A333129 -id:A333129 - cp's OEIS frontend

A333238 Irregular table where row n lists the distinct smallest primes p of prime partitions of n.

2, 3, 2, 2, 5, 2, 3, 2, 7, 2, 3, 2, 3, 2, 3, 5, 2, 3, 11, 2, 3, 5, 2, 3, 13, 2, 3, 7, 2, 3, 5, 2, 3, 5, 2, 3, 5, 17, 2, 3, 5, 7, 2, 3, 5, 19, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 11, 2, 3, 5, 23, 2, 3, 5, 7, 11, 2, 3, 5, 7, 2, 3, 5, 7, 13, 2, 3, 5, 7, 2, 3, 5, 7, 11

Offset: 2

Michael De Vlieger, David James Sycamore, Mar 12 2020

A prime partition of n is an integer partition wherein all parts are prime. For instance, (3 + 2) is a prime partition of the sum 5; for n = 5, (5) is also a prime partition. For 6, we have two prime partitions (3 + 3) and (2 + 2 + 2).
We note that there are no prime partitions for n = 1, therefore the offset of this sequence is 2.
The number of prime partitions of n is shown by A000607(n).
For prime p, row p includes p itself as the largest term, since p is the sum of (p).
The product of all terms in row n gives A333129(n). - Alois P. Heinz, Mar 16 2020
From David James Sycamore, Mar 28 2020: (Start)
In the irregular table below, T(n,k) is either prime(k) or is empty. The former means there is at least one prime partition of n with least part prime(k), the latter means that no such partition exists. T(n,k) empty is not recorded in the data.
Recursion for n >= 4: T(n,k) = prime(k) iff T(n-prime(k), k) = prime(k), or there is a q > k such that T(n-prime(k), q) = prime(q); else T(n,k) is empty. Example: T(17,3) = 5 because T(12,3) = prime(3) = 5. T(10,2) = 3 since although T(7,2) is empty, T(7,4) = prime(4) = 7. (End)

			The least primes among the prime partitions of 5 are 2 and 5, cf. the 2 prime partitions of 5: (5) and (3, 2), thus row 5 lists {2, 5}.
The least primes among the prime partitions of 6 are 2 and 3, cf. the two prime partitions of 6, (3, 3), and (2, 2, 2), thus row 6 lists {2, 3}.
Row 7 contains {2, 7} because there are 3 prime partitions of 7: (7), (5, 2), (3, 2, 2). Note that 2 is the smallest part of the latter two partitions, thus only 2 and 7 are distinct.
Table plotting prime p in row n at pi(p) place, intervening primes missing from row n are shown by "." as a place holder:
n      Primes in row n
----------------------
2:     2
3:     .   3
4:     2
5:     2   .   5
6:     2   3
7:     2   .   .   7
8:     2   3
9:     2   3
10:    2   3   5
11:    2   3   .   .  11
12:    2   3   5
13:    2   3   .   .   .  13
14:    2   3   .   7
15:    2   3   5
16:    2   3   5
17:    2   3   5   .   .   .  17
...

Alois P. Heinz, Rows n = 2..1000, flattened (rows n=2..240 from Michael De Vlieger)
Michael De Vlieger, Labeled plot of p at (pi(p), n), for 2 <= n <= 240, with terms n in A330507 shown in red.

Cf. A330507, A333129, A333259, A333365.

b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
T:= proc(n) option remember; (p-> seq(`if`(isprime(i) and
      coeff(p, x, i)>0, i, [][]), i=2..degree(p)))(b(n, 2, x))
    end:
seq(T(n), n=2..40);  # Alois P. Heinz, Mar 16 2020

Block[{a, m = 20, s}, a = ConstantArray[{}, m]; s = {Prime@ PrimePi@ m}; Do[If[# <= m, If[FreeQ[a[[#]], First@ s], a = ReplacePart[a, # -> Append[a[[#]], Last@ s]], Nothing]; AppendTo[s, Last@ s], If[Last@ s == 2, s = DeleteCases[s, 2]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]]] &@ Total[s], {i, Infinity}]; Union /@ a // Flatten]

A333259 a(n) = Sum_{p in L(n)} 2^(pi(p) - 1) where L(n) is the set of all least primes in partitions of n into prime parts.

