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
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
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.
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
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.
A341109
a(n) = denominator(p(n, x)) / (n!*denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.
Original entry on oeis.org
1, 1, 2, 4, 8, 16, 96, 192, 1152, 768, 1536, 3072, 18432, 36864, 221184, 147456, 884736, 1769472, 10616832, 21233664, 637009920, 424673280, 2548039680, 5096079360, 152882380800, 61152952320, 366917713920, 81537269760, 163074539520, 326149079040, 1956894474240
Offset: 0
- András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
- J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences In: J. W. Brewer, S. Glaz, W. J. Heinzer, B. M. Olberding (eds), Multiplicative Ideal Theory in Commutative Algebra. Springer, Boston, MA., 2006.
- Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, arXiv:1705.03857 [math.NT] 2017, Amer. Math. Monthly.
Cf.
A100655,
A053657 (Minkowski),
A341107,
A341108,
A318256,
A144845,
A163176,
A201637 (Eulerian2),
A036689,
A324370,
A007947,
A324369,
A195441.
-
Epoly := proc(n, x) add(combinat:-eulerian2(n, k)*binomial(x+k, 2*n), k = 0..n) / mul(j-x, j = 1..n): simplify(expand(%)) end:
seq(denom(Epoly(n, x)) / (n!*denom(bernoulli(n, x))), n = 0..30);
-
A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k,0,n}],{p, Prime[Range[n]]}];
A144845[n_] := Denominator[Together[BernoulliB[n, x]]];
A163176[n_] := A053657[n] / n!;
Table[(n + 1) A163176[n + 1] / A144845[n], {n, 0, 30}]
-
def A341109(n): # uses[A341108, A318256]
return A341108(n)//A318256(n)
print([A341109(n) for n in (0..30)])
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