A064538 -id:A064538 - cp's OEIS frontend

A248716 Numerator of (1/e)Sum_{k>=0} (1/k!)(Sum_{j=0..k} j^n).

2, 3, 17, 27, 293, 791, 10583, 25685, 448303, 251411, 4503535, 6331107, 4436875097, 3335427631, 19619696071, 75379875277, 3019260651391, 16773385986619, 3047463007411973, 2732480436961811, 398377271835431771, 173581842021095897, 1716900426430701553, 35001773773285490879, 6684326532123939298051

Offset: 0

Richard R. Forberg, Dec 28 2014

Denominators are given by A130190, which in that entry are associated with the denominators of the z-sequence for a certain Sheffer matrix (triangle), but also apply here. See also the formula given there for the denominator of a certain sum involving the Stirling2 numbers.
Note that a(0) = 2 uses 0^0 := 1. For n >= 1 use triangle A079618 and the formula (1/e)*Sum_{k>=0} ((k+1)^n)/k! = Bell(n+1) = A000110(n+1). - Wolfdieter Lang, Feb 03 2015
If instead of Sum_{j=0..k} j^n one uses the sum with falling factorials, namely F(k, n) := Sum_{j=0..k} A008279(j, n) = A008279(k+1, n+1)/(n+1) the result for the rationals R(n) = (1/e)*Sum_{k>=0} (1/k!)*F(k, n) becomes very simple, namely R(n) = (n+2)/(n+1), n >= 0. - Wolfdieter Lang, Feb 03 2015

			Terms up to n = 10, with denominators, are 2/1, 3/2, 17/6, 27/4, 293/15, 791/12, 10583/42, 25685/24, 448303/90, 251411/10, 4503535/33, ... .
From _Wolfdieter Lang_, Feb 03 2015: (Start)
With triangle A079618, A064538 and the Bell numbers A000110 the rationals r(n) are:
n=4: (1/30)*(-1*1 + 0*2 + 10*5 + 15*15 + 6*52) = 293/15.
n=9: (1/20)*(0*1 + (-3)*2 +  0*5 + 10*15 + 0*52 + (-14)*203 + 0*877 + 15*4140 + 10*21147 + 2*115975) = 251411/10.
(End)

G. C. Greubel, Table of n, a(n) for n = 0..251

Cf. A000110, A064538, A079618, A130190.

Numerator[Table[(1/Exp[1])*Sum[Sum[j^n/k!, {j, 0, k}], {k, 0, Infinity}],
{n, 0, 100}]]

a(n) = numerator(r(n)) with the rationals r(n) = (1/e)*Sum_{k>=0} (1/k!)*(Sum_{j=0..k} j^n), n >= 0, where 0^0 := 1.

a(n) = numerator(r(n)), with r(n) = (1/A064538(n))*Sum_{k=0..n} T(n+1,k+1)*Bell(k+1), with T(n,k) = A079618(n,k) and Bell(n) = A000110(n), for n >= 1. a(0) = 2 using 0^0 := 1. See comments above. - Wolfdieter Lang, Feb 03 2015

Edited. Comment and formula rewritten. Cross references added. - Wolfdieter Lang, Feb 03 2015

A342321 T(n, k) = A343277(n)[x^k] p(n, x) where p(n, x) = (1/(n+1))Sum_{k=0..n} (-1)^kE1(n, k)x^(n - k) / binomial(n, k), and E1(n, k) are the Eulerian numbers A123125. Triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -1, 2, 0, 1, -4, 3, 0, -3, 22, -33, 12, 0, 1, -13, 33, -26, 5, 0, -5, 114, -453, 604, -285, 30, 0, 5, -200, 1191, -2416, 1985, -600, 35, 0, -35, 2470, -21465, 62476, -78095, 42930, -8645, 280, 0, 14, -1757, 21912, -88234, 156190, -132351, 51128, -7028, 126

Offset: 0

Peter Luschny, Mar 09 2021

Conjecture: For even n >= 6 p(n, x)/x and for odd n >= 3 p(n, x)/(x^2 - x) is irreducible.

