A034891 -id:A034891 - cp's OEIS frontend

A379317 Positive integers with a unique even prime index.

3, 6, 7, 12, 13, 14, 15, 19, 24, 26, 28, 29, 30, 33, 35, 37, 38, 43, 48, 51, 52, 53, 56, 58, 60, 61, 65, 66, 69, 70, 71, 74, 75, 76, 77, 79, 86, 89, 93, 95, 96, 101, 102, 104, 106, 107, 112, 113, 116, 119, 120, 122, 123, 130, 131, 132, 138, 139, 140, 141, 142

Offset: 1

Gus Wiseman, Dec 29 2024

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 begin:
   3: {2}
   6: {1,2}
   7: {4}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  19: {8}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  43: {14}
  48: {1,1,1,1,2}

Partitions of this type are counted by A038348 (strict A096911).
For all even parts we have A066207, counted by A035363 (strict A000700).
For no even parts we have A066208, counted by A000009 (strict A035457).
Positions of 1 in A257992.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A349158.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A038550, A087436.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Select[prix[#],EvenQ]]==1&]

A212721 Triangle read by rows: n-th row gives distinct products of partitions of n (A000041).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 27, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14

Offset: 0

Reinhard Zumkeller, Jun 14 2012

A034891(n) = length of n-th row;
A000792(n) = largest term of n-th row;
for n>5: A007918(n) = smallest number <= A000792(n) not occurring in n-th row.

			A000041(6)=11, the 11 partitions and their products of 6:
   1: (1,1,1,1,1,1)   ->   1 * 1 * 1 * 1 * 1 * 1 = 1
   2: (1,1,1,1,2)     ->   1 * 1 * 1 * 1 * 2     = 2
   3: (1,1,1,3)       ->   1 * 1 * 1 * 3         = 3
   4: (1,1,2,2)       ->   1 * 1 * 2 * 2         = 4
   5: (1,1,4)         ->   1 * 1 * 4             = 4
   6: (1,2,3)         ->   1 * 2 * 3             = 6
   7: (1,5)           ->   1 * 5                 = 5
   8: (2,2,2)         ->   2 * 2 * 2             = 8
   9: (2,4)           ->   2 * 4                 = 8
  10: (3,3)           ->   3 * 3                 = 9
  11: (6)             ->                           6,
sorted and duplicates removed: T(6,1..8)=[1,2,3,4,5,6,8,9], A034891(6)=8.
The triangle begins:
   0 |  [1]
   1 |  [1]
   2 |  [1,2]
   3 |  [1,2,3]
   4 |  [1,2,3,4]
   5 |  [1,2,3,4,5,6]
   6 |  [1,2,3,4,5,6,8,9]
   7 |  [1,2,3,4,5,6,7,8,9,10,12]
   8 |  [1,2,3,4,5,6,7,8,9,10,12,15,16,18]
   9 |  [1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,24,27]
  10 |  [1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,25,27,30,32,36].

Reinhard Zumkeller, Rows n = 0..36 of triangle, flattened

Cf. A000041, A000792, A034891.

import Data.List (nub, sort)
a212721 n k = a212721_row n !! (k-1)
a212721_row = nub . sort . (map product) . ps 1 where
   ps x 0 = [[]]
   ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
a212721_tabf = map a212721_row [0..]

row[n_] := Union[Times @@@ IntegerPartitions[n]];
Table[row[n], {n, 0, 10}] (* Jean-François Alcover, Jun 29 2019 *)

[sorted(list(set([mul(p) for p in Partitions(n)]))) for n in range(11)] # Peter Luschny, Dec 13 2015

A159685 Maximal product of distinct primes whose sum is <= n.

Original entry on oeis.org

1, 2, 3, 3, 6, 6, 10, 15, 15, 30, 30, 42, 42, 70, 105, 105, 210, 210, 210, 210, 330, 330, 462, 462, 770, 1155, 1155, 2310, 2310, 2730, 2730, 2730, 2730, 4290, 4290, 6006, 6006, 10010, 15015, 15015, 30030, 30030, 30030, 30030, 39270, 39270, 46410, 46410

Offset: 1

Wouter Meeussen, Apr 19 2009, May 02 2009

nonn

Equivalently, largest value of the LCM of the partitions of n into primes.
Equivalently, maximal number of times a permutation of length n, with prime cycle lengths, can operate on itself before returning to the initial permutation.
If the requirement that primes are distinct is dropped, this becomes A000792. - Charles R Greathouse IV, Jul 10 2012

			A permutation of length 10 can have prime cycle lengths of 2+3+5; so when repeatedly applied to itself, can produce at most 2*3*5 different permutations.
The products of distinct primes whose sum is <= 10 are 1 (the empty product), 2, 3, 5, 7, 2*3=6, 2*5=10, 2*7=14, 3*5=15, 3*7=21, and 2*3*5=30. The maximum is 30, so a(10) = 30. - _Jonathan Sondow_, Jul 06 2012

Alois P. Heinz, Table of n, a(n) for n = 1..10000
M. Deléglise and J.-L. Nicolas, Maximal product of primes whose sum is bounded, arXiv 1207.0603 [math.NT] (2012).
Marc Deléglise and Jean-Louis Nicolas, On the Largest Product of Primes with Bounded Sum, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.8.
Marc Deléglise, Jean-Louis Nicolas, The Landau function and the Riemann hypothesis, Univ. Lyon (France, 2019).

