A070960 -id:A070960 - cp's OEIS frontend

A245334 A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

1, 2, 1, 3, 4, 2, 4, 9, 12, 6, 5, 16, 36, 48, 24, 6, 25, 80, 180, 240, 120, 7, 36, 150, 480, 1080, 1440, 720, 8, 49, 252, 1050, 3360, 7560, 10080, 5040, 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320, 10, 81, 576, 3528, 18144, 75600, 241920, 544320

Offset: 0

Reinhard Zumkeller, Aug 30 2014

row(0) = {1}; row(n+1) = row(n) multiplied by n and prepended with (n+1);
A111063(n+1) = sum of n-th row;
T(2*n,n) = A002690(n), central terms;
T(n,0) = n + 1;
T(n,1) = A000290(n), n > 0;
T(n,2) = A011379(n-1), n > 1;
T(n,3) = A047927(n), n > 2;
T(n,4) = A192849(n-1), n > 3;
T(n,5) = A000142(5) * A027810(n-5), n > 4;
T(n,6) = A000142(6) * A027818(n-6), n > 5;
T(n,7) = A000142(7) * A056001(n-7), n > 6;
T(n,8) = A000142(8) * A056003(n-8), n > 7;
T(n,9) = A000142(9) * A056114(n-9), n > 8;
T(n,n-10) = 11 * A051431(n-10), n > 9;
T(n,n-9) = 10 * A049398(n-9), n > 8;
T(n,n-8) = 9 * A049389(n-8), n > 7;
T(n,n-7) = 8 * A049388(n-7), n > 6;
T(n,n-6) = 7 * A001730(n), n > 5;
T(n,n-5) = 6 * A001725(n), n > 5;
T(n,n-4) = 5 * A001720(n), n > 4;
T(n,n-3) = 4 * A001715(n), n > 2;
T(n,n-2) = A070960(n), n > 1;
T(n,n-1) = A052849(n), n > 0;
T(n,n) = A000142(n);
T(n,k) = A137948(n,k) * A007318(n,k), 0 <= k <= n.

			.  0:   1;
.  1:   2,  1;
.  2:   3,  4,   2;
.  3:   4,  9,  12,    6;
.  4:   5, 16,  36,   48,    24;
.  5:   6, 25,  80,  180,   240,   120;
.  6:   7, 36, 150,  480,  1080,  1440,    720;
.  7:   8, 49, 252, 1050,  3360,  7560,  10080,   5040;
.  8:   9, 64, 392, 2016,  8400, 26880,  60480,  80640,  40320;
.  9:  10, 81, 576, 3528, 18144, 75600, 241920, 544320, 725760, 362880.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A111063 (row sums), A240993 (row products), A002690 (central terms).
Cf. A000290, A011379, A027810, A027818, A047927, A056001, A056003, A056114, A192849.
Cf. A000142, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431, A052849, A070960.
Cf. A007318, A137948.

a245334 n k = a245334_tabl !! n !! k
a245334_row n = a245334_tabl !! n
a245334_tabl = iterate (\row@(h:_) -> (h + 1) : map (* h) row) [1]

Table[(n!)/((n - k)!)*(n + 1 - k), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 10 2017 *)

T(n,k) = n!*(n+1-k)/(n-k)!. - Werner Schulte, Sep 09 2017

A048249 Number of distinct values produced from sums and products of n unity arguments.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 11, 17, 23, 30, 44, 60, 80, 114, 156, 212, 296, 404, 556, 770, 1065, 1463, 2032, 2795, 3889, 5364, 7422, 10300, 14229, 19722, 27391, 37892, 52599, 73075, 101301, 140588, 195405, 271024, 376608, 523518, 726812, 1010576, 1405013, 1952498

Offset: 1

Tony Bartoletti

Values listed calculated by exhaustive search algorithm.
For n+1 operands (n operations) there are (2n)!/((n!)((n+1)!)) possible postfix forms over a single operator. For each such form, there are 2^n ways to assign 2 operators (here, sum and product). Calculate results and eliminate duplicates.
Number of distinct positive integers that can be obtained by iteratively adding or multiplying together parts of an integer partition until only one part remains, starting with 1^n. - Gus Wiseman, Sep 29 2018

			a(3)=3 since (in postfix): 111** = 11*1* = 1, 111*+ = 11*1+ = 111+* = 11+1* = 2 and 111++ = 11+1+ = 3. Note that at n=7, the 11 possible values produced are the set {1,2,3,4,5,6,7,8,9,10,12}. This is the first n for which there are "skipped" values in the set.

