A003101 -id:A003101 - cp's OEIS frontend

A113320 a(1)=1 and a(n) for n>1 has the smallest positive value such that Sum_{i=1..n} a(i)^a(n-i+1) is prime.

1, 1, 1, 2, 2, 4, 4, 4, 6, 2, 6, 4, 18, 6, 4, 20, 6, 30, 4, 40, 30, 8, 18, 16, 40, 128, 24, 40, 58, 194, 78, 84, 56, 56, 72, 112, 98, 300, 444, 54, 978, 1938, 120, 126, 6, 1750

Offset: 1

Jonathan Vos Post, Jan 07 2006

Previous name was: Least integers so ascending descending base exponent transforms all prime.

This is the first sequence submitted as a solution to an "ascending descending base exponent transform inverse problem" where the sequence is iteratively defined such that the transform meets a constraint. The sequence is infinite, but it is hard to characterize the asymptotic cost of adding an n-th term. A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154.

			a(1) = 1 by definition.
a(2) = 1 because 1 is the min such that 1^a(2) + a(2)^1 is prime (p=2).
a(3) = 1 because 1 is the min such that 1^a(3) + 1^1 + a(3)^1 is prime (p=5).
a(4) = 2 because 2 is the min such that 1^a(4) + 1^1 + 3^1 + a(4)^1 is prime (p=7).

Cf. A000040, A005408, A113122, A113153, A113154.

inve[w_] := Total[w^Reverse[w]]; a[1] = 1; a[n_] := a[n] = Block[{k = 0}, While[! PrimeQ[ inve@ Append[Array[a, n-1], ++k]]]; k]; Array[a, 46] (* Giovanni Resta, Jun 13 2016 *)

lista(n)={my(a=vector(n)); a[1]=1; print1(1, ", "); for(n=2, #a, my(t=sum(i=2, n-1, a[i]^a[n-i+1])); my(k=1); while(!ispseudoprime(t+1+k), k++); a[n]=k; print1(k, ", "))} \\ Andrew Howroyd, Jan 03 2020

a(1) = 1. For n>1, a(n) = min {k>0: a(1)^k + k^a(1) + Sum_{i=2..n-1} a(i)^a(n-i+1) is prime}.

Corrected and extended by Giovanni Resta, Jun 13 2016

New name from Giovanni Resta, Jan 03 2020

A113336 Least integers, starting with 2, so ascending descending base exponent transforms all prime.

Original entry on oeis.org

2, 1, 6, 6, 18, 12, 18, 42, 288, 108, 180, 1122, 1458, 660

Offset: 1

Jonathan Vos Post, Jan 07 2006

This is the second sequence submitted as a solution to an "ascending descending base exponent transform inverse problem" where the sequence is iteratively defined such that the transform meets a constraint. The sequence is probably infinite, but it is hard to characterize the asymptotic cost of adding an n-th term (the 9th terms is at least 250). A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154.

			a(1) = 2 by definition.
a(2) = 1 because 1 is the min such that 2^a(2) + a(2)^2 is prime (p=3).
a(3) = 6 because 6 is the min such that 2^a(3) + 1^1 + a(3)^2 is prime (2^6 + 1^1 + 6^1 = 101).
a(4) = 6 because 2^6 + 1^6 + 6^1 + 6^2 = 107 is prime.
a(5) = 18 because 2^18 + 1^6 + 6^6 + 6^1 + 18^2 = 309131 is prime.
a(6) = 12 because 2^12 + 1^18 + 6^6 + 6^6 + 18^1 + 12^2 = 97571 is prime.
a(7) = 18 because 2^18 + 1^12 + 6^18 + 6^6 + 18^6 + 12^1 + 18^2 = 101559990989777 is prime.
a(8) = 42 because 2^42 + 1^18 + 6^12 + 6^18 + 18^6 + 12^6 + 18^1 + 42^2 = 105960216961847 is prime.
a(9) > 250.

Cf. A000040, A005408, A113122, A113153, A113154.

a(1) = 2. For n>1: a(n) = min {n>0: Sum_{i=1..n} a(i)^a(n-i+1) is prime}.

a(9)-a(14) from Giovanni Resta, Jun 13 2016

A234568 Sum_{k=0..n} (n-k)^(2*k).