Cf. A077011, A000793, A034891.

with(numtheory):
b:= proc(n,i) option remember; local p; p:= ithprime(max(i,1));
      `if`(n=0, 1, `if`(i<1, 0,
       max(b(n, i-1), `if`(p>n, 0, b(n-p, i-1)*p))))
    end:
a:= proc(n) option remember;
     `if`(n=0, 1, max(b(n, pi(n)), a(n-1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 04 2012

temp=Series[Times @@ (1/(1-q[ # ] x^#)& /@ Prepend[Prime /@ Range[24],1]),{x,0,Prime[24]}]; Table[Max[List @@ Expand[Coefficient[temp,x^n]]/. q[a_]^_ ->q[a] /.q->Identity],{n,64}]
(* Second program: *)
b[n_, i_] := b[n, i] = Module[{p = Prime[Max[i, 1]]}, If[n == 0, 1, If[i < 1, 0, Max[b[n, i-1], If[p > n, 0, b[n-p, i-1]*p]]]]]; a[n_] := a[n] = If[n == 0, 1, Max[b[n, PrimePi[n]], a[n-1]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 05 2013, translated from Alois P. Heinz's Maple program *)

a(n) <= A002809(n) and A008475(a(n)) <= n (see (1.2) and (1.4) in Deléglise-Nicolas 2012). - Jonathan Sondow, Jul 04 2012.

A339380 Number of partitions of n into an even number of primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 5, 4, 9, 7, 14, 11, 22, 18, 33, 27, 48, 40, 69, 58, 97, 82, 134, 114, 183, 157, 246, 212, 327, 284, 431, 376, 562, 493, 728, 640, 934, 825, 1191, 1056, 1508, 1341, 1899, 1694, 2377, 2126, 2960, 2654, 3668, 3297, 4523, 4075, 5554, 5015, 6792, 6145

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(6) = 5 because we have [5, 1], [3, 3], [3, 1, 1, 1], [2, 2, 1, 1] and [1, 1, 1, 1, 1, 1].

Index entries for sequences related to partitions

Cf. A000607, A008578, A027187, A034891, A184171, A184172, A184198, A184199, A338826, A339381, A339382, A339383.

b:= proc(n, i, t) option remember; (p->
      `if`(n=0, t, `if`(i<0, 0, b(n, i-1, t)+
      `if`(p>n, 0, b(n-p, i, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 02 2020

nmax = 55; CoefficientList[Series[(1/2) ((1/(1 - x)) Product[1/(1 - x^Prime[k]), {k, 1, nmax}] + (1/(1 + x)) Product[1/(1 + x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]
Table[Count[(Boole[PrimeQ/@(IntegerPartitions[n]/.(1->2))]),?(EvenQ[Length[#]] && FreeQ[ #,0]&)],{n,0,60}] (* _Harvey P. Dale, Aug 20 2024 *)

G.f.: (1/2) * ((1/(1 - x)) * Product_{k>=1} 1 / (1 - x^prime(k)) + (1/(1 + x)) * Product_{k>=1} 1 / (1 + x^prime(k))).

a(n) = (A034891(n) + A338826(n)) / 2.

A339381 Number of partitions of n into an odd number of primes (counting 1 as a prime).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 3, 7, 5, 11, 9, 18, 14, 27, 22, 40, 33, 58, 48, 82, 69, 114, 97, 157, 134, 212, 183, 284, 246, 376, 327, 493, 431, 640, 562, 825, 728, 1056, 934, 1341, 1191, 1694, 1508, 2126, 1899, 2654, 2377, 3297, 2960, 4075, 3668, 5015, 4523, 6145, 5554, 7499

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(6) = 3 because we have [3, 2, 1], [2, 2, 2] and [2, 1, 1, 1, 1].