Cf. A000792, A005520, A066739, A070960, A201163, A319850, A318949, A319855, A319856, A319909, A319910, A319911.

b:= proc(n) option remember; `if`(n=1, {1}, {seq(seq(seq(
     [f+g, f*g][], g=b(n-i)), f=b(i)), i=1..iquo(n, 2))})
    end:
a:= n-> nops(b(n)):
seq(a(n), n=1..35);  # Alois P. Heinz, May 05 2019

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
Table[Length[Select[ReplaceListRepeated[{Array[1&,n]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]],{n,10}] (* Gus Wiseman, Sep 29 2018 *)

from functools import cache
@cache
def f(m):
    if m == 1: return {1}
    out = set()
    for j in range(1, m//2+1):
        for x in f(j):
            for y in f(m-j):
                out.update([x + y, x * y])
    return out
def a(n): return len(f(n))
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Aug 03 2022

Equals partial sum of "number of numbers of complexity n" (A005421). - Jonathan Vos Post, Apr 07 2006

More terms from David W. Wilson, Oct 10 2001

a(43)-a(44) from Alois P. Heinz, May 05 2019

A319850 Number of distinct positive integers that can be obtained, starting with the initial interval partition (1, ..., n), by iteratively adding or multiplying together parts until only one part remains.

Original entry on oeis.org

1, 2, 5, 21, 94, 446, 2287, 12568, 78509

Offset: 1

Gus Wiseman, Sep 29 2018

			The n-th row lists all integers that can be obtained starting with (1, ..., n):
  1
  2 3
  5 6 7 8 9
  9 10 11 12 13 14 15 16 17 18 19 20 21 24 25 26 27 28 30 32 36

Cf. A000041, A001055, A001970, A048249, A066739, A066815, A070960, A201163, A318948, A318949, A319841.

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
Table[Length[Select[ReplaceListRepeated[{Range[n]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]],{n,6}]

A319909 Number of distinct positive integers that can be obtained by iteratively adding any two or multiplying any two non-1 parts of an integer partition until only one part remains, starting with 1^n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 4, 5, 10, 15, 21, 34, 49, 68, 101, 142, 197, 280, 387, 538, 751, 1045, 1442, 2010, 2772, 3865, 5339, 7396, 10273, 14201, 19693

Offset: 0

Gus Wiseman, Oct 01 2018

			We have
   7 = 1+1+1+1+1+1+1,
   8 = (1+1)*(1+1+1)+1+1,
   9 = (1+1)*(1+1)*(1+1)+1,
  10 = (1+1+1+1+1)*(1+1),
  12 = (1+1+1)*(1+1+1+1),
so a(7) = 5.

Cf. A000792, A001970, A005520, A048249, A066739, A066815, A070960, A201163, A275870, A319850, A318948, A318949, A319912, A319913.

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
mexos[ptn_]:=If[Length[ptn]==0,{0},Union@@Select[ReplaceListRepeated[{Sort[ptn]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_?(#>1&),mie___,y_?(#>1&),afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]];
Table[Length[mexos[Table[1,{n}]]],{n,30}]

A319855 Minimum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 4, 3, 5, 2, 6, 4, 5, 1, 7, 4, 8, 3, 6, 5, 9, 2, 6, 6, 6, 4, 10, 5, 11, 1, 7, 7, 7, 4, 12, 8, 8, 3, 13, 6, 14, 5, 7, 9, 15, 2, 8, 6, 9, 6, 16, 6, 8, 4, 10, 10, 17, 5, 18, 11, 8, 1, 9, 7, 19, 7, 11, 7, 20, 4, 21, 12, 8, 8, 9, 8, 22, 3, 8

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			a(30) = 5 because the minimum number that can be obtained starting with (3,2,1) is 3+2*1 = 5.

Cf. A000792, A001970, A048249, A056239, A066739, A066815, A070960, A201163, A319850, A318948, A318949, A319841, A319856.

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
nexos[ptn_]:=If[Length[ptn]==0,{0},Union@@Select[ReplaceListRepeated[{Sort[ptn]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]];
Table[Min[nexos[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

a(1) = 0, a(n) = max(A056239(n) - A007814(n), 1). - Charlie Neder, Oct 03 2018

A319856 Maximum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 6, 4, 7, 6, 8, 6, 8, 6, 9, 6, 9, 7, 8, 8, 10, 9, 11, 6, 10, 8, 12, 9, 12, 9, 12, 9, 13, 12, 14, 10, 12, 10, 15, 9, 16, 12, 14, 12, 16, 12, 15, 12, 16, 11, 17, 12, 18, 12, 16, 9, 18, 15, 19, 14, 18, 16, 20, 12, 21, 13

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			a(30) = 9 because the maximum number that can be obtained starting with (3,2,1) is 3*(2+1) = 9.

Cf. A000792, A001970, A048249, A056239, A066739, A066815, A070960, A201163, A319850, A318948, A318949, A319841, A319855.

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
nexos[ptn_]:=If[Length[ptn]==0,{0},Union@@Select[ReplaceListRepeated[{Sort[ptn]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]];
Table[Max[nexos[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

A358805 Numbers k such that k! + (k!/2) + 1 is prime.

