Bernd C. Kellner - cp's OEIS frontend

A377024 Decimal expansion of the constant F(2) related to asymptotic products of factorials.

1, 0, 2, 3, 9, 3, 7, 4, 1, 1, 6, 3, 7, 1, 1, 8, 4, 0, 1, 5, 7, 7, 9, 5, 0, 7, 8, 2, 5, 8, 6, 2, 1, 7, 8, 0, 0, 8, 0, 3, 7, 6, 0, 9, 8, 0, 4, 3, 6, 4, 4, 0, 0, 5, 1, 2, 9, 4, 6, 9, 9, 0, 9, 5, 1, 3, 4, 7, 6, 9, 2, 4, 1, 2, 4, 0, 0, 7, 8, 2, 7, 6, 8, 7, 1, 1, 5, 2, 9, 4, 7, 4, 6, 5, 9, 8, 8, 1, 7, 3, 0, 6, 2, 3, 4, 8, 3, 6, 4, 2, 4

Offset: 1

Bernd C. Kellner, Oct 13 2024

The constants F(1) = A213080, F(2), ... occur in the context of asymptotic constants related to asymptotic products of factorials as well as of binomial and multinomial coefficients. Moreover, the sequence (F(k)){k >= 1} is strictly decreasing with limit 1. For example, for k >= 1 the asymptotic product Prod{v >= 1} (k*v)! has the asymptotic constant F(k)*A^k*(2*Pi)^(1/4), where A = A074962 denotes the Glaisher-Kinkelin constant. Let gamma = A001620 be Euler's constant and Gamma(x) be the gamma function.
For k >= 1, the constants F(k) can be computed by an explicit formula and a divergent series expansion, as follows. We have log(F(k)) = (1/(12*k))*(1-log(k)) + (k/4)*log(2*Pi) - ((k^2+1)/k)*log(A) - Sum_{v=1..k-1} (v/k)*log(Gamma(v/k)) = gamma/(12*k) - t*zeta(3)/(360*k^3) with some t in (0,1), respectively.
It follows that log(F(2)) = 1/24 + log(2*Pi)/4 + (5/24)*log(2) - (5/2)*log(A) = gamma/24 - t*zeta(3)/2880 with some t in (0,1), and so F(2) lies in the interval (1.023914..., 1.024342...) (see Kellner 2009 and 2024).

			1.02393741163711840157795078258621780080376098043644005129469909513476924124007...

Bernd C. Kellner, Table of n, a(n) for n = 1..10000
Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers 9 (2009), Article #A08, 83-106; alternative link; arXiv:0604505 [math.NT], 2006.
Bernd C. Kellner, Asymptotic products of binomial and multinomial coefficients revisited, Integers 24 (2024), Article #A59, 10 pp.; arXiv:2312.11369 [math.CO], 2023.

Cf. A001620, A002117, A074962, A213080, A377023.

exp(-1/6+5/2*Zeta(1, -1))*(2*Pi)^(1/4)*2^(5/24); evalf(%, 100);

RealDigits[Exp[1/24] (2 Pi)^(1/4) 2^(5/24) / Glaisher^(5/2), 10, 100][[1]]

default(realprecision, 100);
exp(-1/6+5/2*zeta'(-1))*(2*Pi)^(1/4)*2^(5/24)

import mpmath
mpmath.mp.pretty = True; mpmath.mp.dps = 100
mpmath.exp(-1/6+5/2*mpmath.zeta(-1, 1, 1))*(2*pi)^(1/4)*2^(5/24)

Equals exp(1/24)*(2*Pi)^(1/4)*2^(5/24)/A^(5/2) where A = A074962.

Equals exp(-1/6+(5/2)*zeta'(-1))*(2*Pi)^(1/4)*2^(5/24).

A377023 Decimal expansion of the asymptotic constant of the product of binomial coefficients in a row of Pascal's triangle.