Original entry on oeis.org

1, 1, 2, 6, 27, 163, 1268, 12344, 145653, 2036149, 33192790, 622384730, 13263528351, 318121600695, 8517247764136, 252725694989612, 8258153081400857, 295515712276222953, 11523986940937975402, 487562536078882116718, 22291094729329088403299, 1097336766599161926448779

Offset: 0

Paul D. Hanna, Dec 28 2013

nonn

			O.g.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 27*x^4 + 163*x^5 + 1268*x^6 +...
O.g.f.: A(x) = 1 + x/(1-x) + x^2/(1-4*x) + x^3/(1-9*x) + x^4/(1-16*x) +...
E.g.f.: E(x) = 1 + x + 2*x^2/2! + 6*x^3/3! + 27*x^4/4! + 163*x^5/5! +...
where the e.g.f. is a series involving iterated integration:
E(x) = 1 + Integral exp(x) dx + Integral^2 exp(4*x) dx^2 + Integral^3 exp(9*x) dx^3 + Integral^4 exp(16*x) dx^4 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A003101, A026898, A349880, A349881.

Flatten[{1,Table[Sum[(n-k)^(2*k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 23 2014 *)

a(n)=sum(k=0, n, (n-k)^(2*k))
for(n=0, 20, print1(a(n), ", "))

/* From o.g.f. Sum_{n>=0} x^n/(1-n^2*x): */
{a(n)=polcoeff(sum(m=0, n, x^m/(1-m^2*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

/* From e.g.f. involving iterated integration: */
INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G
a(n)=my(A=1+x);A=1+sum(k=1,n,INTEGRATE(k,exp(k^2*x+x*O(x^n))));n!*polcoeff(A,n)
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Dec 28 2013

O.g.f.: Sum_{n>=0} x^n / (1 - n^2*x).
E.g.f.: Sum_{n>=0} Integral^n exp(n^2*x) dx^n, where integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration.
a(n) ~ sqrt(Pi) * (n/LambertW(exp(1)*n))^(1/2 + 2*n - 2*n/LambertW(exp(1)*n)) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

A105795 Shallow diagonal sums of the triangle k!*Stirling2(n,k): a(n) = Sum_{k=0..floor(n/2)} T(n-k,k) where T is A019538.

Original entry on oeis.org

1, 0, 1, 1, 3, 7, 21, 67, 237, 907, 3741, 16507, 77517, 385627, 2024301, 11174587, 64673997, 391392667, 2470864941, 16237279867, 110858862477, 784987938907, 5755734591981, 43636725010747, 341615028340557, 2758165832945947, 22940755633301421, 196354180631212027

Offset: 0

Paul Barry, Apr 20 2005

From Gus Wiseman, Jan 08 2019: (Start)
Also the number of set partitions of {1,...,n} into blocks of sizes > 1 whose minima form an initial interval of positive integers. For example, the a(5) = 7 set partitions are:
  {{1,2,3,4,5}}
  {{1,3},{2,4,5}}
  {{1,4},{2,3,5}}
  {{1,5},{2,3,4}}
  {{1,3,4},{2,5}}
  {{1,3,5},{2,4}}
  {{1,4,5},{2,3}}
Also the number of ordered set partitions of {1,...,n-k} of length k, for any 0 <= k <= n. For example, the a(5) = 7 ordered set partitions are:
  {{1,2,3,4}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1,2},{3}}
  {{1,3},{2}}
  {{2,3},{1}}
(End)

			a(8) = 1!*Stirling2(7,1) + 2!*Stirling2(6,2) + 3!*Stirling2(5,3) + 4!*Stirling2(4,4) = 1 + 62 + 150 + 24 = 237. - _Peter Bala_, Jul 09 2014

Alois P. Heinz, Table of n, a(n) for n = 0..599
Giulio Cerbai, Pattern-avoiding modified ascent sequences, arXiv:2401.10027 [math.CO], 2024. See p. 13.

