A324319 -id:A324319 - cp's OEIS frontend

A324370 Product of all primes p not dividing n such that the sum of the base-p digits of n is at least p, or 1 if no such prime exists.

1, 1, 2, 1, 6, 1, 6, 3, 10, 1, 6, 1, 210, 15, 2, 3, 30, 5, 210, 21, 110, 15, 30, 5, 546, 21, 14, 1, 30, 1, 462, 231, 1190, 105, 6, 1, 51870, 1365, 70, 21, 2310, 55, 2310, 105, 322, 105, 210, 35, 6630, 663, 286, 33, 330, 55, 798, 57, 290, 15, 30, 1, 930930, 15015, 1430, 2145, 1122, 85, 82110, 2415, 70, 3, 330, 55, 21111090, 285285

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 24 2019

The product is finite, as the sum of the base-p digits of n is n if p > n.
a(198) = 2465 is the only term below 10^6 that is a Carmichael number (A002997).
It appears that a(n)=1 if and only if n is in A094960. - Robert Israel, Mar 30 2020
It turns out that a(n) equals the denominator of the first derivative of the Bernoulli polynomial B(n,x). So a(n)=1 if and only if n is in A094960, also impyling that n+1 is prime. A324370 is also involved in such formulas regarding higher derivatives. See Kellner 2023. - Bernd C. Kellner, Oct 12 2023

			For p = 2, 3, and 5, the sum of the base p digits of 7 is 1+1+1 = 3 >= 2, 2+1 = 3 >= 3, and 1+2 = 3 < 5, respectively, so a(7) = 2*3 = 6.

Robert Israel, Table of n, a(n) for n = 1..5000
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, 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. A002997, A094960, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324371, A324404, A324405, A366168, A366169, A366186, A366187, A366188.

N:= 100: # for a(1)..a(N)
V:= Vector(N,1):
p:= 1:
for iter from 1 do
   p:= nextprime(p);
   if p >= N then break fi;
   for n from p+1 to N do
     if n mod p <> 0 and convert(convert(n,base,p),`+`)>= p then
       V[n]:= V[n]*p
     fi
od od:
convert(V,list); # Robert Israel, Mar 30 2020
# Alternatively, note that this formula is suggesting offset 0 and a(0) = 1:
seq(denom(diff(bernoulli(n, x), x)), n = 1..51); # Peter Luschny, Oct 13 2023

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
DD2[n_] := Times @@ Select[Prime[Range[PrimePi[(n+1)/(2+Mod[n+1, 2])]]], !Divisible[n, #] && SD[n, #] >= # &];
Table[DD2[n], {n, 1, 100}]
(* From Bernd C. Kellner, Oct 12 2023 (Start) *)
(* Denominator of first derivative of BP *)
k = 1; Table[Denominator[Together[D[BernoulliB[n, x], {x, k}]]], {n, 1, 100}]
(* End *)

from math import prod
from sympy.ntheory import digits
from sympy import primefactors, primerange
def a(n):
    nonpf = set(primerange(1, n+1)) - set(primefactors(n))
    return prod(p for p in nonpf if sum(digits(n, p)[1:]) >= p)
print([a(n) for n in range(1, 75)]) # Michael S. Branicky, Jul 03 2022

a(n) * A324369(n) = A195441(n-1) = denominator(Bernoulli_n(x) - Bernoulli_n).
a(n) * A324369(n) * A324371(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)).
a(n+1) = A195441(n)/A324369(n+1) = A144845(n)/A007947(n+1) = A318256(n). Essentially the same as A318256. - Peter Luschny, Mar 05 2019
From Bernd C. Kellner, Oct 12 2023: (Start)
a(n) = denominator(Bernoulli_n(x)').
k-th derivative: let (n)_m be the falling factorial.
For n > k, a(n-k+1)/gcd(a(n-k+1), (n)_{k-1}) = denominator(Bernoulli_n(x)^(k)). Otherwise, the denominator equals 1. (End)

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).

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.

Original entry on oeis.org

231, 561, 1001, 1045, 1105, 1122, 1155, 1729, 2002, 2093, 2145, 2465, 2821, 3003, 3315, 3458, 3553, 3570, 3655, 3927, 4186, 4199, 4522, 4774, 4845, 4862, 5005, 5187, 5565, 5642, 5681, 6006, 6118, 6270, 6279, 6545, 6601, 6670, 6734, 7337, 7395, 7735, 8177, 8211, 8265, 8294, 8323, 8463, 8645, 8789, 8855, 8911, 9282, 9361, 9435, 9690, 9867

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 21 2019

The sequence is infinite, because it contains all Carmichael numbers (A002997).
If m is a term and p is a prime factor of m, then p <= a*sqrt(m) with a = sqrt(11/21) = 0.7237..., where the bound is sharp.
A term m must have at least 3 prime factors if m is odd, and must have at least 4 prime factors if m is even.
m is a term if and only if m > 1 divides denominator(Bernoulli_m(x) - Bernoulli_m) = A195441(m-1).
A term m is a Carmichael number iff s_p(m) == 1 (mod p-1) whenever prime p divides m, where s_p(m) is the sum of the base p digits of m.
See Kellner and Sondow 2019.

