Displaying 1-10 of 49 results found.
2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557, 74878248419470886233, 1394313503224447816939
MATHEMATICA
Select[PartitionsP[Range[1000]], PrimeQ] (* Harvey P. Dale, Mar 11 2013 *)
PROG
(PARI) for(n=0, 10^3, my(p=numbpart(n)); if(isprime(p), print1(p, ", "))); \\ Joerg Arndt, May 09 2013
Number of partitions-into-distinct-parts of n ( A000009) is a prime.
+10
23
3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, 3335367, 16168775, 37472505, 52940251, 78840125, 81191852
EXAMPLE
Strict partitions of a(1) = 3 through a(4) = 7:
(3) (4) (5) (7)
(2,1) (3,1) (3,2) (4,3)
(4,1) (5,2)
(6,1)
(4,2,1)
(End)
CROSSREFS
The version for powers of 2 instead of primes is A331022. The version for factorizations instead of strict partitions is A330991. The version for strict factorizations instead of strict partitions is A331201.
KEYWORD
nonn,nice,hard,more,changed
Positive integers whose number of factorizations into factors > 1 ( A001055) is a prime number ( A000040).
+10
17
4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 24, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 46, 49, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 69, 70, 74, 77, 78, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 110, 111, 114, 115, 118, 119
EXAMPLE
Factorizations of selected terms:
(4) (8) (16) (24) (60) (96)
(2*2) (2*4) (2*8) (3*8) (2*30) (2*48)
(2*2*2) (4*4) (4*6) (3*20) (3*32)
(2*2*4) (2*12) (4*15) (4*24)
(2*2*2*2) (2*2*6) (5*12) (6*16)
(2*3*4) (6*10) (8*12)
(2*2*2*3) (2*5*6) (2*6*8)
(3*4*5) (3*4*8)
(2*2*15) (4*4*6)
(2*3*10) (2*2*24)
(2*2*3*5) (2*3*16)
(2*4*12)
(2*2*3*8)
(2*2*4*6)
(2*3*4*4)
(2*2*2*12)
(2*2*2*2*6)
(2*2*2*3*4)
(2*2*2*2*2*3)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[100], PrimeQ[Length[facs[#]]]&]
CROSSREFS
Numbers whose number of strict integer partitions is prime are A035359. Numbers whose number of integer partitions is prime are A046063. Numbers whose number of set partitions is prime are A051130. Numbers whose number of factorizations is a power of 2 are A330977. The least number with prime(n) factorizations is A330992(n).
Numbers with an odd number of partitions.
+10
15
0, 1, 3, 4, 5, 6, 7, 12, 13, 14, 16, 17, 18, 20, 23, 24, 29, 32, 33, 35, 36, 37, 38, 39, 41, 43, 44, 48, 49, 51, 52, 53, 54, 56, 60, 61, 63, 67, 68, 69, 71, 72, 73, 76, 77, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 93, 95, 99, 102, 104, 105, 107, 111, 114, 115, 118, 119, 121
COMMENTS
Also, numbers having an odd number of partitions into distinct odd parts; that is, numbers m such that A000700(m) is odd. For example, 16 is in the list since 16 has 5 partitions into distinct odd parts, namely, 1 + 15, 3 + 13, 5 + 11, 7 + 9 and 1 + 3 + 5 + 7. See Formula section for a proof. - Peter Bala, Jan 22 2017
FORMULA
Sum_{n>=0} x^a(n) = (1 + x)*(1 + x^3)*(1 + x^5)*... taken modulo 2. Proof: Product_{n>=1} 1 + x^(2*n-1) = Product_{n>=1} (1 - x^(4*n-2))/(1 - x^(2*n-1)) = Product_{n>=1} (1 - x^(2*n))*(1 - x^(4*n-2))/( (1 - x^(2*n)) * (1 - x^(2*n-1)) ) = ( 1 + 2*Sum_{n>=1} (-1)^n*x^(2*n^2) )/(Product_{n>=1} (1 - x^n)) == 1/( Product_{n>=1} (1 - x^n) ) (mod 2). (End)
EXAMPLE
The partitions of the initial terms are:
(1) (3) (4) (5) (6) (7)
(21) (22) (32) (33) (43)
(111) (31) (41) (42) (52)
(211) (221) (51) (61)
(1111) (311) (222) (322)
(2111) (321) (331)
(11111) (411) (421)
(2211) (511)
(3111) (2221)
(21111) (3211)
(111111) (4111)
(22111)
(31111)
(211111)
(1111111)
(End)
MAPLE
N:= 1000: # to get all terms <= N
V:= Vector(N+1):
V[1]:= 1:
for i from 1 to (N+1)/2 do
V[2*i..N+1]:= V[2*i..N+1] + V[1..N-2*i+2] mod 2
od:
MATHEMATICA
f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
PROG
(PARI) for(n=0, 200, if(numbpart(n)%2==1, print1(n", "))) \\ Altug Alkan, Nov 02 2015 (Haskell)
import Data.List (findIndices)
a052002 n = a052002_list !! (n-1)
a052002_list = findIndices odd a000041_list
CROSSREFS
The version for prime instead of odd numbers is A046063. The version for squarefree instead of odd numbers is A038630. The version for set partitions appears to be A032766. The version for factorizations is A331050. The version for strict factorizations is A331230.