Cf. A000110, A000296, A000670, A003101, A008277, A019538, A026898, A287215.

a:= n-> add(Stirling2(n-k, k)*k!, k=0..n/2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 09 2014

Table[Sum[StirlingS2[n-k, k]*k!, {k,0,n/2}],{n,0,20}] (* Vaclav Kotesovec, Jul 16 2014 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Length[Join@@Permutations/@Select[sps[Range[n-k]],Length[#]==k&]],{k,0,n}],{n,0,10}] (* Gus Wiseman, Jan 08 2019 *)

/* From Paul Barry's formula: */
{a(n)=sum(k=0,floor(n/2), sum(i=0,k,(-1)^i*binomial(k, i)*(k-i)^(n-k)))} \\ Paul D. Hanna, Dec 28 2013
for(n=0,30,print1(a(n),", "))

/* From e.g.f. series involving iterated integration: */
{INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G}
{a(n)=local(A=1+x);A=1+sum(k=1,n,INTEGRATE(k,(exp(x+x*O(x^n))-1)^k ));n!*polcoeff(A,n)} \\ Paul D. Hanna, Dec 28 2013

a(n) = sum{k=0..floor(n/2), sum{(-1)^i*binomial(k, i)*(k-i)^(n-k)}}.
E.g.f.: Sum_{n>=0} Integral^n (exp(x) - 1)^n dx^n, where integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration. - Paul D. Hanna, Dec 28 2013
Formal o.g.f.: 1/(1 + x)*sum {n >= 0} 1/(1 - n*x)*(x/(1 + x))^n = 1 + x^2 + x^3 + 3*x^4 + 7*x^5 + .... Cf. A229046. - Peter Bala, Jul 09 2014

A062810 a(n) = Sum_{i=1..n} i^(n - i) + (n - i)^i.

Original entry on oeis.org

1, 3, 7, 17, 45, 131, 419, 1465, 5561, 22755, 99727, 465537, 2303829, 12037571, 66174411, 381560425, 2301307841, 14483421859, 94909491607, 646309392369, 4565559980989, 33401808977411, 252713264780595, 1974606909857945

Offset: 1

Olivier Gérard, Jun 23 2001

Seiichi Manyama, Table of n, a(n) for n = 1..600

Cf. A003101, A026898, A062811, A062812.

Mathematica
```
Sum[i^(n - i) + (n - i)^i, {i, 1, n}]
```

a(n) = sum(i=1, n, i^(n-i) + (n-i)^i); \\ Michel Marcus, Mar 24 2019

a(n) = 2 * A026898(n-1) - 1.

a(n) = 2 * A003101(n-1) + 1.

A247358 Triangle read by rows: n-th row contains powers b^e with b + e = n + 1 in natural order.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 8, 9, 1, 5, 16, 16, 27, 1, 6, 25, 32, 64, 81, 1, 7, 36, 64, 125, 243, 256, 1, 8, 49, 128, 216, 625, 729, 1024, 1, 9, 64, 256, 343, 1296, 2187, 3125, 4096, 1, 10, 81, 512, 512, 2401, 6561, 7776, 15625, 16384, 1, 11, 100, 729, 1024, 4096, 16807, 19683, 46656, 65536, 78125

Offset: 1

Reinhard Zumkeller, Sep 14 2014

Sorted rows of triangle A051129.

			.  1  |  1                            |  1^1
.  2  |  1 2                          |  1^2 2^1
.  3  |  1 3  4                       |  1^3 3^1 2^2
.  4  |  1 4  8   9                   |  1^4 4^1 2^3 3^2
.  5  |  1 5 16  16  27               |  1^5 5^1 2^4 4^2 3^3
.  6  |  1 6 25  32  64  81           |  1^6 6^1 5^2 2^5 4^3 3^4
.  7  |  1 7 36  64 125 243 256       |  1^7 7^1 6^2 2^6 5^3 3^5 4^4
.  8  |  1 8 49 128 216 625 729 1024  |  1^8 8^1 7^2 2^7 6^3 5^4 3^6 4^5 .

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A051129, A003101 (row sums), A247363 (central terms), A003320 (row maxima).

import Data.List (sort)
a247358 n k = a247358_tabl !! (n-1) !! (k-1)
a247358_row n = a247358_tabl !! (n-1)
a247358_tabl = map sort a051129_tabl

Table[Table[k^(n-k+1), {k, 1, n}] // Sort, {n, 1, 11}] // Flatten (* Jean-François Alcover, Nov 18 2019 *)

row(n) = vecsort(vector(n, k, k^(n-k+1))); \\ Michel Marcus, Jan 24 2022

from itertools import chain
A247358_list = list(chain.from_iterable(sorted((b+1)**(n-b) for b in range(n)) for n in range(1,8))) # Chai Wah Wu, Sep 14 2014

A062809 a(n) = Sum_{i = 1..n} (n - i)^(1 + i).