Original entry on oeis.org

0, 0, 1, 2, 1, 5, 3, 9, 3, 3, 7, 19, 7, 35, 11, 7, 7, 71, 15, 135, 15, 15, 23, 263, 31, 15, 47, 15, 31, 527, 63, 1039, 47, 31, 95, 31, 111, 2079, 143, 63, 95, 4127, 191, 8255, 63, 63, 351, 16447, 223, 63, 191, 127, 319, 32895, 383, 127, 191, 255, 639, 65663

Offset: 0

Michael De Vlieger, Mar 16 2020

In other words, convert the indices of primes p_i in row n of A333238 to 1s in the (i - 1)-th place to create a binary number m; convert m to decimal.
The number of prime partitions of n is shown by A000607(n), which in terms of this sequence equates to the number of 1s in a(n), written in binary.
For prime p, row p of A333238 includes p itself as the largest term, since p is the sum of (p); here we find a(p) >= 2^(p - 1). More specifically, a(2) = 1, a(p) > 2^(p - 1) for p odd.
For n = A330507(m), a(n) = 2^m - 1, the smallest n with this value in this sequence.

			The least primes among the prime partitions of 5 are 2 and 5, cf. the 2 prime partitions of 5: (5) and (3, 2), thus row 5 of A333238 lists {2, 5}. Convert these to their indices gives us {1, 3}, take the sum of 2^(1 - 1) and 2^(3 - 1) = 2^0 + 2^2 = 1 + 4 = 5, thus a(5) = 5.
The least primes among the prime partitions of 6 are 2 and 3, cf. the two prime partitions of 6, (3, 3), and (2, 2, 2), thus row 6 of A333238 lists {2, 3}. Convert these to their indices: {1, 2}, take the sum of 2^(1 - 1) and 2^(2 - 1) = 2^0 + 2^1 = 1 + 2 = 3, thus a(6) = 3.
Row 7 of A333238 contains {2, 7} because there are 3 prime partitions of 7: (7), (5, 2), (3, 2, 2). Note that 2 is the smallest part of the latter two partitions, thus only 2 and 7 are distinct. Convert to indices: {1, 4}, sum 2^(1 - 1) and 2^(4 - 1) = 2^0 + 2^3 = 1 + 8 = 9, therefore a(7) = 9.
Table plotting prime p in row n of A333238 at pi(p) place, intervening primes missing from row n are shown by "." as a place holder. We convert the indices of these primes into a binary number to obtain the terms of this sequence:
    n      Row n of A333238    binary   a(n)
    ---------------------------------------
    2:     2                 =>     1 =>  1
    3:     .   3             =>    10 =>  2
    4:     2                 =>     1 =>  1
    5:     2   .   5         =>   101 =>  5
    6:     2   3             =>    11 =>  3
    7:     2   .   .   7     =>  1001 =>  9
    8:     2   3             =>    11 =>  3
    9:     2   3             =>    11 =>  3
    10:    2   3   5         =>   111 =>  7
    11:    2   3   .   .  11 => 10011 => 19
    12:    2   3   5         =>   111 =>  7
    ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000607, A000720, A330507, A333129, A333238.

b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
a:= proc(n) option remember; (p-> add(`if`(isprime(i) and coeff(p, x,
      i)>0, 2^(numtheory[pi](i)-1), 0), i=2..degree(p)))(b(n, 2, x))
    end:
seq(a(n), n=0..63);  # Alois P. Heinz, Mar 16 2020

Block[{a, m = 59, s}, a = ConstantArray[{}, m]; s = {Prime@ PrimePi@ m}; Do[If[# <= m, If[FreeQ[a[[#]], Last@ s], a = ReplacePart[a, # -> Append[a[[#]], Last@ s]], Nothing]; AppendTo[s, Last@ s], If[Last@ s == 2, s = DeleteCases[s, 2]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]]] &@ Total[s], {i, Infinity}]; {0, 0}~Join~Map[Total[2^(-1 + PrimePi@ #)] &, Rest[Union /@ a]]]

