A000009 -id:A000009 - cp's OEIS frontend

A233417 a(n) = |{0 < k <= n/2: q(k)*q(n-k) + 1 is prime}|, where q(.) is the strict partition function (A000009).

0, 1, 1, 2, 2, 2, 2, 3, 2, 2, 4, 5, 3, 1, 5, 7, 1, 3, 4, 4, 3, 2, 5, 3, 6, 6, 1, 6, 8, 6, 6, 4, 7, 7, 3, 5, 5, 6, 6, 5, 5, 3, 7, 8, 7, 7, 8, 8, 6, 4, 8, 8, 5, 3, 8, 8, 5, 15, 6, 8, 3, 9, 5, 6, 7, 9, 4, 6, 8, 9, 5, 4, 7, 8, 7, 6, 10, 9, 9, 8, 6, 6, 9, 9, 7, 12, 5, 10, 7, 7, 5, 3, 8, 10, 7, 5, 9, 7, 4, 5

Offset: 1

Zhi-Wei Sun, Dec 09 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Similarly, for any integer n > 5, there is a positive integer k < n with q(k)*q(n-k) - 1 prime.

(ii) Let n > 1 be an integer. Then p(k) + q(n-k)^2 is prime for some 0 < k < n, where p(.) is the partition function (A000041). If n is not equal to 8, then k^3 + q(n-k)^2 is prime for some 0 < k < n.

			a(14) = 1 since q(1)*q(13) + 1 = 1*18 + 1 = 19 is prime.
a(17) = 1 since q(4)*q(13) + 1 = 2*18 + 1 = 37 is prime.
a(27) = 1 since q(13)*q(14) + 1 = 18*22 + 1 = 397 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.

Cf. A000009, A000040, A232504, A233307, A233346, A233359, A233390, A233393.

a[n_]:=Sum[If[PrimeQ[PartitionsQ[k]*PartitionsQ[n-k]+1],1,0],{k,1,n/2}]
Table[a[n],{n,1,100}]

A235343 a(n) = |{0 < k < n: f(n,k) - 1, f(n,k) + 1 and q(f(n,k)) + 1 are all prime with f(n,k) = phi(k) + phi(n-k)/4}|, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 06 2014

nonn

Conjecture: (i) a(n) > 0 for all n >= 60.
(ii) For any integer n > 1234, there is a positive integer k < n such that g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime, where g(n,k) = phi(k) + phi(n-k)/8.
Clearly, part (i) implies that there are infinitely many primes of the form q(m) + 1 with m - 1 and m + 1 also prime, and part (ii) implies that there are infinitely many primes of the form q(m) - 1 with m - 1 and m + 1 also prime. As log q(m) is asymptotically equivalent to  pi*sqrt(m/3), the conjecture is much stronger than the twin prime conjecture.
We have verified parts (i) and (ii) for n up to 100000 and 60000 respectively.

			a(50) = 1 since phi(34) + phi(16)/4 = 18 with 18 - 1, 18 + 1 and q(18) + 1 = 47 all prime.
a(215) = 1 since phi(87) + phi(128)/4 = 72 with 72 - 1, 72 + 1 and q(72) + 1 = 36353 all prime.
a(645) = 1 since phi(365) + phi(280)/4 = 312 with 312 - 1, 312 + 1 and q(312) + 1 = 207839472391 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000010, A000040, A001359, A006512, A014574, A234514, A234567, A234615, A235344, A235346, A235356, A235357, A235358.

f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/4
p[n_,k_]:=PrimeQ[f[n,k]-1]&&PrimeQ[f[n,k]+1]&&PrimeQ[PartitionsQ[f[n,k]]+1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A235356 Primes of the form q(m) + 1 with m - 1 and m + 1 both prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

3, 5, 47, 1427, 36353, 525017, 24782061071, 46193897033, 207839472391, 58195383726460417, 20964758762885249107969, 47573613463034233651201, 35940172290335689735986241, 39297101749677990678763409480449, 538442167350331131544523981355841

Offset: 1

Zhi-Wei Sun, Jan 07 2014

nonn

Though the primes in this sequence are very rare, by part (i) of the conjecture in A235343 there should be infinitely many such primes.
See A235344 for a list of known numbers m with m - 1, m + 1 and q(m) + 1 all prime.
See also A235357 for a similar sequence.

			a(1) = 3 since 3 = q(4) + 1 with 4 - 1 and 4 + 1 both prime.
a(2) = 5 since 5 = q(6) + 1 with 6 - 1 and 6 + 1 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..30

Cf. A000009, A000040, A014574, A235343, A235344, A235346, A235357.

f[n_]:=A235344(n)
Table[PartitionsQ[f[n]]+1,{n,1,15}]

a(n) = A000009(A235344(n)) + 1.

