A046063 -id:A046063 - cp's OEIS frontend

A330994 Numerator of P(n)/Q(n) = A000041(n)/A000009(n).

1, 1, 2, 3, 5, 7, 11, 3, 11, 15, 21, 14, 77, 101, 135, 176, 231, 297, 385, 245, 627, 198, 1002, 1255, 1575, 979, 812, 1505, 1859, 4565, 1401, 3421, 2783, 1449, 6155, 4961, 17977, 21637, 26015, 31185, 1778, 2123, 26587, 63261, 75175, 44567, 17593, 8911, 49091

Offset: 0

Gus Wiseman, Jan 08 2020

An integer partition of n is a finite, nonincreasing sequence of positive integers (parts) summing to n. It is strict if the parts are all different. Integer partitions and strict integer partitions are counted by A000041 and A000009 respectively.

The denominators are A330995.
The rounded quotients are A330996.
The same for factorizations is A331023.
Cf. A000009, A000041, A003238, A005117, A035359, A046063, A331022.

Table[PartitionsP[n]/PartitionsQ[n],{n,0,100}]//Numerator

A330994/A330995 = A000041/A000009.

A330995 Denominator P(n)/Q(n) = A000041(n)/A000009(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 1, 3, 4, 5, 3, 15, 18, 22, 27, 32, 38, 46, 27, 64, 19, 89, 104, 122, 71, 55, 96, 111, 256, 74, 170, 130, 64, 256, 195, 668, 760, 864, 982, 53, 60, 713, 1610, 1816, 1024, 384, 185, 970, 3264, 1829, 4097, 4582, 5120, 5718, 3189, 7108, 2639

Offset: 0

Gus Wiseman, Jan 08 2020

An integer partition of n is a finite, nonincreasing sequence of positive integers (parts) summing to n. It is strict if the parts are all different. Integer partitions and strict integer partitions are counted by A000041 and A000009 respectively.

Conjecture: The only 1's occur at n = 0, 1, 2, 7.

The numerators are A330994.
The rounded quotients are A330996.
The same for factorizations is A331024.
Cf. A000009, A000041, A001055, A003238, A005117, A035359, A045778, A046063, A331022.

Table[PartitionsP[n]/PartitionsQ[n],{n,0,100}]//Denominator

A330994/A330995 = A000041/A000009.

A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 12, 13, 21, 22, 25, 33, 38, 41, 45, 46, 49, 50, 55, 57, 58, 63

Offset: 1

Gus Wiseman, Jan 07 2020

This multiset (row k of A305936) is generally not the same as the multiset of prime indices of k. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Also numbers whose inverse prime shadow has a prime number of factorizations. A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798. The inverse prime shadow of k is the least number whose prime exponents are the prime indices of k.

			The multiset partitions for n = 1..6:
  {11}    {12}    {111}      {1111}        {123}      {1112}
  {1}{1}  {1}{2}  {1}{11}    {1}{111}      {1}{23}    {1}{112}
                  {1}{1}{1}  {11}{11}      {2}{13}    {11}{12}
                             {1}{1}{11}    {3}{12}    {2}{111}
                             {1}{1}{1}{1}  {1}{2}{3}  {1}{1}{12}
                                                      {1}{2}{11}
                                                      {1}{1}{1}{2}
The factorizations for n = 1..8:
  4    6    8      16       30     24       32         60
  2*2  2*3  2*4    2*8      5*6    3*8      4*8        2*30
            2*2*2  4*4      2*15   4*6      2*16       3*20
                   2*2*4    3*10   2*12     2*2*8      4*15
                   2*2*2*2  2*3*5  2*2*6    2*4*4      5*12
                                   2*3*4    2*2*2*4    6*10
                                   2*2*2*3  2*2*2*2*2  2*5*6
                                                       3*4*5
                                                       2*2*15
                                                       2*3*10
                                                       2*2*3*5

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

The same for powers of 2 (instead of primes) is A330990.
Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of integer partitions is prime are A046063.
Numbers whose number of strict integer partitions is prime are A035359.
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).
Factorizations of a number's inverse prime shadow are A318284.
Numbers with a prime number of factorizations are A330991.
Cf. A033833, A045783, A056239, A181819, A181821, A305936, A318286, A325755, A330972, A330973, A330998.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
unsh[n_]:=Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[30],PrimeQ[Length[facs[unsh[#]]]]&]

A001055(A181821(a(n))) belongs to A000040.

