+15+0 = 19, so 1729 is a member.

Bernd C. Kellner, Table of n, a(n) for n = 1..10000 (computed by using Pinch's database, see link below)
Bernd C. Kellner, On primary Carmichael numbers, #A38 Integers 22 (2022), 39 p.; arXiv:1902.11283 [math.NT], 2019.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, #A52 Integers 21 (2021), 21 p.; arXiv:1902.10672 [math.NT], 2019.
R. G. E. Pinch, The Carmichael numbers up to 10^18, 2008.
Index entries for sequences related to Carmichael numbers.

Subsequence of A002997, A324315.
Least primary Carmichael number with n prime factors is A306657.
Cf. also A005117, A195441, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405, A324973, A324976, A001235.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestCP[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] == # &];
Select[Range[1, 10^7, 2], TestCP[#] &]

use ntheory ":all"; my $m; forsquarefree { $m=$; say if @ > 2 && is_carmichael($m) && vecall { $ == vecsum(todigits($m,$)) } @; } 1e7; # _Dana Jacobsen, Mar 28 2019

from sympy import factorint
from sympy.ntheory import digits
def ok(n):
    pf = factorint(n)
    if n < 2 or max(pf.values()) > 1: return False
    return all(sum(digits(n, p)[1:]) == p for p in pf)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 03 2022

a_1 + a_2 + ... + a_k = p if p is prime and m = a_1 * p + a_2 * p^2 + ... + a_k * p^k with 0 <= a_i <= p-1 for i = 1, 2, ..., k (note that a_0 = 0).

A090466 Regular figurative or polygonal numbers of order greater than 2.

Original entry on oeis.org

6, 9, 10, 12, 15, 16, 18, 21, 22, 24, 25, 27, 28, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 69, 70, 72, 75, 76, 78, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 105, 106, 108, 111, 112, 114, 115, 117, 118

Offset: 1

Robert G. Wilson v, Dec 01 2003

The sorted k-gonal numbers of order greater than 2. If one were to include either the rank 2 or the 2-gonal numbers, then every number would appear.
Number of terms less than or equal to 10^k for k = 1,2,3,...: 3, 57, 622, 6357, 63889, 639946, 6402325, 64032121, 640349979, 6403587409, 64036148166, 640362343980, ..., . - Robert G. Wilson v, May 29 2014
The n-th k-gonal number is 1 + k*n(n-1)/2 - (n-1)^2 = A057145(k,n).
For all squares (A001248) of primes p >= 5 at least one a(n) exists with p^2 = a(n) + 1. Thus the subset P_s(3) of rank 3 only is sufficient. Proof: For p >= 5, p^2 == 1 (mod {3,4,6,8,12,24}) and also P_s(3) + 1 = 3*s - 2 == 1 (mod 3). Thus the set {p^2} is a subset of {P_s(3) + 1}; Q.E.D. - Ralf Steiner, Jul 15 2018
For all primes p > 5, at least one polygonal number exists with P_s(k) + 1 = p when k = 3 or 4, dependent on p mod 6. - Ralf Steiner, Jul 16 2018
Numbers m such that r = (2*m/d - 2)/(d - 1) is an integer for some d, where 2 < d < m is a divisor of 2*m. If r is an integer, then m is the d-th (r+2)-gonal number. - Jianing Song, Mar 14 2021

Albert H. Beiler, Recreations In The Theory Of Numbers, The Queen Of Mathematics Entertains, Dover, NY, 1964, pp. 185-199.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1000 terms are from T. D. Noe)
Felix Ponomarenko and Vadim Ponomarenko, Polygonal Simplex Numbers, San Diego State Univ., 2025.
Eric Weisstein's World of Mathematics, Figurate Number
Index to sequences related to polygonal numbers

Cf. A057145, A001248, A177028 (A342772, A342805), A177201, A316676, A364693 (characteristic function).
Complement is A090467.
Sequence A090428 (excluding 1) is a subsequence of this sequence. - T. D. Noe, Jun 14 2012
Other subsequences: A324972 (squarefree terms), A324973, A342806, A364694.
Cf. also A275340.

isA090466 := proc(n)
    local nsearch,ksearch;
    for nsearch from 3 do
        if A057145(nsearch,3) > n then
            return false;
        end if;
        for ksearch from 3 do
            if A057145(nsearch,ksearch) = n then
                return true;
            elif A057145(nsearch,ksearch) > n then
                break;
            end if;
        end do:
    end do:
end proc:
for n from 1 to 1000 do
    if isA090466(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 28 2016