			231 = 3 * 7 * 11 is squarefree, and 231 in base 3 is 22120_3 = 2 * 3^4 + 2 * 3^3 + 1 * 3^2 + 2 * 3 + 0 with 2+2+1+2+0 = 7 >= 3, and 231 = 450_7 with 4+5+0 = 9 >= 7, and 231 = 1a0_11 with 1+a+0 = 1+10+0 = 11 >= 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. A002997, A005117, A195441, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
Select[Range[10^4], TestS[#] &]

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**4) if ok(k)]) # Michael S. Branicky, Jul 03 2022

a_1 + a_2 + ... + a_k >= p for 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).

A324369 Product of all primes p dividing n such that the sum of the base p digits of n is at least p, or 1 if no such prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 15, 2, 1, 6, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 3, 1, 5, 6, 1, 2, 3, 10, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 1, 2, 1, 2, 5, 2, 3, 2, 1, 10, 7, 2, 3, 2, 5, 6, 1

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 24 2019

a(n) = n iff n divides denominator(Bernoulli_n(x) - Bernoulli_n) (see A195441).
a(n) = n iff n = 1 or n is in A324315.
a(n) = n if n is a Carmichael number (A002997).
See the section on Bernoulli polynomials in Kellner and Sondow 2019.

			6 = 2 * 3, and 6 = 110_2 in base 2 with 1+1+0 >= 2, but 6 = 20_3 in base 3 with 2+0 = 2 < 3, so a(6) = 2.

Amiram Eldar, 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 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, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A007947, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324370, A324371, A324404, A324405.

g:= proc(n,p) convert(convert(n,base,p),`+`) >= p end proc:
f:= proc(n) local p;
      convert(select(p -> g(n,p), numtheory:-factorset(n)),`*`)
end proc:
map(f, [$1..100]); # Robert Israel, Feb 28 2019

SD[n_, p_] := If[n < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
DD1[n_] := Times @@ Select[LP[n], SD[n, #] >= # &];
Table[DD1[n], {n, 1, 100}]

from math import prod
from sympy.ntheory import digits
from sympy import primefactors as pf
def a(n): return prod(p for p in pf(n) if sum(digits(n, p)[1:]) >= p)
print([a(n) for n in range(1, 98)]) # Michael S. Branicky, Jul 03 2022

a(n) * A324371(n) = A007947(n) = radical(n).
a(n) * A324370(n) = A195441(n-1) = denominator(Bernoulli_n(x) - Bernoulli_n).
a(n) * A324370(n) * A324371(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)).

A324371 Product of all primes p dividing n such that the sum of the base p digits of n is less than p, or 1 if no such prime.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 1, 5, 13, 3, 7, 29, 15, 31, 2, 11, 17, 35, 3, 37, 19, 13, 5, 41, 7, 43, 11, 1, 23, 47, 1, 7, 5, 17, 13, 53, 3, 55, 7, 19, 29, 59, 5, 61, 31, 7, 2, 13, 11, 67, 17, 23, 7, 71, 1, 73, 37, 5, 19, 77, 13, 79, 5, 3, 41, 83, 21

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 25 2019

Does not contain any elements of A324315, and thus none of the Carmichael numbers A002997.

See the section on Bernoulli polynomials in Kellner and Sondow 2019.

			For p = 2 and 3, the sum of the base p digits of 6 is 1+1+0 = 2 >= 2 and 2+0 = 2 < 3, respectively, so a(6) = 3.

Robert Israel, 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 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, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A002997, A007947, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324404, A324405.

f:= n -> convert(select(p -> convert(convert(n,base,p),`+`)Robert Israel, Apr 26 2020

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
DD3[n_] := Times @@ Select[LP[n], SD[n, #] < # &];
Table[DD3[n], {n, 1, 100}]

from math import prod
from sympy.ntheory import digits
from sympy import primefactors as pf
def a(n): return prod(p for p in pf(n) if sum(digits(n, p)[1:]) < p)
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jul 03 2022

a(n) * A324369(n) = A007947(n) = radical(n).

a(n) * A195441(n) = a(n) * A324369(n) * A324370(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)).

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

A324404 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 2 (mod p-1), where s_p(m) is the sum of the base p digits of m.

Original entry on oeis.org

1122, 3458, 5642, 6734, 11102, 13202, 17390, 17822, 21170, 22610, 27962, 31682, 46002, 58682, 61778, 79730, 82082, 93314, 105266, 106262, 125490, 127946, 136202, 150722, 153254, 177122, 182002, 202202, 203870, 214370, 231842, 252434, 274298, 278462, 305102, 315282

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 26 2019

For d >= 1 define S_d = (terms m in A324315 such that s_p(m) == d (mod p-1) if prime p divides m). Then S_1 is precisely the Carmichael numbers (A002997), S_2 is A324404, S_3 is A324405, and the union of all S_d for d >= 1 is A324315.
Subsequence of the 2-Knödel numbers (A050990). Generally, for d > 1 the terms of S_d that are greater than d form a subsequence of the d-Knödel numbers.
See Kellner and Sondow 2019.