A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A080257 in lacking 60.

			Strict factorizations of selected terms:
  (6)    (12)   (24)     (48)     (216)
  (2*3)  (2*6)  (3*8)    (6*8)    (3*72)
         (3*4)  (4*6)    (2*24)   (4*54)
                (2*12)   (3*16)   (6*36)
                (2*3*4)  (4*12)   (8*27)
                         (2*3*8)  (9*24)
                         (2*4*6)  (12*18)
                                  (2*108)
                                  (3*8*9)
                                  (4*6*9)
                                  (2*3*36)
                                  (2*4*27)
                                  (2*6*18)
                                  (2*9*12)
                                  (3*4*18)
                                  (3*6*12)
                                  (2*3*4*9)

The version for strict integer partitions is A035359.
The version for integer partitions is A046063.
The version for set partitions is A051130.
The non-strict version is A330991.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
Numbers whose number of strict factorizations is odd are A331230.
Numbers whose number of strict factorizations is even are A331231.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A318286, A328966, A330992, A330993, A330997, A331023/A331024, A331050, A331051, A331200, A331232.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],PrimeQ[Length[strfacs[#]]]&]

A051143 Numbers k such that the k-th prime is a partition number.

Original entry on oeis.org

1, 2, 3, 4, 5, 26, 2061, 702993, 307058572, 3350187739, 9088200428, 43794115173, 51932790219, 378210209388, 521301342188, 297064987225918, 19677201507658441, 437852535314831447, 1673669998972800207, 29252504332047744188, 42842701894337201916

Offset: 1

G. L. Honaker, Jr.

Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000040, A000041, A000720, A046063, A049575.

PrimePi@Select[PartitionsP@Range@301, PrimeQ@# &] (* Robert G. Wilson v, Nov 14 2005 *)

a(n) = A000720(A049575(n)).

a(16)-a(21) using Kim Walisch's primecount, from Amiram Eldar, Jul 26 2019

A113499 Numbers n such that P(11*n) is prime where P(n) is the partition number.

Original entry on oeis.org

7, 12, 40, 75, 163, 228, 261, 288, 322, 331, 618, 678, 768, 926, 2990, 3821, 4852, 5726, 10802, 11710, 12006, 12635, 14470, 18097, 22156, 25776, 29142, 32692, 36965, 48830, 51821, 56433, 58008, 63757, 64433, 67545, 68391, 69850, 73723, 77498, 77770

Offset: 1

Parthasarathy Nambi, Jan 10 2006

nonn

n belongs to this sequence if and only if 11*n belongs to A046063.

			If n=163 then P(11*n) = 11820527237297139926370474832027317722017807 (prime).

Max Alekseyev, Table of n, a(n) for n = 1..467

Cf. A111036, A111045, A111389.

For[n = 1, n < 1000, n++, If[PrimeQ[PartitionsP[11*n]], If[ProvablePrimeQ[PartitionsP[11*n]], Print[n]]]] (* Stefan Steinerberger *)
Do[ If[ PrimeQ@ PartitionsP[11n], Print@n], {n, 3000}] (* Robert G. Wilson v *)

More terms from Stefan Steinerberger and Robert G. Wilson v, Jan 12 2006

Terms a(28) onward from Max Alekseyev, Dec 18 2011

A121992 List of Eisenstein triples: {a,b,c} such that {a^2 + b^2 - a*b - c^2 = 0} and abs(a - b) > 0, sorted by greatest a.

Original entry on oeis.org

3, 8, 7, 5, 8, 7, 5, 21, 19, 6, 16, 14, 7, 15, 13, 8, 3, 7, 8, 5, 7, 8, 15, 13, 9, 24, 21, 10, 16, 14, 15, 7, 13, 15, 8, 13, 15, 24, 21, 16, 6, 14, 16, 10, 14, 16, 21, 19, 21, 5, 19, 21, 16, 19, 24, 9, 21, 24, 15, 21

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 10 2006

			Grouped as threes: {{3, 8, 7}, {5, 8, 7}, {5, 21, 19}, {6, 16, 14}, {7, 15, 13}, {8, 3, 7}, {8, 5, 7}, {8, 15, 13}, {9, 24, 21}, {10, 16, 14}, {15, 7, 13}, {15, 8, 13}, {15, 24, 21}, {16,6, 14}, {16, 10, 14}, {16, 21, 19}, {21, 5, 19}, {21, 16, 19}, {24, 9, 21}, {24, 15, 21}}

Ross Honsberger, "Mathematical Delights", MAA, 2004, p. 64.