Original entry on oeis.org

0, 1, 5, 18, 60, 203, 725, 2772, 11368, 49853, 232757, 1151902, 6018772, 33087191, 190780197, 1150653904, 7241710912, 47454745785, 323154696165, 2282779990474, 16700904488684, 126356632390275, 987303454928949

Offset: 1

Olivier Gérard, Jun 23 2001

Seiichi Manyama, Table of n, a(n) for n = 1..600

Cf. A003101, A349878, A349879.

a:=n->sum((n-j)^j,j=2..n): seq(a(n), n=2..24); # Zerinvary Lajos, Jun 07 2008

Mathematica
```
Sum[(n - i)^(1 + i), {i, 1, n}]
```

a(n) = sum(i=1, n, (n-i)^(1+i)); \\ Michel Marcus, Mar 24 2019

a(n) ~ sqrt(2*Pi) * ((n + 1)/LambertW(exp(1)*(n + 1)))^(1/2 + (n + 1)*(1 - 1/LambertW(exp(1)*(n + 1)))) / sqrt(1 + LambertW(exp(1)*(n + 1))). - Vaclav Kotesovec, Dec 04 2021

A113173 Ascending descending base exponent transform of semiprimes (A001358).

Original entry on oeis.org

256, 5392, 315361, 11667713, 717360537, 83932270482, 27775696582531, 22260761742531649, 109563850113131234720, 2013390472722146301196, 1899501614194512059559835, 85600281199526209989968735

Offset: 1

Jonathan Vos Post, Jan 07 2006

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. a(7) is itself semiprime. The smallest primes in this sequence are a(3) = 315361 and a(4) = 11667713. What is the next prime?

			a(1) = 256 because semiprime(1)^semiprime(1) = 4^4 = 256.
a(2) = 5392 because prime(1)^prime(2) + prime(2)^prime(1) = 4^6 + 6^4 = 5392.
a(3) = 315361 because 4^9 + 6^6 + 9^4 = 315361.
a(4) = 11667713 = 4^10 + 6^9 + 9^6 + 10^4.
a(5) = 717360537 = 4^14 + 6^10 + 9^9 + 10^6 + 14^4.
a(6) = 83932270482 = 4^15 + 6^14 + 9^10 + 10^9 + 14^6 + 15^4.
a(7) = 27775696582531 = 4^21 + 6^15 + 9^14 + 10^10 + 14^9 + 15^6 + 21^4.
a(8) = 22260761742531649 = 4^22 + 6^21 + 9^15 + 10^14 + 14^10 + 15^9 + 21^6 + 22^4.
a(9) = 109563850113131234720 = 4^25 + 6^22 + 9^21 + 10^15 + 14^14 + 15^10 + 21^9 + 22^6 + 25^4.

G. C. Greubel, Table of n, a(n) for n = 1..100

Cf. A001358, A005408, A113122, A113153, A113154.

A001358[A001358%5Bk%5D%5B%5Bk%5D%5D)%5E((A001358%5Bn%20-%20k%20+%201%5D%5B%5Bn%20-%20k%20+%201%5D%5D)),%20%7Bk,%201,%20n%7D%5D,%20%7Bn,%201,%2010%7D%5D%20(*%20_G.%20C.%20Greubel">] := Select[Range[100], PrimeOmega[#] == 2 &]; Table[Sum[(A001358[k][[k]])^((A001358[n - k + 1][[n - k + 1]])), {k, 1, n}], {n, 1, 10}] (* _G. C. Greubel, May 19 2017 *)

a(n) = Sum_{i=1..n} (semiprime(i))^(semiprime(n-i+1)).

a(n) = Sum_{i=1..n} (A001358(i))^(A001358(n-i+1)).