Original entry on oeis.org

6, 0, 3, 6, 4, 8, 6, 7, 6, 0, 3, 6, 0, 1, 0, 3, 1, 9, 6, 7, 0, 7, 0, 2, 1, 1, 8, 0, 4, 2, 0, 5, 2, 6, 8, 3, 0, 6, 7, 0, 4, 4, 6, 3, 0, 4, 0, 7, 0, 1, 7, 0, 0, 7, 4, 0, 5, 8, 5, 8, 0, 3, 6, 2, 1, 9, 1, 7, 7, 8, 3, 7, 5, 6, 0, 3, 3, 9, 6, 7, 0, 6, 5, 4, 9, 7, 3, 0, 3, 7, 2, 3, 0, 1, 3, 5, 7, 4, 0, 0, 0, 5, 7, 9, 0

Offset: 0

Bernd C. Kellner, Oct 13 2024

The asymptotic product of binomial coefficients in the n-th row of Pascal's triangle as n goes to infinity provides an asymptotic constant C. This constant must lie in the interval [0.590727...,0.631618...), where the interval is derived from asymptotic products of binomial coefficients over the rows. Indeed, the constant C can also be derived from a limiting case of the latter products (see Kellner 2024).

The constant C is involved with a certain constant F(1) = A213080. The constants F(1), F(2), ... occur in the context of asymptotic constants related to asymptotic products of factorials as well as of binomial and multinomial coefficients. Moreover, the sequence (F(k))_{k >= 1} is strictly decreasing with limit 1. By a divergent series expansion, it follows that F(1) lies in the interval (1.0457...,1.0492...) (see Kellner 2009 and 2024).

			0.60364867603601031967070211804205268306704463040701700740585803621917783756033...

Bernd C. Kellner, Table of n, a(n) for n = 0..10000
Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers 9 (2009), Article #A08, 83-106; alternative link; arXiv:0604505 [math.NT], 2006.
Bernd C. Kellner, Asymptotic products of binomial and multinomial coefficients revisited, Integers 24 (2024), Article #A59, 10 pp.; arXiv:2312.11369 [math.CO], 2023.

Cf. A001620, A002117, A074962, A213080, A377024.

exp(1/12-2*Zeta(1, -1))/(2*Pi)^(1/2); evalf(%, 100);

RealDigits[Glaisher^2/(Exp[1/12] (2 Pi)^(1/2)), 10, 100][[1]]

default(realprecision, 100);
exp(1/12-2*zeta'(-1))/(2*Pi)^(1/2)

import mpmath
mpmath.mp.pretty = True; mpmath.mp.dps = 100
mpmath.exp(1/12-2*mpmath.zeta(-1, 1, 1))/(2*pi)^(1/2)

Let A = A074962 denote the Glaisher-Kinkelin constant.
Equals 1/(A213080*(2*Pi)^(1/4)).
Equals A^2/(exp(1/12)*(2*Pi)^(1/2)).
Equals exp(1/12-2*zeta'(-1))/(2*Pi)^(1/2).

A366169 Positive integers k such that the second derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 16, 21, 25, 28, 29, 30, 31, 36, 37, 55, 57, 60, 61, 70, 121, 190

Offset: 1

Bernd C. Kellner, Oct 02 2023

The sequence is finite and is a supersequence of A094960. The terms are those numbers k where the denominator A366168(k) = 1. It remains to show that 190 is the last term. This is very likely, since the terms depend on the estimation of a product of primes satisfying certain p-adic conditions that is connected with A324370. A proven asymptotic formula related to that product implies that this sequence is finite. See Kellner 2017, 2023, and BLMS 2018.

			B(5,x) = x^5 - (5x^4)/2 + (5 x^3)/3 - x/6 and B''(5,x) = 20x^3 - 30x^2 + 10x, so 5 is a term.

Olivier Bordellès, Florian Luca, Pieter Moree, and Igor E. Shparlinski, Denominators of Bernoulli polynomials, Mathematika 64 (2018), 519-541.
Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
Bernd C. Kellner, On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients, J. Integer Seq. 27 (2024), Article 24.2.8, 11 pp.; arXiv:2310.01325 [math.NT], 2023.
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, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.
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.