Park City Mathematics Institute, Session 13 Number Theory, Summer 2001. A similar factoring allows for the generation of Eisenstein triples, which are numbers that form the sides of a triangle with a 60-degree angle.

Cf. A046063.

f[a_, b_, c_] = If[c^2 - a^2 - b^2 + a*b == 0 && Abs[a - b] > 0, {a, b, c}, {}] a0 = Flatten[Delete[Union[Table[Delete[Union[Table[Flatten[Table[f[a, b, c], {c, 1, 25}]], {b, 1, 25}]], 1], {a, 1, 25}]], 1], 1] b0 = Sort[a0] Flatten[b0]

T(n) = {a(n), b(n), c(n)} such that a(n)^2 + b(n)^2 - a(n)*b(n) - c(n)^2 = 0 and abs(a(n) - b(n)) > 0.

A330996 Nearest integer to P(n)/Q(n) = A000041(n)/A000009(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 30, 32, 34, 35, 37, 39, 41, 44, 46, 48, 51, 53, 56, 59, 61, 64, 68, 71, 74, 78, 81, 85, 89, 93, 97, 101, 106, 111, 115, 120, 126

Offset: 0

Gus Wiseman, Jan 08 2020

nonn

Conjecture: This sequence is nondecreasing. More generally, the rational sequence A000041(n)/A000009(n) is nondecreasing for n > 5.

The numerators are A330994.
The denominators are A330995.
The same for factorizations is A331048.
Cf. A000009, A000041, A001055, A003238, A035359, A045778, A046063, A331022.

Table[Round[PartitionsP[n]/PartitionsQ[n]],{n,0,100}]

A355704 Indices k of partition function p where p(k) and p(k) + 2 are twin primes.

Original entry on oeis.org

3, 4, 6, 13, 2335166

Offset: 1

Serge Batalov, Jul 14 2022

Because asymptotically size of partitions number function p(n) ~ O(exp(sqrt(n))), and probability of primality of p(n) ~ O(1/sqrt(n)) and combined probability of primality of p(n) and p(n)+-2 is ~ O(1/n), the sum of the prime probabilities is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

a(6) > 10^7.

			13 is a term because A000041(13) = 101 is prime and 103 is prime.

Cf. A000041, A046063, A072213, A285086, A285087, A285088, A284594, A355705, A355706.

for(n=1, 2500, if(ispseudoprime(p=numbpart(n))&&ispseudoprime(p+2), print1(n,", ")))

A355706 Indices k of partition function p where p(k) is a twin prime.

Original entry on oeis.org

3, 4, 5, 6, 13, 186, 3542, 2335166

Offset: 1

Serge Batalov, Jul 15 2022

Because asymptotically the size of the partition number function p(n) is ~ O(exp(sqrt(n))), and the probability of primality of p(n) is ~ O(1/sqrt(n)) and the combined probability of primality of p(n) and p(n)+-2 is ~ O(1/n), the sum of the prime probabilities is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
p(2335166), a 1696-digit number, was known to be prime and proven prime by F. Morain using his software (ca. April 2001), but the primality of p(2335166)+2 was found by targeted search (for this sequence) in July 2022.
a(9) > 10^7.

			13 is a term because A000041(13) = 101 is prime and 101 and 103 are twin primes.

Subsequence of A046063.
Union of A355704 and A355705.
Cf. A000041, A072213, A285086, A285087, A285088, A284594.

for(n=1, 3600, if(ispseudoprime(p=numbpart(n))&&(ispseudoprime(p-2)||ispseudoprime(p+2)), print1(n, ", ")))

cp's OEIS Frontend

A330994 Numerator of P(n)/Q(n) = A000041(n)/A000009(n).

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A330995 Denominator P(n)/Q(n) = A000041(n)/A000009(n).

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A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

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A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

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A051143 Numbers k such that the k-th prime is a partition number.

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A113499 Numbers n such that P(11*n) is prime where P(n) is the partition number.

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A121992 List of Eisenstein triples: {a,b,c} such that {a^2 + b^2 - a*b - c^2 = 0} and abs(a - b) > 0, sorted by greatest a.

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A330996 Nearest integer to P(n)/Q(n) = A000041(n)/A000009(n).

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A355704 Indices k of partition function p where p(k) and p(k) + 2 are twin primes.