A333365 T(n,k) is the number of times that prime(k) is the least part in a partition of n into prime parts; triangle T(n,k), n >= 0, 1 <= k <= max(1,A000720(A331634(n))), read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 0, 1, 2, 1, 3, 1, 3, 1, 1, 4, 1, 0, 0, 1, 5, 1, 1, 6, 2, 0, 0, 0, 1, 7, 2, 0, 1, 9, 2, 1, 10, 3, 1, 12, 3, 1, 0, 0, 0, 1, 14, 3, 1, 1, 17, 4, 1, 0, 0, 0, 0, 1, 19, 5, 1, 1, 23, 5, 1, 1, 26, 6, 2, 0, 1, 30, 7, 2, 0, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, Mar 16 2020

			In the A000607(11) = 6 partitions of 11 into prime parts, (11), 335, 227, 2225, 2333, 22223 the least parts are 11 = prime(5) (once), 3 = prime(2)(once), and 2 = prime(1) (four times), whereas 5 and 7 (prime(3) and prime(4)) do not occur. Thus row 11 is [4,1,0,0,1].
Triangle T(n,k) begins:
   0    ;
   0    ;
   1    ;
   0, 1    ;
   1       ;
   1, 0, 1    ;
   1, 1       ;
   2, 0, 0, 1    ;
   2, 1          ;
   3, 1          ;
   3, 1, 1       ;
   4, 1, 0, 0, 1    ;
   5, 1, 1          ;
   6, 2, 0, 0, 0, 1    ;
   7, 2, 0, 1          ;
   9, 2, 1             ;
  10, 3, 1             ;
  12, 3, 1, 0, 0, 0, 1    ;
  14, 3, 1, 1             ;
  17, 4, 1, 0, 0, 0, 0, 1    ;
  19, 5, 1, 1                ;
  ...

Alois P. Heinz, Rows n = 0..1000, flattened

Columns k=1-2 give: A000607(n-2) for n>1, A099773(n-3) for n>2.
Row sums give A000607 for n>0.
Length of n-th row is A000720(A331634(n)) for n>1.
Indices of rows without 1's: A330433.
Cf. A000040, A000720, A010051, A333129, A333238, A333259.

b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
T:= proc(n) option remember; (p-> seq(`if`(isprime(i),
      coeff(p, x, i), [][]), i=2..max(2,degree(p))))(b(n, 2, x))
    end:
seq(T(n), n=0..23);

b[n_, p_, t_] := b[n, p, t] = If[n == 0, 1, If[p > n, 0, With[{q = NextPrime[p]}, Sum[b[n - p*j, q, 1], {j, 1, n/p}]*t^p + b[n, q, t]]]];
T[n_] := If[n < 2, {0}, MapIndexed[If[PrimeQ[#2[[1]]], #1, Nothing]&, Rest @ CoefficientList[b[n, 2, x], x]]];
T /@ Range[0, 23] // Flatten (* Jean-François Alcover, Mar 30 2021, after Alois P. Heinz *)

T(n,pi(n)) = A010051(n) for n > 1.
T(p,pi(p)) = 1 if p is prime.
T(prime(k),k) = 1 for k >= 1.
Recursion: T(n,k) = Sum_{q=k..pi(n-p)} T(n-p, q) with p := prime(k) and T(n,k) = 0 if n < p, or 1 if n = p. - David James Sycamore, Mar 28 2020

A333266 a(n) is the smallest number k such that for all m >= k there is at least one prime partition of m with prime(n) as least part.

Original entry on oeis.org

4, 8, 15, 24, 39, 49, 67, 83, 89, 115, 127, 143, 163, 179, 193, 223, 235, 249, 271, 281, 295, 333, 349, 363, 387, 403, 409, 427, 461, 483, 515, 535, 545, 565, 595, 625, 643, 659, 685, 703, 725, 733, 759, 805, 813, 835, 851, 895, 907, 923, 937, 965, 989, 1033

Offset: 1

David James Sycamore, Mar 13 2020

nonn

a(n) is a term if and only if it is the smallest number such that there exists a prime partition of every m in the interval [a(n), 2*a(n)] with least part prime(n). There is no prime partition of a(n)-1 having prime(n) as least part, and this is the greatest such number.

