A000009 -id:A000009 - cp's OEIS frontend

A266232 Binomial transform of the number of partitions into distinct parts (A000009).

1, 2, 4, 9, 21, 49, 114, 265, 615, 1422, 3272, 7493, 17090, 38850, 88065, 199097, 448953, 1009788, 2265642, 5071611, 11328395, 25254093, 56195143, 124829822, 276839061, 612991848, 1355268779, 2992016128, 6596222234, 14522634554, 31933047707, 70130243427

Offset: 0

Vaclav Kotesovec, Dec 25 2015

nonn

Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then
Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where
g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).
Special cases:
p < 1/2, g(n) = 0
p = 1/2, g(n) = r^2/16
p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81
p = 3/4, g(n) = 9*r^2*sqrt(n)/(64*sqrt(2)) - 27*r^3*n^(1/4)/(2048*2^(1/4)) + 81*r^4/65536
p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))
p = 4/5, g(n) = 2^(7/5)*r^2*n^(3/5)/25 - 4*2^(3/5)*r^3*n^(2/5)/625 + 8*2^(4/5)*r^4*n^(1/5)/15625 - 32*r^5/390625

Vaclav Kotesovec, Table of n, a(n) for n = 0..3200

Cf. A000009, A294467, A293467, A294468.

Cf. A218481, A294466, A281425, A095051, A294500.

Table[Sum[Binomial[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}]
nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k] * x^k / (1-x)^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2022 *)

a(n) ~ 2^(n-5/4) * exp(Pi*sqrt(n/6) + Pi^2/48) / (3^(1/4)*n^(3/4)).

G.f.: (1/(1 - x))*Product_{k>=1} (1 + x^k/(1 - x)^k). - Ilya Gutkovskiy, Aug 19 2018

A129519 First differences of the binomial transform of the distinct partition numbers (A000009).

Original entry on oeis.org

1, 1, 2, 5, 12, 28, 65, 151, 350, 807, 1850, 4221, 9597, 21760, 49215, 111032, 249856, 560835, 1255854, 2805969, 6256784, 13925698, 30941050, 68634679, 152009239, 336152787, 742276931, 1636747349, 3604206106, 7926412320, 17410413153

Offset: 0

Paul D. Hanna, Apr 18 2007

nonn

			Product formula is illustrated by:
A(x) = [1 + x + x^2 + x^3 + x^4 + x^5 +...]*
[1 + x^2 + 2x^3 + 3x^4 + 4x^5 + 5x^6 +...]*
[1 + x^3 + 3x^4 + 6x^5 + 10x^6 + 15x^7 +...]*
[1 + x^4 + 4x^5 + 10x^6 + 20x^7 + 35x^8 +...]*
[1 + x^5 + 5x^6 + 15x^7 + 35x^8 + 70x^9 +...]*...*
[1 + Sum_{k>=n+1} C(k-1,n)*x^k ]*...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A218482, A266232, A307501.

Flatten[{1, Differences[Table[Sum[Binomial[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 40}]]}] (* Vaclav Kotesovec, Oct 30 2017 *)

{a(n)=polcoeff(prod(k=0,n,1+sum(i=k+1,n,binomial(i-1,k)*x^i +x*O(x^n))),n)}

G.f.: A(x) = Product_{n>=1} [1 + x^n/(1-x)^n].
a(n) = A266232(n) - A266232(n-1), for n>0. - Vaclav Kotesovec, Oct 30 2017
a(n) ~ exp(Pi*sqrt(n/6) + Pi^2/48) * 2^(n - 9/4) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 30 2017

A233346 Primes of the form p(k)^2 + q(m)^2 with k > 0 and m > 0, where p(.) is the partition function (A000041), and q(.) is the strict partition function (A000009).

Original entry on oeis.org

2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 101, 109, 113, 137, 149, 157, 193, 229, 241, 349, 373, 509, 709, 733, 1033, 1049, 1213, 1249, 1453, 1493, 1669, 1789, 2141, 2237, 2341, 2917, 3037, 3137, 3361, 4217, 5801, 5897, 6029, 6073, 8821, 10301, 10937, 11057, 18229, 18289, 19249, 20173, 20341, 20389, 21017, 24001, 30977, 36913, 42793

Offset: 1

Zhi-Wei Sun, Dec 07 2013

nonn

Conjecture: The sequence contains infinitely many terms.

This follows from part (i) of the conjecture in A233307. Similarly, the conjecture in A232504 implies that there are infinitely many primes of the form p(k) + q(m) with k and m positive integers.

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..176
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.

Cf. A000009, A000040, A000041, A002313, A232504, A233307.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0
Do[If[Mod[Prime[m]+1,4]>0,Do[If[PartitionsP[j]>=Sqrt[Prime[m]],Goto[aa],
If[SQ[Prime[m]-PartitionsP[j]^2]==False,Goto[bb],Do[If[PartitionsQ[k]^2==Prime[m]-PartitionsP[j]^2,
n=n+1;Print[n," ",Prime[m]];Goto[aa]];If[PartitionsQ[k]^2>Prime[m]-PartitionsP[j]^2,Goto[bb]];Continue,{k,1,2*Sqrt[Prime[m]]}]]];
Label[bb];Continue,{j,1,Sqrt[Prime[m]]}]];
Label[aa];Continue,{m,1,4475}]

