A000607 -id:A000607 - cp's OEIS frontend

A331418 If A331417(n) is the maximum sum of primes of the parts of an integer partition of n, then a(n) = A331417(n) - n + 1.

1, 2, 3, 4, 5, 7, 8, 11, 12, 15, 20, 21, 26, 29, 30, 33, 38, 43, 44, 49, 52, 53, 58, 61, 66, 73, 76, 77, 80, 81, 84, 97, 100, 105, 106, 115, 116, 121, 126, 129, 134, 139, 140, 149, 150, 153, 154, 165, 176, 179, 180, 183, 188, 189, 198, 203, 208, 213, 214, 219

Offset: 0

Gus Wiseman, Jan 17 2020

nonn

For n > 4, a(n) = A014692(n).

Converges to A014692.
Row lengths of A331385.
Sum of prime factors is A001414.
Partitions into primes are A000607.
Partitions whose sum of primes is divisible by their sum are A331379.
Cf. A000040, A014689, A056239, A330950, A330953, A330954, A331378, A331381, A331383, A331387, A331415, A331416.

Table[Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}]-n+1,{n,0,30}]

a(n) = A331417(n) - n + 1.

A338902 Number of integer partitions of the n-th semiprime into semiprimes.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 7, 7, 10, 17, 25, 21, 34, 34, 73, 87, 103, 149, 176, 206, 281, 344, 479, 725, 881, 1311, 1597, 1742, 1841, 2445, 2808, 3052, 3222, 6784, 9298, 11989, 14533, 15384, 17414, 18581, 19680, 28284, 35862, 38125, 57095, 60582, 64010, 71730, 76016

Offset: 1

Gus Wiseman, Nov 24 2020

nonn

A semiprime (A001358) is a product of any two prime numbers.

			The a(1) = 1 through a(33) = 17 partitions of 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, where A-Z = 10-35:
  4  6  9  A   E    F   L     M      P      Q       X
           64  A4   96  F6    994    FA     M4      EA9
               644      966   A66    L4     AA6     F99
                        9444  E44    A96    E66     FE4
                              6664   F64    9944    L66
                              A444   9664   A664    P44
                              64444  94444  E444    9996
                                            66644   AA94
                                            A4444   E964
                                            644444  F666
                                                    FA44
                                                    L444
                                                    96666
                                                    A9644
                                                    F6444
                                                    966444
                                                    9444444

A002100 counts partitions into squarefree semiprimes.
A056768 uses primes instead of semiprimes.
A101048 counts partitions into semiprimes.
A338903 is the squarefree version.
A339112 includes the Heinz numbers of these partitions.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A037143 lists primes and semiprimes.
A084126 and A084127 give the prime factors of semiprimes.
A320655 counts factorizations into semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes, with sum/difference/product A176504/A176506/A087794.
A338899/A270650/A270652 give prime indices of squarefree semiprimes.
Cf. A000041, A000607, A006881, A065516, A115392, A128301, A320656, A320732, A320892, A320912, A338915, A338916, A339113.

nn=100;Table[Length[IntegerPartitions[n,All,Select[Range[nn],PrimeOmega[#]==2&]]],{n,Select[Range[nn],PrimeOmega[#]==2&]}]

a(n) = A101048(A001358(n)).

A352492 Powerful numbers whose prime indices are all prime numbers.

Original entry on oeis.org

1, 9, 25, 27, 81, 121, 125, 225, 243, 289, 625, 675, 729, 961, 1089, 1125, 1331, 1681, 2025, 2187, 2601, 3025, 3125, 3267, 3375, 3481, 4489, 4913, 5625, 6075, 6561, 6889, 7225, 7803, 8649, 9801, 10125, 11881, 11979, 14641, 15125, 15129, 15625, 16129, 16875

Offset: 1

Gus Wiseman, Mar 24 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices (not prime factors) begin:
    1: {}
    9: {2,2}
   25: {3,3}
   27: {2,2,2}
   81: {2,2,2,2}
  121: {5,5}
  125: {3,3,3}
  225: {2,2,3,3}
  243: {2,2,2,2,2}
  289: {7,7}
  625: {3,3,3,3}
  675: {2,2,2,3,3}
  729: {2,2,2,2,2,2}
  961: {11,11}
For example, 675 = prime(2)^3 prime(3)^2 = 3^3 * 5^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Powerful numbers are A001694, counted by A007690.
The version for prime exponents instead of indices is A056166, counted by A055923.
This is the powerful case of A076610 (products of A006450), counted by A000607.
The partitions with these Heinz numbers are counted by A339218.
A000040 lists primes.
A031368 lists primes of odd index, products A066208.
A101436 counts exponents in prime factorization that are themselves prime.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A053810 lists all numbers p^q with p and q prime, counted by A230595.
A257994 counts prime indices that are themselves prime, complement A330944.
Cf. A000720, A000961, A001597, A007821, A007916, A052485, A109297, A140319, A164336, A181819, A325131, A330945, A346068.