Original entry on oeis.org

4, 5, 7, 11, 12, 14, 18, 28, 30, 62, 135, 153, 275, 584, 630, 1424, 1493, 4419, 8492, 10950

Offset: 1

Arsen Vardanyan, Dec 01 2022

Numbers k such that A070960(k)+1 is prime.

No more terms < 10000. - Vaclav Kotesovec, Dec 12 2022

Table of n, a(n) for n = 1..19
Arsen Vardanyan, A list of prime numbers

Cf. A070960, A358878.

PARI
```
is(k) = isprime(k!+(k!/2)+1);
```

a(18)-a(19) from Vaclav Kotesovec, Dec 09 2022

a(20) from Michael S. Branicky, Aug 02 2024

A071107 a(n) is the greatest integer that can be obtained from the integers {1, 2, 3, ..., n} using each number at most once and the operators +,-,*,/,^.

Original entry on oeis.org

1, 3, 27, 115792089237316195423570985008687907853269984665640564039457584007913129639936

Offset: 1

Koksal Karakus (karakusk(AT)hotmail.com), May 27 2002

nonn

a(4) = 2^(4^4) = 2^256 = 115792089237316195423570985008687907853269984665640564039457584007913129639936 with 78 digits. a(5) = 2^(3^(4^6)) with more than 10^1950 digits.

			a(3) = 27 because 3^(2+1) = 27 is the greatest integer that can be obtained by using 1, 2, 3 once and the operations +, -, *, /, ^.

Cf. A070960.

A358878 Numbers k such that k! + (k!/2) - 1 is prime.

Original entry on oeis.org

2, 5, 7, 15, 20, 47, 84, 138, 169, 257, 263, 431, 559, 2939, 4403, 4870, 5273

Offset: 1

Arsen Vardanyan, Dec 04 2022

Numbers k such that A070960(k) - 1 is prime.

No more terms < 10000. - Vaclav Kotesovec, Dec 12 2022

Cf. A070960, A358805.

PARI
```
is(k) = isprime(k!+(k!/2)-1);
```

a(14)-a(17) from Vaclav Kotesovec, Dec 07 2022

A071603 Number of different positive integers that we can obtain from the integers {1,2,...,n} using each number at most once and the operators +, -, *, /, where intermediate subexpressions must be integers.

Original entry on oeis.org

1, 3, 9, 31, 121, 542, 2868, 16329, 106762, 758155, 6142570

Offset: 1

Koksal Karakus (karakusk(AT)hotmail.com), Jun 02 2002

			a(4)=31 because we can obtain the positive integers 1,2,...,28 and 30,32,36 by using the integers {1, 2, 3, 4} at most once and the four operations. For example 30 = 3*2*(4+1).

Index entries for similar sequences

Cf. A060315, A070960, A071848.

def a(n):
    R = dict() # index of each reachable subset is [card(s)-1][s]
    for i in range(n): R[i] = dict()
    for i in range(1, n+1): R[0][(i,)] = {i}
    reach = set(range(1, n+1))
    for j in range(1, n):
        for i in range((j+1)//2):
            for s1 in R[i]:
                for s2 in R[j-1-i]:
                    if set(s1) & set(s2) == set():
                        s12 = tuple(sorted(set(s1) | set(s2)))
                        if s12 not in R[len(s12)-1]:
                            R[len(s12)-1][s12] = set()
                        for a in R[i][s1]:
                            for b in R[j-1-i][s2]:
                                allowed = [a+b, a*b, a-b, b-a]
                                if a!=0 and b%a==0: allowed.append(b//a)
                                if b!=0 and a%b==0: allowed.append(a//b)
                                R[len(s12)-1][s12].update(allowed)
                                reach.update(allowed)
    return len(set(r for r in reach if r > 0 and r.denominator == 1))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Jul 01 2022

a(10)-a(11) from Michael S. Branicky, Jul 01 2022

cp's OEIS Frontend

A245334 A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

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A048249 Number of distinct values produced from sums and products of n unity arguments.

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A319850 Number of distinct positive integers that can be obtained, starting with the initial interval partition (1, ..., n), by iteratively adding or multiplying together parts until only one part remains.

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A319909 Number of distinct positive integers that can be obtained by iteratively adding any two or multiplying any two non-1 parts of an integer partition until only one part remains, starting with 1^n.

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A319855 Minimum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

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A319856 Maximum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

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Mathematica

A358805 Numbers k such that k! + (k!/2) + 1 is prime.

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PARI

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A071107 a(n) is the greatest integer that can be obtained from the integers {1, 2, 3, ..., n} using each number at most once and the operators +,-,*,/,^.

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A358878 Numbers k such that k! + (k!/2) - 1 is prime.