A242431 Triangle read by rows: T(n, k) = (k + 1)*T(n-1, k) + Sum_{j=k..n-1} T(n-1, j) for k < n, T(n, n) = 1. T(n, k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 10, 4, 1, 43, 35, 17, 5, 1, 144, 128, 74, 26, 6, 1, 523, 491, 329, 137, 37, 7, 1, 2048, 1984, 1498, 730, 230, 50, 8, 1, 8597, 8469, 7011, 3939, 1439, 359, 65, 9, 1, 38486, 38230, 33856, 21568, 9068, 2588, 530, 82, 10, 1

Offset: 0

Peter Luschny, May 14 2014

			0|    1;
1|    2,    1;
2|    5,    3,    1;
3|   14,   10,    4,   1;
4|   43,   35,   17,   5,   1;
5|  144,  128,   74,  26,   6,  1;
6|  523,  491,  329, 137,  37,  7, 1;
7| 2048, 1984, 1498, 730, 230, 50, 8, 1;

Peter Luschny, Rows n = 0..50, flattened.
Mathew Englander, Comments on A101494 and A089246, and related sequences

Cf. A003101, A026898, A047969, A047970, A101494, A089246.

T := proc(n, k) option remember; local j;
    if k=n then 1
  elif k>n then 0
  else (k+1)*T(n-1, k) + add(T(n-1, j), j=k..n)
    fi end:
seq(print(seq(T(n,k), k=0..n)), n=0..7);

def A242431_rows():
    T = []; n = 0
    while True:
        T.append(1)
        yield T
        for k in (0..n):
            T[k] = (k+1)*T[k] + add(T[j] for j in (k..n))
        n += 1
a = A242431_rows()
for n in range(8): next(a)

T(n, 0) = A047970(n).
Sum_{k=0..n} T(n, k) = A112532(n+1).
From Mathew Englander, Feb 25 2021: (Start)
T(n,k) = 1 + Sum_{i = k+1..n} i*(i+1)^(n-i).
T(n,k) = T(n,k+1) + (k+1)*(k+2)^(n-k-1) for 0 <= k < n.
T(n,k) = T(n,k+1) + (k+2)*(T(n-1,k) - T(n-1,k+1)) for 0 <= k <= n-2.
T(n,k) = Sum_{i = 0..n-k} (k+2)^i*A089246(n-k,i).
Sum_{i = k..n} T(i,k) = Sum_{i = 0..n-k} (n+2-i)^i = Sum_{i = 0..n-k} A101494(n-k,i)*(k+2)^i. (End)

A349878 Expansion of Sum_{k>=0} k^3 * x^k/(1 - k * x).

Original entry on oeis.org

0, 1, 9, 44, 178, 689, 2723, 11304, 49772, 232657, 1151781, 6018628, 33087022, 190780001, 1150653679, 7241710656, 47454745496, 323154695841, 2282779990113, 16700904488284, 126356632389834, 987303454928465, 7957133905608283, 66071772829246808

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A026898, A003101, A062809, A349879.

Cf. A349854.

Table[Sum[k^(n - k + 3), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 04 2021 *)

a(n, s=3, t=1) = sum(k=0, n, k^(t*(n-k)+s));

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^3*x^k/(1-k*x))))

a(n) = Sum_{k=0..n} k^(n-k+3).

a(n) ~ sqrt(2*Pi) * ((n + 3)/LambertW(exp(1)*(n + 3)))^(1/2 + (n + 3)*(1 - 1/LambertW(exp(1)*(n + 3)))) / sqrt(1 + LambertW(exp(1)*(n + 3))). - Vaclav Kotesovec, Dec 04 2021

cp's OEIS Frontend

A113320 a(1)=1 and a(n) for n>1 has the smallest positive value such that Sum_{i=1..n} a(i)^a(n-i+1) is prime.

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A113336 Least integers, starting with 2, so ascending descending base exponent transforms all prime.

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A234568 Sum_{k=0..n} (n-k)^(2*k).

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A105795 Shallow diagonal sums of the triangle k!*Stirling2(n,k): a(n) = Sum_{k=0..floor(n/2)} T(n-k,k) where T is A019538.

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A062810 a(n) = Sum_{i=1..n} i^(n - i) + (n - i)^i.

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A247358 Triangle read by rows: n-th row contains powers b^e with b + e = n + 1 in natural order.

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A062809 a(n) = Sum_{i = 1..n} (n - i)^(1 + i).

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