Cf. A094960, A144845, A195441, A324370, A366168.

aList := len -> select(n -> denom(diff(diff(bernoulli(n, x), x), x)) = 1, [seq(1..len)]): aList(200);  # Peter Luschny, Oct 03 2023

(* k-th derivative of BP *)
k = 2; Select[Range[1000], Denominator[Together[D[BernoulliB[#, x],{x, k}]]] == 1&]
(* exact denominator formula *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
DBP[n_, k_] := Module[{m = n-k+1, fac = FactorialPower[n, k]}, If[n < 1 || k < 1 || n <= k, Return[1]]; Times@@Select[Prime[Range[PrimePi[(m+1)/(2 + Mod[m+1, 2])]]], !Divisible[fac, #] && SD[m, #] >= #&]];
k = 2; Select[Range[1000], DBP[#, k] == 1&]

isok(k) = #select(x->denominator(x)>1, Vec(deriv(deriv(bernpol(k))))) == 0; \\ Michel Marcus, Oct 03 2023

from itertools import count, islice
from sympy import Poly, diff, bernoulli
from sympy.abc import x
def A366169_gen(): # generator of terms
    return filter(lambda k:k<=2 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x,2)).coeffs()),count(1))
A366169_list = list(islice(A366169_gen(),20)) # Chai Wah Wu, Oct 03 2023

k is a term if A366168(k) = 1.

A366168 Denominator of the second derivative of the n-th Bernoulli polynomial B(n,x).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 3, 5, 5, 21, 1, 5, 15, 5, 1, 21, 7, 1, 1, 1, 1, 231, 7, 35, 3, 1, 1, 1365, 35, 7, 21, 55, 55, 105, 7, 7, 105, 35, 5, 663, 13, 11, 33, 55, 1, 57, 1, 5, 15, 1, 1, 15015, 715, 715, 33, 17, 85, 2415, 35, 1, 3, 55, 55, 285285, 19019, 1001

Offset: 1

Bernd C. Kellner, Oct 02 2023

The sequence consists only of odd numbers. The denominators are connected with A324370, from which an explicit formula follows as given below. See Kellner 2023.

			B(5,x) = x^5 - (5x^4)/2 + (5 x^3)/3 - x/6 and B''(5,x) = 20x^3 - 30x^2 + 10x, so a(5) = 1.
a(14) = A324370(13)/gcd(A324370(13), 14) = 210/gcd(210, 14) = 15.

Bernd C. Kellner, Table of n, a(n) for n = 1..10000
Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
Bernd C. Kellner, On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients, J. Integer Seq. 27 (2024), Article 24.2.8, 11 pp.; arXiv:2310.01325 [math.NT], 2023.
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, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.
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.

Cf. A094960, A144845, A195441, A324370, A366169.

(* k-th derivative of BP *)
k = 2; Table[Denominator[Together[D[BernoulliB[n, x], {x, k}]]], {n, 1, 100}]
(* exact denominator formula *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
DBP[n_, k_] := Module[{m = n-k+1, fac = FactorialPower[n, k]}, If[n < 1 || k < 1 || n <= k, Return[1]]; Times@@Select[Prime[Range[PrimePi[(m+1)/(2 + Mod[m+1, 2])]]], !Divisible[fac, #] && SD[m, #] >= #&]];
k = 2; Table[DBP[n, k], {n, 1, 100}]

from math import lcm
from sympy import Poly, diff, bernoulli
from sympy.abc import x
def A366168(n): return lcm(*(c.q for c in Poly(diff(bernoulli(n,x),x,2)).coeffs())) if n>=3 else 1 # Chai Wah Wu, Oct 04 2023

Let (n)_k be the falling factorial. Let s_p(n) be the sum of the p-adic digits of n.

a(1) = 1, and for n > 1, a(n) = A324370(n-1)/gcd(A324370(n-1), n) = Product_{prime p <= n/(2+(n mod 2)): gcd(p,(n)_2)=1, s_p(n-1) >= p} p.

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

A324973 Special polygonal numbers.

Original entry on oeis.org

6, 15, 66, 70, 91, 190, 231, 435, 561, 703, 715, 782, 861, 946, 1045, 1105, 1426, 1653, 1729, 1770, 1785, 1794, 1891, 2035, 2278, 2465, 2701, 2821, 2926, 3059, 3290, 3367, 3486, 3655, 4371, 4641, 4830, 5005, 5083, 5151, 5365, 5551, 5565, 5995, 6441, 6545, 6601

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Mar 21 2019

nonn

Squarefree polygonal numbers P(r,p) = (p^2*(r-2)-p*(r-4))/2 whose greatest prime factor is p >= 3, and whose rank (or order) is r >= 3 (see A324974).