Numbers n such that p(6n) is prime, where p(n) is the number of partitions of n.
+10
14
1, 6, 22, 28, 31, 36, 61, 83, 91, 181, 216, 263, 356, 417, 418, 428, 528, 557, 777, 1133, 1243, 1408, 2170, 2708, 3046, 3867, 5100, 5540, 5662, 7418, 9397, 12110, 12797, 14787, 16161, 16482, 18022, 19431, 19667, 21180, 22011, 22720, 23560, 27903
EXAMPLE
If n=91 then p(6n) = 27833079238879849385687 (prime).
CROSSREFS
Cf. A000041, A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170.
Numbers n such that P(4n) is prime, where P(m) is the number of partitions of m.
+10
14
1, 9, 33, 42, 47, 53, 54, 110, 324, 534, 627, 642, 683, 728, 792, 1114, 2112, 2228, 2323, 2770, 3007, 3255, 3368, 3760, 4062, 4569, 6139, 7650, 7939, 8138, 8310, 8493, 8674, 9122, 9407, 10345, 11127, 13343, 14713, 15442, 15632, 16358, 16904, 18165, 19303
EXAMPLE
If n=110 then P(4*n) = 74878248419470886233 (prime).
CROSSREFS
Cf. A000041, A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170.
Numbers n such that p(3n) is prime, where p(n) is the number of partitions of n.
+10
14
1, 2, 12, 44, 56, 62, 72, 122, 139, 166, 175, 182, 245, 251, 275, 362, 432, 526, 712, 831, 834, 836, 856, 909, 957, 1009, 1056, 1114, 1554, 2266, 2486, 2816, 3967, 4340, 5416, 6092, 6837, 6959, 7215, 7255, 7439, 7734, 9655, 10200, 11080, 11324, 11361, 12819
FORMULA
Elements of A046063 which are == 0 (mod 3) divided by 3
EXAMPLE
If n=72 then p(3n) = 15285151248481 (prime).
CROSSREFS
Cf. A000041, A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170.
Numbers n such that p(2n) is prime, where p(n) is the number of partitions of n.
+10
13
1, 2, 3, 18, 66, 84, 93, 94, 106, 108, 151, 183, 220, 249, 273, 329, 543, 648, 789, 793, 1068, 1251, 1254, 1284, 1366, 1456, 1549, 1584, 1671, 1771, 2059, 2131, 2228, 2331, 2501, 3399, 3729, 4224, 4456, 4646, 4999, 5093, 5540, 6014, 6510, 6736, 7520, 8124
COMMENTS
2n-th partition number ( A000041(2n)) is prime.
MATHEMATICA
Select[ Range[9137], PrimeQ[ PartitionsP[2# ]] &]
CROSSREFS
Cf. A000041, A046063, A068413, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170.
Numbers n such that p(5n) is prime, where p(n) is the number of partitions of n.
+10
13
1, 88, 105, 147, 165, 217, 1481, 2216, 2579, 2604, 3008, 3658, 3694, 4329, 4353, 4447, 4534, 5074, 5793, 6120, 6578, 6648, 7861, 7994, 8276, 8851, 9421, 10371, 12350, 12359, 12389, 13010, 13345, 13479, 14532, 14727, 16461, 19313, 19466, 20354
MATHEMATICA
Select[ Range[20780], PrimeQ[PartitionsP[5# ]] &]
CROSSREFS
Cf. A000041, A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170.
Numbers n such that p(7n) is prime, where p(n) is the number of partitions of n.
+10
13
11, 24, 75, 78, 94, 105, 155, 211, 293, 416, 506, 666, 1860, 3013, 3508, 3811, 4869, 5615, 5710, 8824, 8841, 8998, 10380, 11014, 11779, 13795, 14276, 15285, 18014, 19456, 19855, 22435, 23343, 23391, 26328, 30608, 31380, 32074, 32810, 33459
MATHEMATICA
Select[ Range[28571], PrimeQ[PartitionsP[7# ]] &]
CROSSREFS
Cf. A000041, A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170.
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