Select[Range[1000],#==1||And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&&Min@@Last/@FactorInteger[#]>1&]

Intersection of A001694 and A076610.

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/(p*(p-1))) = 1.24410463... - Amiram Eldar, May 04 2022

A046676 Expansion of 1 + Sum_{k>=1} x^(p_1+p_2+...+p_k)/((1-x)(1-x^2)(1-x^3)...(1-x^k)) (where p_i are the primes).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 41, 46, 54, 60, 69, 78, 89, 99, 113, 126, 143, 159, 179, 199, 224, 248, 277, 307, 343, 378, 421, 464, 515, 567, 628, 690, 763, 837, 923, 1012, 1115, 1219, 1340, 1465, 1607

Offset: 0

N. J. A. Sloane

nonn

Ramanujan considered that this could equal the prime parts partition numbers A000607, but they differ from the 20th term on, cf. A192541. See A238804 for a correct variant, where the coefficient and power of x^{...} are adjusted to match A000607. - M. F. Hasler, Mar 06 2014

B. C. Berndt and B. M. Wilson, Chapter 5 of Ramanujan's second notebook, pp. 49-78 of Analytic Number Theory (Philadelphia, 1980), Lect. Notes Math. 899, 1981, see Entry 29.

George E. Andrews, Arnold Knopfmacher, John Knopfmacher, Engel expansions and the Rogers-Ramanujan identities J. Number Theory 80 (2000), 273-290. See Eq. 2.1.

Differs from A000607 at the 20th term. Cf. A192541.

t3:=1+add(q^sum(ithprime(i),i=1..j)/mul(1-q^i,i=1..j), j=1..51);
t4:=series(t3,q,50);
t5:=seriestolist(%);

Vec(sum(i=0,25,x^sum(k=1,i,prime(k))/prod(k=1,i,1-x^k),O(x^99))) \\ M. F. Hasler, Mar 05 2014

A046676(n,S=1,P=1+O(x^(n+1)))={for(k=1,n, nM. F. Hasler, Mar 05 2014

A199118 Number of partitions of n into terms of (1,3)-Ulam sequence, cf. A002859.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 7, 10, 13, 17, 21, 28, 34, 42, 52, 65, 78, 96, 113, 138, 165, 196, 231, 276, 322, 379, 442, 518, 600, 698, 803, 931, 1071, 1231, 1407, 1615, 1839, 2099, 2384, 2712, 3069, 3478, 3923, 4434, 4991, 5618, 6303, 7083, 7928, 8878, 9916, 11081

Offset: 0

Reinhard Zumkeller, Nov 03 2011

nonn

			The first terms of A002859 are 1, 3, 4, 5, 6, 8, 10, 12, 17, 21, ...
a(7) = #{6+1, 5+1+1, 4+3, 4+1+1+1, 3+3+1, 3+1+1+1+1, 7x1} = 7;
a(8) = #{8, 6+1+1, 5+3, 5+1+1+1, 4+4, 4+3+1, 4+1+1+1+1, 3+3+1+1, 3+1+1+1+1+1, 8x1} = 10.

Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers

Cf. A000607; A199119, A199016, A199120, A199122.

a199118 = p a002859_list where
   p _ 0 = 1
   p us'@(u:us) m | m < u     = 0
                  | otherwise = p us' (m - u) + p us m

A199120 Number of partitions of n into terms of (1,4)-Ulam sequence, cf. A003666.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 30, 38, 44, 54, 63, 77, 90, 107, 124, 148, 171, 202, 231, 271, 310, 360, 412, 477, 542, 622, 705, 809, 915, 1042, 1175, 1335, 1501, 1699, 1905, 2148, 2403, 2702, 3018, 3383, 3768, 4212, 4682, 5223, 5794, 6445

Offset: 0

Reinhard Zumkeller, Nov 03 2011

nonn

			The first terms of A003666 are 1, 4, 5, 6, 7, 8, 10, 16, 18, 19, ...
a(7) = #{7, 6+1, 5+1+1, 4+1+1+1, 7x1} = 5;
a(8) = #{8, 7+1, 6+1+1, 5+1+1+1, 4+4, 4+1+1+1+1, 8x1} = 7;
a(9) = #{8+1, 7+1+1, 6+1+1+1, 5+4, 5+1+1+1+1, 4+4+1, 4+5x1, 9x1} = 8.

Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers

Cf. A000607; A199016, A199118, A199121, A199122.

a199120 = p a003666_list where
   p _ 0 = 1
   p us'@(u:us) m | m < u     = 0
                  | otherwise = p us' (m - u) + p us m

A199122 Number of partitions of n into terms of (2,3)-Ulam sequence, cf. A001857.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 11, 14, 16, 20, 23, 29, 33, 39, 47, 54, 64, 75, 86, 101, 117, 135, 155, 179, 204, 236, 268, 306, 349, 397, 450, 511, 577, 653, 736, 831, 934, 1050, 1179, 1322, 1478, 1657, 1848, 2065, 2302, 2562, 2852, 3172, 3518, 3909

Offset: 0

Reinhard Zumkeller, Nov 03 2011

nonn

			The first terms of A001857 are 2, 3, 5, 7, 8, 9, 13, 14, 18, 19, ...
a(10) = #{8+2, 7+3, 5+5, 5+3+2, 3+3+2+2, 2+2+2+2+2} = 6;
a(11) = #{9+2, 8+3, 7+2+2, 5+3+3, 5+2+2+2, 3+3+3+2, 3+2+2+2+2} = 7;
a(12) = #{9+3, 8+2+2, 7+5, 7+3+2, 5+5+2, 5+3+2+2, 3+3+3+3, 3+3+2+2+2, 6x2} = 9.

Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers

Cf. A000607; A199123, A199016, A199118, A199120.

a199122 = p a001857_list where
   p _ 0 = 1
   p us'@(u:us) m | m < u     = 0
                  | otherwise = p us' (m - u) + p us m

nmax = 60;
U = {2, 3};
Do[AppendTo[U, k = Last[U]; While[k++; Length[DeleteCases[Intersection[U, k - U], k/2, 1, 1]] != 2]; k], {nmax}];
a[n_] := IntegerPartitions[n, All, Select[U, # <= n &]] // Length;
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 12 2021 *)

A223893 Number of partitions of n into at most three distinct primes.

Original entry on oeis.org

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

Offset: 1

Frank M Jackson, Mar 28 2013

nonn

The sequence shows a stronger version of the Goldbach conjecture that for n > 6, n has partitions with at most three distinct primes.

			a(21)=3 as 21 = 2+19 = 3+5+13 = 3+7+11.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000586, A000607, A071335, A185101, A218007, A218469.

a[n_] := Length@Select[IntegerPartitions[n, 3, Prime@Range@PrimePi@n],
Sort@#==Union@# &]; Array[a, 100] (* Giovanni Resta, Mar 29 2013 *)

A333129 Product of all distinct least part primes from all partitions of n into prime parts.

Original entry on oeis.org

1, 1, 2, 3, 2, 10, 6, 14, 6, 6, 30, 66, 30, 78, 42, 30, 30, 510, 210, 570, 210, 210, 330, 690, 2310, 210, 2730, 210, 2310, 6090, 30030, 6510, 2730, 2310, 39270, 2310, 46410, 85470, 3990, 30030, 39270, 94710, 570570, 1291290, 30030, 30030, 903210, 1411410, 746130

Offset: 0

David James Sycamore, Mar 08 2020

nonn

For all n, omega(a(n)) = Omega(a(n)). The prime factorization of each term gives the least part primes of all partitions of n into prime parts.

Product of all terms in row n of A333238. - Alois P. Heinz, Mar 16 2020

			a(2) = 2 because [2] is the only prime partition of 2. a(5) = 10 because the prime partitions of 5 are [2,3] and [5], so the products of all distinct least part primes is 2*5 = 10.

Alois P. Heinz, Table of n, a(n) for n = 0..6269

Cf. A000040, A000607, A001221, A001222, A005117, 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:= n-> (p-> mul(`if`(coeff(p, x, i)>0, i, 1), i=2..n))(b(n, 2, x)):
seq(a(n), n=0..55);  # Alois P. Heinz, Mar 12 2020

a[0] = 1; a[n_] := Times @@ Union[Min /@ IntegerPartitions[n, All, Prime[ Range[PrimePi[n]]]]];
a /@ Range[0, 55] (* Jean-François Alcover, Nov 01 2020 *)

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]]]

cp's OEIS Frontend

A331418 If A331417(n) is the maximum sum of primes of the parts of an integer partition of n, then a(n) = A331417(n) - n + 1.

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A338902 Number of integer partitions of the n-th semiprime into semiprimes.

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A352492 Powerful numbers whose prime indices are all prime numbers.

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A046676 Expansion of 1 + Sum_{k>=1} x^(p_1+p_2+...+p_k)/((1-x)*(1-x^2)*(1-x^3)*...*(1-x^k)) (where p_i are the primes).

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PARI

PARI

A199118 Number of partitions of n into terms of (1,3)-Ulam sequence, cf. A002859.

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A199120 Number of partitions of n into terms of (1,4)-Ulam sequence, cf. A003666.

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Haskell

A199122 Number of partitions of n into terms of (2,3)-Ulam sequence, cf. A001857.

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Haskell

Mathematica

A223893 Number of partitions of n into at most three distinct primes.

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A046676 Expansion of 1 + Sum_{k>=1} x^(p_1+p_2+...+p_k)/((1-x)(1-x^2)(1-x^3)...(1-x^k)) (where p_i are the primes).