A235357 Primes of the form q(m) - 1 with m - 1 and m + 1 both prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

3, 4919887991, 28253252977151, 20964758762885249107967, 47573613463034233651199, 12796446358667905839216959, 10712934162879755412803989317623807, 33014011446550388413724585366558782455972162239

Offset: 1

Zhi-Wei Sun, Jan 07 2014

Though the primes in this sequence are very rare, by part (ii) of the conjecture in A235343, there should be infinitely many such primes.
See A235346 for a list of known numbers m with m - 1, m + 1 and q(m) - 1 all prime.
See also A235356 for a similar sequence.

			a(1) = 3 since 3 = q(6) - 1 with 6 - 1 and 6 + 1 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..25

Cf. A000009, A000040, A014574, A235343, A235344, A235346, A235356.

g[n_]:=A235346(n)
Table[PartitionsQ[g[n]]-1,{n,1,10}]

a(n) = A000009(A235346(n)) - 1.

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

Original entry on oeis.org

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.

A152827 Partial products of PartitionsQ numbers (A000009).

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 48, 240, 1440, 11520, 115200, 1382400, 20736000, 373248000, 8211456000, 221709312000, 7094697984000, 269598523392000, 12401532076032000, 669682732105728000, 42859694854766592000

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 13 2008

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..190

Cf. A000009, A000041, A058694.

Table[Product[PartitionsQ[k],{k,0,n}],{n,0,33}]
FoldList[Times,1,PartitionsQ[Range[20]]] (* Harvey P. Dale, Sep 16 2011 *)

log(a(n)) ~ 2*Pi*n^(3/2)/(3*sqrt(3)) * (1 - 9*sqrt(3)*log(n)/(8*Pi*sqrt(n)) + 3*sqrt(3)*(3 - 8*log(2) - log(3))/(8*Pi*sqrt(n)) + (13/16 - 27/(8*Pi^2))/n). - Vaclav Kotesovec, Nov 30 2015

Offset corrected by Vaclav Kotesovec, Nov 29 2015

A236412 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with p(m)^2 + q(m)^2 prime}|, where phi(.) is Euler's totient, p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 0, 1, 2, 3, 3, 4, 5, 3, 4, 4, 7, 4, 5, 5, 3, 3, 4, 5, 4, 3, 6, 8, 3, 3, 3, 7, 3, 7, 4, 5, 3, 6, 3, 2, 3, 6, 3, 3, 2, 5, 1, 4, 6, 4, 3, 3, 7, 5, 3, 3, 3, 4, 1, 5, 4, 3, 2, 4, 3, 6, 2, 5, 6, 4, 5, 2, 1, 6, 4, 4, 2, 11, 1, 6, 3, 5, 6, 7, 2, 4, 4, 2, 3, 2

Offset: 1

Zhi-Wei Sun, Jan 24 2014

nonn

Conjecture: a(n) > 0 for all n > 17.
We have verified this for n up to 65000.
The conjecture implies that there are infinitely positive integers m with p(m)^2 + q(m)^2 prime. See A236413 for a list of such numbers m. See also A236414 for primes of the form p(m)^2 + q(m)^2.

			a(15) = 1 since phi(2)/2 + phi(13)/8 = 1/2 + 12/8 = 2 with p(2)^2 + q(2)^2 = 2^2 + 1^2 = 5 prime.
a(69) = 1 since phi(5)/2 + phi(64)/8 = 2 + 4 = 6 with p(6)^2 + q(6)^2 = 11^2 + 4^2 = 137 prime.
a(89) = 1 since phi(73)/2 + phi(16)/8 = 36 + 1 = 37 with p(37)^2 + q(37)^2 = 21637^2 + 760^2 = 468737369 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000010, A000040, A000041, A233307, A236374, A236413, A236414.