Take[Union[Flatten[Table[1+k*n (n-1)/2-(n-1)^2,{n,3,100},{k,3,40}]]],67] (* corrected by Ant King, Sep 19 2011 *)
mx = 150; n = k = 3; lst = {}; While[n < Floor[mx/3]+2, a = PolygonalNumber[n, k]; If[a < mx+1, AppendTo[ lst, a], (n++; k = 2)]; k++]; lst = Union@ lst (* Robert G. Wilson v, May 29 2014 and updated Jul 23 2018; PolygonalNumber requires version 10.4 or higher *)

list(lim)=my(v=List()); lim\=1; for(n=3,sqrtint(8*lim+1)\2, for(k=3,2*(lim-2*n+n^2)\n\(n-1), listput(v, 1+k*n*(n-1)/2-(n-1)^2))); Set(v); \\ Charles R Greathouse IV, Jan 19 2017

is(n)=for(s=3,n\3+1,ispolygonal(n,s)&&return(s)); \\ M. F. Hasler, Jan 19 2017

isA090466(m) = my(v=divisors(2*m)); for(i=3, #v, my(d=v[i]); if(d==m, return(0)); if((2*m/d - 2)%(d - 1)==0, return(1))); 0 \\ Jianing Song, Mar 14 2021

Integer k is in this sequence iff A176774(k) < k. - Max Alekseyev, Apr 24 2018

Verified by Don Reble, Mar 12 2006

A324319 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also hexagonal numbers (A000384) with index equal to their largest prime factor.

Original entry on oeis.org

231, 561, 3655, 5565, 8911, 10585, 13695, 23653, 32131, 45451, 59685, 74305, 108345, 115921, 157641, 243253, 248865, 302253, 314821, 334153, 371091, 392055, 417241, 458403, 505515, 546535, 688551, 702705, 795691, 821121, 915981, 932295, 1004653, 1145341, 1181953

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 23 2019

561, 8911, and 10585 are also Carmichael numbers (A002997).
The smallest primary Carmichael number (A324316) in the sequence is 8801128801 = 181 * 733 * 66337 = A000384(66337).
See the section on polygonal numbers in Kellner and Sondow 2019.
Subsequence of the special polygonal numbers A324973. - Jonathan Sondow, Mar 27 2019

			A324315(1) = 231 = 3 * 7 * 11 = 11 * (2 * 11 - 1) = A000384(11), so 231 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, #A52 Integers 21 (2021), 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A000384, A002997, A195441, A324315, A324316, A324317, A324318, A324320, A324369, A324370, A324371, A324404, A324405, A324973.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
HN[n_] := n(2n - 1);
TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
Select[HN@ Prime[Range[100]], TestS[#] &]

More terms from Amiram Eldar, Dec 05 2020

A324320 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also octagonal numbers (A000567) with index equal to their largest prime factor.

Original entry on oeis.org

1045, 2465, 2821, 15841, 20501, 34133, 51221, 68101, 89441, 116033, 118405, 162401, 170885, 216545, 300833, 364705, 439301, 472033, 530881, 642181, 687365, 746005, 970145, 976981, 997633, 1104133, 1148245, 1193221, 1231361, 1239061, 1398101, 1654661, 1971541

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 23 2019

2465 is also a Carmichael number (A002997).
2821 is also a primary Carmichael number (A324316).
See the section on polygonal numbers in Kellner and Sondow 2019.
Subsequence of the special polygonal numbers A324973. - Jonathan Sondow, Mar 27 2019

			A324315(4) = 1045 = 5 * 11 * 19 = 19 * (3 * 19 - 2) = A000567(19), so 1045 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, #A52 Integers 21 (2021), 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A000567, A002997, A324315, A324316, A324317, A324318, A324319, A324369, A324370, A324371, A324404, A324405, A324973.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
ON[n_] := n(3n - 2);
TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
Select[ON@ Prime[Range[100]], TestS[#] &]

More terms from Amiram Eldar, Dec 05 2020

A324972 Squarefree polygonal numbers P(s,n) with s >= 3 and n >= 3.

Original entry on oeis.org

6, 10, 15, 21, 22, 30, 33, 34, 35, 39, 42, 46, 51, 55, 57, 58, 65, 66, 69, 70, 78, 82, 85, 87, 91, 93, 94, 95, 102, 105, 106, 111, 114, 115, 118, 123, 129, 130, 133, 138, 141, 142, 145, 154, 155, 159, 165, 166, 174, 177, 178, 183, 185, 186, 190, 195, 201, 202

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Mar 21 2019

nonn

The main entry for this sequence is A090466 = polygonal numbers of order (or rank) greater than 2.