The Carmichael numbers A002997 and primary Carmichael numbers A324316 are subsequences. See Kellner and Sondow 2019.

			P(3,5) = 15 is squarefree, and its greatest prime factor is 5, so 15 is a member.
More generally, if p is an odd prime and P(3,p) is squarefree, then P(3,p) is a member, since P(3,p) = (p^2+p)/2 = p*(p+1)/2, so p is its greatest prime factor.
CAUTION: P(6,7) = 91 = 7*13 is a member even though 7 is NOT its greatest prime factor, as P(6,7) = P(3,13) and 13 is its greatest prime factor.

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.

Subsequence of A324972 = intersection of A005117 and A090466.
A002997, A324316, A324319 and A324320 are subsequences.
Cf. A324974, A324975, A324976.

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[#]] &];
Take[Union[Table[Last[t], {t, T}]], 47]

is(k) = if(issquarefree(k) && k>1, my(p=vecmax(factor(k)[, 1]), r); p>2 && (r=2*(k/p-1)/(p-1)) && denominator(r)==1, 0); \\ Jinyuan Wang, Feb 18 2021

Several missing terms inserted by Jinyuan Wang, Feb 18 2021

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.

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

Original entry on oeis.org

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

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.

A324460 Numbers m > 1 that have a strict s-decomposition.

Original entry on oeis.org

45, 96, 225, 325, 405, 576, 637, 640, 891, 1225, 1377, 1408, 1536, 1701, 1729, 2025, 2541, 2821, 3321, 3751, 3825, 4225, 4608, 4961, 6400, 6517, 6525, 7381, 7840, 8125, 8281, 9216, 9537, 9801, 10625, 10935, 12025, 12288, 12825, 12936, 13125, 13312, 13357

Offset: 1

Bernd C. Kellner, Feb 28 2019

The sequence contains the primary Carmichael numbers A324316.
The sequence is infinite. If f(x) counts such numbers m below x, then f(x) > 1/11 x^(1/3) - 1/3 for x >= 1.
A number m > 1 has a strict s-decomposition if there exists a decomposition in n proper factors g_k with exponents e_k >= 1 (the factors g_k being strictly increasing but not necessarily coprime) such that
    m = g_1^e_1 * ... * g_n^e_n, where s_{g_k}(m) = g_k for all k,
  and s_g(m) gives the sum of the base-g digits of m.
A term m has the following properties:
m must have at least 2 factors g_k. If m = g_1^e_1 * g_2^e_2 with exactly two factors, then e_1 + e_2 >= 3.
Each factor g_k of m satisfies the inequalities 1 < g_k < m^(1/(ord_{g_k}(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.
See Kellner 2019.

			Since 576 = 2^4 * 6^2 with s_2(576) = 2 and s_6(576) = 6, 576 is a member.

Bernd C. Kellner, Table of n, a(n) for n = 1..254
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.

Subsequences are A324316, A324458. Subsequence of A324459.

Cf. A324455, A324456, A324457.

s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
HasDecompS[m_] := Module[{E0, EV, G, R, k, n, v},
If[m < 1 || !CompositeQ[m], Return[False]];
G = Select[Divisors[m], s[m, #] == # &];
n = Length[G]; If[n < 2, Return[False]];
E0 = Array[0 &, n]; EV = Array[v, n];
R = Solve[Product[G[[k]]^EV[[k]], {k, 1, n}] == m && EV >= E0, EV, Integers]; Return[R != {}]];
Select[Range[10^4], HasDecompS[#] &]

cp's OEIS Frontend

User: Bernd C. Kellner

A377024 Decimal expansion of the constant F(2) related to asymptotic products of factorials.

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A377023 Decimal expansion of the asymptotic constant of the product of binomial coefficients in a row of Pascal's triangle.

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A366169 Positive integers k such that the second derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.

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A366168 Denominator of the second derivative of the n-th Bernoulli polynomial B(n,x).

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A324976 Rank of the n-th primary Carmichael number.

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A324973 Special polygonal numbers.

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