			Triangle starts:
[n]                T(n, k)                      A343277(n)
----------------------------------------------------------
[0] 1;                                                 [1]
[1] 0,  1;                                             [2]
[2] 0, -1,     2;                                      [6]
[3] 0,  1,    -4,     3;                              [12]
[4] 0, -3,    22,   -33,    12;                       [60]
[5] 0,  1,   -13,    33,   -26,     5;                [30]
[6] 0, -5,   114,  -453,   604,  -285,    30;        [210]
[7] 0,  5,  -200,  1191, -2416,  1985,  -600,  35;   [280]
.
The coefficients of the polynomials p(n, x) = (Sum_{k = 0..n} T(n, k) x^k) / A343277(n) for the first few n:
[0] 1;
[1] 0,   1/2;
[2] 0,  -1/6,    1/3;
[3] 0,  1/12,   -1/3,    1/4;
[4] 0, -1/20,   11/30, -11/20,    1/5;
[5] 0,  1/30,  -13/30,  11/10,  -13/15,  1/6.

Peter Luschny, Illustration of the polynomials.
Peter Luschny, Eulberian polynomials, A notebook companion to A342321 and A356601, 2022.

Cf. A343277, A123125.
Sequences of rational polynomials p(n, x) with p(n, 1) = Bernoulli(n, 1):
A342312/A342313, A342321/A342320, A342322/A064538.

CoeffList := p -> op(PolynomialTools:-CoefficientList(p,x)):
E1 := (n, k) -> combinat:-eulerian1(n, k):
poly := n -> (1/(n+1))*add((-1)^k*E1(n,k)*x^(n-k)/binomial(n,k), k=0..n):
Trow := n -> denom(poly(n))*CoeffList(poly(n)): seq(Trow(n), n = 0..9);

Poly342321[n_, x_] := If[n == 0, 1, Sum[x^k*k!*Sum[(-1)^(n - j)*StirlingS2[n, j] /((k - j)!(n - j + 1) Binomial[n + 1, j]), {j, 0, k}], {k, 1, n}]];
Table[A343277[n] CoefficientList[Poly342321[n, x], x][[k+1]], {n, 0, 9}, {k, 0, n}] // Flatten

An alternative representation of the sequence of rational polynomials is:
p(n, x) = Sum_{k=1..n} x^k*k!*Sum_{j=0..k} (-1)^(n-j)*Stirling2(n, j)/((k - j)!(n - j + 1)*binomial(n + 1, j)) for n >= 1 and p(0, x) = 1.
(Sum_{k = 0..n} T(n, k)) / A343277(n) = Bernoulli(n, 1).

A299145 Primes of the form j^k + (j-1)^k + ... + 2^k, for j > 1 and k > 0.

Original entry on oeis.org

2, 5, 13, 29, 97, 139, 353, 4889, 72353, 353815699, 42065402653, 84998999651, 102769130749, 15622297824266188673, 28101527071305611527, 20896779938941631284493075599148668795944697935466419104293, 105312291668560568089831550410013687058921146068446092937783402353

Offset: 1

Gionata Neri, Feb 03 2018

nonn

Except for the terms 2, 5, 13, 29, 139, the exponent k satisfies k >= 4. More generally, if Q(j) = j^k + (j-1)^k + ... + 2^k is a term, then j-1 is a divisor of A064538(k). This is because (j-1) is a factor of Q(j) and thus Q(j) is prime only if j-1 is a divisor of the denominator of Q(j), i.e. A064538(k). Thus for each k there is only a finite number of values of j to check. This provides an efficient algorithm to find terms of this sequence by looking only for primes in the numbers H_{j,-k} - 1 = j^k + (j-1)^k + ... + 2^k for j-1 a divisor of A064538(k). - Chai Wah Wu, Mar 06 2018

			2 = 2^1;
5 = 3^1 + 2^1;
13 = 3^2 + 2^2;
29 = 4^2 + 3^2 + 2^2;
97 = 3^4 + 2^4;
139 = 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2;
353 = 4^4 + 3^4 + 2^4;
4889 = 4^6 + 3^6 + 2^6;
72353 = 4^8 + 3^8 + 2^8;

Chai Wah Wu, Table of n, a(n) for n = 1..45 (all terms < 10^1000).

Cf. A007689, A074526, A064538.