The special polygonal numbers A324973 form a subsequence that contains all Carmichael numbers A002997. See Kellner and Sondow 2019.

			P(3,3) = 6 which is squarefree, so a(1) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv preprint, arXiv:1902.10672 [math.NT], 2019-2021.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv preprint, arXiv:1902.11283 [math.NT], 2019-2022.
Wikipedia, Polygonal number.

Intersection of A005117 and A090466.

Includes A324973 which contains A002997.

mx = 250; n = s = 3; lst = {};
While[s < Floor[mx/3] + 2, a = (n^2 (s - 2) - n (s - 4))/2;
If[a < mx + 1, AppendTo[lst, a], (s++; n = 2)]; n++]; lst = Union@lst;
Select[lst, SquareFreeQ]

isok(n) = if (!issquarefree(n), return (0)); for(s=3, n\3+1, ispolygonal(n, s) && return(s)); \\ Michel Marcus, Mar 24 2019

Squarefree P(s,n) = (n^2*(s-2)-n*(s-4))/2 with s >= 3 and n >= 3.

A324975 Rank of the n-th Carmichael number.

Original entry on oeis.org

6, 10, 12, 8, 8, 10, 6, 6, 8, 18, 52, 12, 12, 18, 98, 164, 22, 6, 50, 8, 96, 34, 52, 46, 52, 6, 6, 156, 20, 46, 36, 32, 16, 8, 304, 36, 20, 36, 10, 316, 76, 468, 8, 30, 24, 1580, 84, 54, 8, 12, 250, 28, 92, 36, 20, 418, 456, 928, 188, 16, 8, 276, 284, 56, 144

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Mar 24 2019

nonn

See A324974 for definition and explanation of rank of a special polygonal number, hence of rank of a Carmichael number A002997 by Kellner and Sondow 2019.

The ranks of the primary Carmichael numbers A324316 form the subsequence A324976.

			If m = A002997(1) = 561 = 3*11*17, then p = 17, so a(1) = 2+2*((561/17)-1)/(17-1) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Wikipedia, Polygonal number

Subsequence of A324974.
A324976 is a subsequence.
Cf. also A002997, A324316, A324972, A324973, A324977.

T = Cases[Range[1, 10000000, 2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]];
GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
Table[2 + 2*(T[[i]]/GPF[T[[i]]] - 1)/(GPF[T[[i]]] - 1), {i, Length[T]}]

a(n) = 2+2*((m/p)-1)/(p-1), where m = A002997(n) and p is its greatest prime factor. (See Formula in A324974.) Hence a(n) is even, by Carmichael's theorem that p-1 divides (m/p)-1, for any prime factor p of a Carmichael number m.

A324976 Rank of the n-th primary Carmichael number.

Original entry on oeis.org

12, 8, 18, 12, 52, 52, 20, 32, 16, 54, 8, 36, 124, 34, 12, 72, 96, 26, 28, 76, 98, 1804, 108, 124, 18, 72, 172, 120, 10, 104, 32, 244, 130, 376, 18, 92, 780, 36, 172, 92, 284, 24, 198, 12, 244, 64, 234, 340, 100, 284, 24, 124, 44, 518, 364, 16, 82, 148, 8, 206

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Mar 24 2019

See A324974 for definition and explanation of rank of a special polygonal number, hence of rank of a primary Carmichael number A324316 by Kellner and Sondow 2019.

			If m = A324316(1) = 1729 = 7*13*19, then p = 19, so a(1) = 2+2*((1729/19)-1)/(19-1) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Wikipedia, Polygonal number

Subsequence of A324975 (rank of the n-th Carmichael number A002997) and of A324974 (rank of the n-th special polygonal number A324973).

Cf. also A324316, A324972.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestCP[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] == # &];
T = Select[Range[1, 10^7, 2], TestCP[#] &];
GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
Table[2 + 2*(T[[i]]/GPF[T[[i]]] - 1)/(GPF[T[[i]]] - 1), {i, Length[T]}]

a(n) = 2+2*((m/p)-1)/(p-1), where m = A324316(n) and p is its greatest prime factor. Hence a(n) is even; see Formula in A324975.

More terms from Amiram Eldar, Mar 27 2019

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A324974 Rank of the n-th special polygonal number A324973(n).

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A002997 Carmichael numbers: composite numbers k such that a^(k-1) == 1 (mod k) for every a coprime to k.

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A324316 Primary Carmichael numbers.

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A090466 Regular figurative or polygonal numbers of order greater than 2.

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A324319 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also hexagonal numbers (A000384) with index equal to their largest prime factor.

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A324320 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also octagonal numbers (A000567) with index equal to their largest prime factor.

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