			For any k >= 4 there exists a prime partition of k having least part 2, hence a(1)=4.

Cf. A333129, A333238, A330507, A000607, A333259.

More terms from Alois P. Heinz, Mar 16 2020

A333417 a(n) is the greatest number k having for every prime <= prime(n) at least one prime partition with least part p, and no such partition having least part > prime(n). If no such k exists then a(n) = 0.

Original entry on oeis.org

4, 9, 16, 27, 35, 49, 63, 65, 85, 95, 105, 121, 135, 145, 169, 175, 187, 203, 207, 221, 253, 265, 273, 289, 301, 305, 319, 351, 369, 387, 403, 407, 425, 445, 473, 485, 495, 517, 529, 545, 551, 567, 611, 615, 629, 637, 671, 679, 693, 697, 725, 747, 781, 793, 799

Offset: 1

David James Sycamore, Mar 20 2020

nonn

Alternatively a(n) is the greatest number whose product of distinct least part primes from all prime partitions of n, is equal to primorial(n). Companion sequence to A330507.
From Michael De Vlieger, Mar 20 2020: (Start)
a(n) = 0 for n = {90, 151, 349, 352, 444, ...}, cf. the comment from Alois P. Heinz at A330507.
Index m of last instance of A002110(n) in A333129 as m increases.
Last row n in A333238 that contains the consecutive primes (1...n).
Last index of the occurrence of 2^n - 1 in A333259, which is the decimal value of the characteristic function of primes in A333238 interpreted as a binary number. (End)

			a(1) = 4 because [2,2] is the only prime partition of 4, and no greater number n has only 2 as least part in any partition of n into primes.
From _Michael De Vlieger_, Mar 20 2020: (Start)
Looking at this sequence as the first position of 2^n - 1 in A333259, which in binary is a k-bit repunit, we look for the last occasion of such in A333259, indicated by the arrows. a(k) = n for rows n that have an arrow. In the chart, we reverse the portrayal of the binary rendition of A333259(n), replacing zeros with "." for clarity:
   n   A333259(n)            k
------------------------------
   2   1                     1
   3   . 1
   4   1                  -> 1
   5   1 . 1
   6   1 1                   2
   7   1 . . 1
   8   1 1                   2
   9   1 1                -> 2
  10   1 1 1                 3
  11   1 1 . . 1
  12   1 1 1                 3
  13   1 1 . . . 1
  14   1 1 . 1
  15   1 1 1                 3
  16   1 1 1              -> 3
  17   1 1 1 . . . 1
  18   1 1 1 1               4
  19   1 1 1 . . . . 1
  20   1 1 1 1               4
  ... (End)

Cf. A000040, A002110, A051034, A331634, A332861, A330507, A333129, A333238, A333259, A333365.

With[{s = TakeWhile[Import["https://oeis.org/A333259/b333259.txt", "Data"], Length@ # > 0 &][[All, -1]]}, Array[If[Length[#] == 0, 0, #[[-1, 1]] - 1] &@ Position[s, 2^# - 1] &, 55]] (* Michael De Vlieger, Mar 20 2020, using the b-file at A333259 *)

More terms from Michael De Vlieger, Mar 20 2020

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A333238 Irregular table where row n lists the distinct smallest primes p of prime partitions of n.

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A333259 a(n) = Sum_{p in L(n)} 2^(pi(p) - 1) where L(n) is the set of all least primes in partitions of n into prime parts.

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A333365 T(n,k) is the number of times that prime(k) is the least part in a partition of n into prime parts; triangle T(n,k), n >= 0, 1 <= k <= max(1,A000720(A331634(n))), read by rows.

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A333266 a(n) is the smallest number k such that for all m >= k there is at least one prime partition of m with prime(n) as least part.

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A333417 a(n) is the greatest number k having for every prime <= prime(n) at least one prime partition with least part p, and no such partition having least part > prime(n). If no such k exists then a(n) = 0.

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