			1122 = 2*3*11*17 is squarefree and equals 10001100010_2, 1112120_3, 930_11, and 3f0_17 in base p = 2, 3, 11, and 17. Then s_2(1122) = 1+1+1+1 = 4 >= 2, s_3(1122) = 1+1+1+2+1+2 = 8 >= 3, s_11(1122) = 9+3 = 12 >= 11, and s_17(1122) = 3+f = 3+15 = 18 >= 17. Also, s_2(1122) = 4 == 2 (mod 1), s_3(1122) = 8 == 2 (mod 2), s_11(1122) = 12 == 2 (mod 10), and s_17(1122) = 18 == 2 (mod 16), so 1122 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..2200
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 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, Integers 21 (2021), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019-2021.

Cf. A002997, A050990, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324405.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestSd[n_, d_] := (n > 1) && (d > 0) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # && Mod[SD[n, #] - d, # - 1] == 0 &];
Select[Range[200000], TestSd[#, 2] &]

More terms from Amiram Eldar, Dec 05 2020

A324405 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 3 (mod p-1), where s_p(m) is the sum of the base p digits of m.

Original entry on oeis.org

3003, 3315, 5187, 7395, 8463, 14763, 19803, 26733, 31755, 47523, 50963, 58035, 62403, 88023, 105339, 106113, 123123, 139971, 152643, 157899, 166611, 178923, 183183, 191919

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 26 2019

For d >= 1 define S_d = (terms m in A324315 such that s_p(m) == d (mod p-1) if prime p divides m). Then S_1 is precisely the Carmichael numbers (A002997), S_2 is A324404, S_3 is A324405, and the union of all S_d for d >= 1 is A324315.
Subsequence of the 3-Knödel numbers (A033553). Generally, for d > 1 the terms of S_d that are greater than d form a subsequence of the d-Knödel numbers.
See Kellner and Sondow 2019.

			3003 = 3*7*11*13 is squarefree and equals 11010020_3, 11520_7, 2290_11, and 14a0_13 in base p = 3, 7, 11, and 13. Then s_3(3003) = 1+1+1+2 = 5 >= 3, s_7(3003) = 1+1+5+2 = 9 >= 7, s_11(3003) = 2+2+9 = 13 >= 11, and s_13(3003) = 1+4+a = 1+4+10 = 15 >= 13. Also, s_3(3003) = 5 == 3 (mod 2), s_7(3003) = 9 == 3 (mod 6), s_11(3003) = 13 == 3 (mod 10), and s_13(3003) = 15 == 3 (mod 12), so 3003 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..2000
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, Integers 21 (2021), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019-2021.

Cf. A002997, A033553, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestSd[n_, d_] := (n > 1) && (d > 0) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # && Mod[SD[n, #] - d, # - 1] == 0 &];
Select[Range[200000], TestSd[#, 3] &]

A324317 Number of primary Carmichael numbers (A324316) less than 10^n.

Original entry on oeis.org

0, 0, 0, 2, 4, 9, 19, 51, 107, 219, 417, 757, 1470, 2666, 5040, 9280, 17210, 32039, 59762, 111811, 210627, 397968

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 22 2019

The number of Carmichael numbers (A002997) less than 10^n is 0, 0, 1, 7, 16, 43, 105, 255, 646, 1547, 3605, 8241, 19279, 44706, 105212, 246683, 585355, 1401644, ... (see A055553).
The terms up to a(10) are given in Table 1 of Kellner and Sondow 2019. The terms up to a(18) and related results are given in Table 1.5 of Kellner 2019.
All computations depend on Pinch's database.

			There are two primary Carmichael numbers less than 10^4, namely, 1729 and 2821, so a(4) = 2.

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv preprint, 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, 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.
R. G. E. Pinch, Tables relating to Carmichael numbers (The Carmichael numbers up to 10^18, 2008).
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A055553, A324315, A324316, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405.

a(11)-a(18) from Amiram Eldar, Mar 01 2019
a(19) from Amiram Eldar, Dec 05 2020
a(20)-a(22) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 22 2024

A324318 Number of terms in 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) less than 10^n.

Original entry on oeis.org

0, 0, 2, 57, 636, 7048, 75150, 801931, 8350039, 86361487

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 23 2019

The number of squarefree integers less than 10^n is 0, 6, 61, 608, 6083, 60794, 607926, 6079291, 60792694, 607927124, ... (see A053462).

			There are two terms of A324315 less than 10^3, namely, 231 and 561, so a(3) = 2.

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. A053462, A324315, A324316, A324317, A324319, A324320, A324369, A324370, A324371, A324404, A324405.

cp's OEIS Frontend

A324370 Product of all primes p not dividing n such that the sum of the base-p digits of n is at least p, or 1 if no such prime exists.

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A324316 Primary Carmichael numbers.

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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.

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A324369 Product of all primes p dividing n such that the sum of the base p digits of n is at least p, or 1 if no such prime.

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A324371 Product of all primes p dividing n such that the sum of the base p digits of n is less than p, or 1 if no such prime.

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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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A324404 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 2 (mod p-1), where s_p(m) is the sum of the base p digits of m.

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