With[{nn = 350}, Sort@ Flatten@ Map[Select[#, PrimeQ] &, Table[Total[Range[j, 1, -1]^k] - 1, {j, 2, nn}, {k, nn - j}]]] (* Michael De Vlieger, Feb 03 2018 *)

limit=100000; v=vector(limit); for(n=1, ceil((-1+(1+8*limit)^(1/2))/2), for(k=1, logint(limit, n+0^(n-1)), a=sum(i=1,n,i^k)-1;if(isprime(a)&&a

a(10)-a(15) from Michael De Vlieger, Feb 03 2018

a(16)-a(17) from Chai Wah Wu, Mar 07 2018

A202533 For a polynomial P(m) with rational coefficients, denote by lcmd(P) the LCM of the denominators of all its coefficients. Then a(n) = lcmd(Sum_{i=1..m} (i^n*Sum_{j=1..i} j^n))/ lcmd((Sum_{i=1..m} i^n)^2).

Original entry on oeis.org

2, 6, 10, 42, 2, 22, 130, 10, 34, 798, 70, 230, 2, 6, 58, 4774, 154, 14, 962, 26, 82, 602, 42, 658, 34, 374, 5830, 6270, 38, 118, 7930, 390, 26, 3082, 46, 7810, 1606, 22, 22, 158, 306, 2822, 12818, 754, 2314, 168454, 12958, 418, 2134, 66, 1010, 7210, 14, 214, 104858

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Dec 20 2011

nonn

Note also that Sum_{i=1..m} i^n is a polynomial in m of degree n+1.
One can prove that all terms of the sequence are integers.
The sequence appears to possess an astonishing property: every odd prime p is the maximal prime divisor of a((p-1)/2).

			Let n=1. Since lcmd(Sum_{i=1..m} i) = 2, lcmd(Sum_{i=1..m} i^2) = 6, lcmd(Sum_{i=1..m} i^3) = 4, then lcmd(Sum_{i=1..m} i*Sum_{j=1..i} j) = lcmd(Sum_{i=1..m} i^2*(i+1)/2) = 24, therefore, a(1) = 24/4 = 6.
Let p=53. Then a(26) = 5830 = 2*5*11*53 has maximal prime divisor 53.

Cf. A064538.

LCMD[P_, m_] := LCM @@ Denominator[CoefficientList[P // FunctionExpand, m] ]; a[n_] := LCMD[Sum[i^n*HarmonicNumber[i, -n], {i, 1, m}], m]/ LCMD[ HarmonicNumber[m, -n]^2, m]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 60}] (* Jean-François Alcover, Feb 18 2016 *)

sp(p) = x * Polrev(vector(p+1, k, (-1)^(k==p)*binomial(p+1, k)*bernfrac(p+1-k))/(p+1));
lcmd(pol) = lcm(apply(x->denominator(x), Vec(pol)));
a(n) = {pol = x^n*sp(n); pnum = sum(i=0, poldegree(pol), polcoeff(pol, i)*sp(i)); lcmd(pnum)/lcmd(sp(n)^2);} \\ Michel Marcus, Feb 17 2016

More terms from Michel Marcus, Feb 17 2016

cp's OEIS Frontend

A248716 Numerator of (1/e)*Sum_{k>=0} (1/k!)*(Sum_{j=0..k} j^n).

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A342321 T(n, k) = A343277(n)*[x^k] p(n, x) where p(n, x) = (1/(n+1))*Sum_{k=0..n} (-1)^k*E1(n, k)*x^(n - k) / binomial(n, k), and E1(n, k) are the Eulerian numbers A123125. Triangle read by rows, for 0 <= k <= n.

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A299145 Primes of the form j^k + (j-1)^k + ... + 2^k, for j > 1 and k > 0.

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A202533 For a polynomial P(m) with rational coefficients, denote by lcmd(P) the LCM of the denominators of all its coefficients. Then a(n) = lcmd(Sum_{i=1..m} (i^n*Sum_{j=1..i} j^n))/ lcmd((Sum_{i=1..m} i^n)^2).

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A248716 Numerator of (1/e)Sum_{k>=0} (1/k!)(Sum_{j=0..k} j^n).

A342321 T(n, k) = A343277(n)[x^k] p(n, x) where p(n, x) = (1/(n+1))Sum_{k=0..n} (-1)^kE1(n, k)x^(n - k) / binomial(n, k), and E1(n, k) are the Eulerian numbers A123125. Triangle read by rows, for 0 <= k <= n.