Index entries for sequences related to partitions

Cf. A000607, A008578, A027193, A034891, A184171, A184172, A184198, A184199, A338826, A339380, A339382, A339383.

b:= proc(n, i, t) option remember; (p->
      `if`(n=0, t, `if`(i<0, 0, b(n, i-1, t)+
      `if`(p>n, 0, b(n-p, i, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 02 2020

nmax = 55; CoefficientList[Series[(1/2) ((1/(1 - x)) Product[1/(1 - x^Prime[k]), {k, 1, nmax}] - (1/(1 + x)) Product[1/(1 + x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: (1/2) * ((1/(1 - x)) * Product_{k>=1} 1 / (1 - x^prime(k)) - (1/(1 + x)) * Product_{k>=1} 1 / (1 + x^prime(k))).

a(n) = (A034891(n) - A338826(n)) / 2.

A298602 Expansion of (1 - x)*Product_{k>=1} (1 - x^prime(k)).

Original entry on oeis.org

1, -1, -1, 0, 1, 0, 0, 0, 1, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, -1, 1, 0, 0, -1, 2, 0, -1, -1, 1, -1, 1, -1, 2, 0, 0, -2, 2, -2, 0, 0, 3, -2, 1, -2, 2, -1, 0, -3, 5, -1, 0, -3, 3, -3, 3, -3, 3, -1, 2, -5, 6, -4, 1, -2, 6, -5, 3, -6, 5, -2, 4, -8, 9, -5, 3, -5, 7, -8, 7, -8, 8, -4, 5

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

sign

The difference between the number of partitions of n into an even number of distinct prime parts (including 1) and the number of partitions of n into an odd number of distinct prime parts (including 1).

Convolution inverse of A034891.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000586, A000607, A034891, A036497, A046675 (partial sums).

nmax = 82; CoefficientList[Series[(1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: (1 - x)*Product_{k>=1} (1 - x^prime(k)).

A341719 Number of partitions of n into 9 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 11, 11, 16, 15, 23, 21, 30, 27, 39, 35, 51, 44, 63, 54, 78, 67, 97, 81, 116, 96, 139, 115, 166, 133, 194, 155, 227, 180, 265, 206, 305, 236, 351, 271, 403, 305, 460, 346, 522, 391, 592, 438, 668, 489, 751, 551, 844, 608, 942, 674, 1050, 750

Offset: 9

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A034891, A259200, A341945, A341946, A341947, A341948, A341949, A341950, A341951, A341972.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 10)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 9):
seq(a(n), n=9..68);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 10}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 9];
Table[a[n], {n, 9, 68}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

A341972 Number of partitions of n into 10 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 11, 11, 17, 16, 24, 23, 32, 30, 43, 39, 56, 51, 71, 63, 89, 78, 111, 97, 134, 116, 164, 139, 197, 166, 232, 194, 275, 227, 324, 265, 374, 305, 438, 351, 505, 403, 578, 460, 665, 522, 760, 592, 859, 668, 978, 751, 1105, 844, 1239, 942, 1394

Offset: 10

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A034891, A259201, A341719, A341945, A341946, A341947, A341948, A341949, A341950, A341951.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 11)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 10):
seq(a(n), n=10..68);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 11}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 10];
Table[a[n], {n, 10, 68}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

A347622 Number of partitions of n into at most 2 prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 3, 2, 3, 1, 3, 1, 4, 1, 3, 0, 2, 1, 4, 2, 3, 1, 4, 0, 4, 1, 3, 1, 3, 1, 5, 2, 4, 1, 4, 1, 6, 1, 4, 0, 3, 1, 6, 1, 3, 0, 4, 1, 7, 2, 4, 1, 5, 0, 6, 1, 3, 1, 5, 1, 7, 2, 6, 1, 5

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A008578, A034891, A218007, A341945, A347552, A347623, A347624, A347625.

A347623 Number of partitions of n into at most 4 prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 9, 10, 10, 12, 11, 14, 13, 15, 14, 18, 15, 20, 17, 22, 18, 25, 19, 26, 20, 26, 21, 30, 23, 33, 24, 35, 26, 38, 27, 40, 28, 41, 31, 48, 32, 50, 34, 53, 36, 58, 37, 60, 39, 61, 40, 67, 40, 68, 41, 72, 45, 79, 47, 82, 49

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A008578, A034891, A218007, A341947, A347578, A347622, A347624, A347625.

cp's OEIS Frontend

A379317 Positive integers with a unique even prime index.

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A212721 Triangle read by rows: n-th row gives distinct products of partitions of n (A000041).

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A159685 Maximal product of distinct primes whose sum is <= n.

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Maple

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A339380 Number of partitions of n into an even number of primes (counting 1 as a prime).

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Maple

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A339381 Number of partitions of n into an odd number of primes (counting 1 as a prime).

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Maple

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A298602 Expansion of (1 - x)*Product_{k>=1} (1 - x^prime(k)).

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Mathematica

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A341719 Number of partitions of n into 9 primes (counting 1 as a prime).

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Maple

Mathematica

A341972 Number of partitions of n into 10 primes (counting 1 as a prime).

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