p[n_]:=IntegerQ[n]&&PrimeQ[PartitionsP[n]^2+PartitionsQ[n]^2]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236413 Positive integers m with p(m)^2 + q(m)^2 prime, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

Original entry on oeis.org

1, 2, 3, 4, 6, 17, 24, 37, 44, 95, 121, 162, 165, 247, 263, 601, 714, 742, 762, 804, 1062, 1144, 1149, 1323, 1508, 1755, 1833, 1877, 2330, 2380, 2599, 3313, 3334, 3368, 3376, 3395, 3504, 3688, 3881, 4294, 4598, 4611, 5604, 5696, 5764, 5988, 6552, 7206, 7540, 7689

Offset: 1

Zhi-Wei Sun, Jan 24 2014

nonn

According to the conjecture in A236412, this sequence should have infinitely many terms.
See A236414 for primes of the form p(m)^2 + q(m)^2.
See also A236440 for a similar sequence.

			a(1) = 1 since p(1)^2 + q(1)^2 = 1^2 + 1^2 = 2 is prime.
a(2) = 2 since p(2)^2 + q(2)^2 = 2^2 + 1^2 = 5 is prime.
a(3) = 3 since p(3)^2 + q(3)^2 = 3^2 + 2^2 = 13 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000010, A000040, A233346, A236412, A236414, A236417, A236418, A236419, A236440.

pq[n_]:=PrimeQ[PartitionsP[n]^2+PartitionsQ[n]^2]
n=0;Do[If[pq[m],n=n+1;Print[n," ",m]],{m,1,10000}]

A282893 The difference between the number of partitions of 2n into odd parts (A000009) and the number of partitions of 2n into even parts (A035363).

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 7, 10, 16, 22, 33, 45, 64, 87, 120, 159, 215, 283, 374, 486, 634, 814, 1049, 1335, 1700, 2146, 2708, 3390, 4243, 5276, 6552, 8095, 9989, 12266, 15044, 18375, 22409, 27235, 33049, 39974, 48281, 58148, 69923, 83871, 100452, 120027, 143214, 170515, 202731, 240567, 285073, 337195

Offset: 0

Robert G. Wilson v, Feb 24 2017

nonn

The even bisection of A282892. The other bisection is A078408.

			G.f. = x^3 + x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 10*x^8 + 16*x^9 + 22*x^10 + 33*x^11 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000041, A035294, A035363, A078408, A214264, A282892.

with(numtheory):
b:= proc(n, t) option remember; `if`(n=0, 1, add(add(`if`(
      (d+t)::odd, d, 0), d=divisors(j))*b(n-j, t), j=1..n)/n)
    end:
a:= n-> b(2*n, 0) -b(2*n, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 24 2017

f[n_] := Length[ IntegerPartitions[n, All, 2Range[n] -1]] - Length[ IntegerPartitions[n, All, 2 Range[n]]]; Array[ f[2#] &, 52]
a[ n_] := SeriesCoefficient[ Sum[ Sign @ SquaresR[1, 16 k + 1] x^k, {k, n}] / QPochhammer[x], {x, 0, n}]; (* Michael Somos, Feb 24 2017 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=1, n, issquare(16*k + 1)*x^k, A) / eta(x + A), n))}; /* Michael Somos, Feb 24 2017 */

a(n) = A282892(2n).
Expansion of (f(x^3, x^5) - 1) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Feb 24 2017
a(n) = A035294(n) - A000041(n). - Michael Somos, Feb 24 2017

cp's OEIS Frontend

A233417 a(n) = |{0 < k <= n/2: q(k)*q(n-k) + 1 is prime}|, where q(.) is the strict partition function (A000009).

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A235343 a(n) = |{0 < k < n: f(n,k) - 1, f(n,k) + 1 and q(f(n,k)) + 1 are all prime with f(n,k) = phi(k) + phi(n-k)/4}|, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).

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A235356 Primes of the form q(m) + 1 with m - 1 and m + 1 both prime, where q(.) is the strict partition function (A000009).

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A235357 Primes of the form q(m) - 1 with m - 1 and m + 1 both prime, where q(.) is the strict partition function (A000009).

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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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A152827 Partial products of PartitionsQ numbers (A000009).

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A236412 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with p(m)^2 + q(m)^2 prime}|, where phi(.) is Euler's totient, p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

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