A234514 Number of ways to write n = k + m with k > 0 and m > 0 such that p = k + phi(m)/2 and q(p) + 1 are both prime, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 1, 1, 1, 0, 1, 0, 2, 2, 2, 3, 4, 2, 4, 2, 3, 3, 3, 2, 2, 3, 1, 4, 2, 1, 4, 2, 4, 2, 5, 3, 4, 1, 5, 6, 4, 2, 5, 5, 5, 3, 5, 4, 6, 3, 5, 7, 10, 2, 4, 5, 6, 5, 5, 2, 3, 5, 6, 6, 4, 2, 5, 3, 7, 4, 5, 3, 8, 7, 2, 5, 9, 3, 3, 2, 9, 9, 6, 6, 7, 6, 9, 4, 7, 4, 10, 8, 6, 11, 11, 4, 6, 4, 9, 7

Offset: 1

Zhi-Wei Sun, Dec 27 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 12.
(ii) For any integer n > 4, there is a prime p < n - 2 such that q(p + phi(n-p)/2) + 1 is prime.
Clearly, part (i) of the conjecture implies that there are infinitely many primes p with q(p) + 1 prime (cf. A234530).
We have verified part (i) for n up to 10^5.

			a(11) = 1 since 11 = 1 + 10 with 1 + phi(10)/2 = 3 and q(3) + 1 = 3 both prime.
a(27) = 1 since 27 = 7 + 20 with 7 + phi(20)/2 = 11 and q(11) + 1 = 13 both prime.
a(30) = 1 since 30 = 8 + 22 with 8 + phi(22)/2 = 13 and q(13) + 1 = 19 both prime.
a(38) = 1 since 38 = 21 + 17 with 21 + phi(17)/2 = 29 and q(29) + 1 = 257 both prime.
a(572) = 1 since 572 = 77 + 495 with 77 + phi(495)/2 = 197 and q(197) + 1 = 406072423 both prime.
a(860) = 1 since 860 = 523 + 337 with 523 + phi(337)/2 = 691 and q(691) + 1 = 712827068077888961 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000009, A000010, A000040, A229835, A233307, A233390, A233417, A234451, A234470, A234475, A234503, A234504, A234530

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

A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 2, 4, 2, 2, 3, 5, 3, 4, 3, 6, 4, 4, 3, 5, 8, 5, 6, 6, 5, 10, 6, 8, 6, 5, 7, 12, 8, 10, 9, 10, 7, 15, 10, 12, 12, 10, 7, 11, 18, 12, 16, 15, 15, 14, 11, 22, 15, 20, 18, 20, 14, 11, 15

Offset: 0

Gary W. Adamson, Mar 27 2010

Row sums = A000041, the partition numbers.
The current triangle is the 2nd in an infinite set, followed by A174714 (k=3), and A174715, (k=4); in which row sums of each triangle = A000041.
k-th triangle in the infinite set can be defined as having the sequence:
"Euler transform of ones: (1,1,1,...) interleaved with (k-1) zeros"; shifted down k times (except column 0) in successive columns, then multiplied * triangle A174712, the diagonalized variant of A000041, A174713 begins with A000009 shifted down twice (triangle A173305); where A000009 = the Euler transform of period 2 sequence: [1,0,1,0,...].
Similarly, triangle A174714 begins with A000716 shifted down thrice; where A000716 = the Euler transform of period 3 series: [1,1,0,1,1,0,...]. Then multiply the latter as an infinite lower triangular matrix * A174712, the diagonalized variant of A000041, obtaining triangle A174714 with row sums = A000041.
Case k=4 = triangle A174715 which begins with the Euler transform of period 4 series: [1,1,1,0,1,1,1,0,...], shifted down 4 times in successive columns then multiplied * A174712, the diagonalized variant of A000041.
All triangles in the infinite set have row sums = A000041.
The sequences: "Euler transform of ones interleaved with (k-1) zeros" have the following properties, beginning with k=2:
...
k=2, A000009: = Euler transform of [1,0,1,0,1,0,...] and satisfies
.....A000009. = p(x)/p(x^2), where p(x) = polcoeff A000041; and A000041 =
.....A000009(x) = r(x), then p(x) = r(x) * r(x^2) * r(x^4) * r(x^8) * ...
...
k=3, A000726: = Euler transform of [1,1,0,1,1,0,...] and satisfies
.....A000726(x): = p(x)/p(x^3), and given s(x) = polcoeff A000726, we get
.....A000041(x) = p(x) = s(x) * s(x^3) * s(x^9) * s(x^27) * ...
...
k=4, A001935: = Euler transform of [1,1,1,0,1,1,1,0,...] and satisfies
.....A001935(x) = p(x)/p(x^4) and given t(x) = polcoeff A001935, we get
.....A000041(x) = p(x) = t(x) * t(x^4) * t(x^16) * t(x^64) * ...
...
Also the number of integer partitions of n whose even parts sum to k, for k an even number from zero to n. The version including odd k is A113686. - Gus Wiseman, Oct 23 2023

			First few rows of the triangle =
1;
1;
1, 1;
2, 1;
2, 1, 2;
3, 2, 2;
4, 2, 2, 3;
5, 3, 4, 3;
6, 4, 4, 3, 5;
8, 5, 6, 6, 5;
10, 6, 8, 6, 5, 7;
12, 8, 10, 9, 10, 7;
15, 10, 12, 12, 10, 7, 11;
18, 12, 16, 15, 15, 14, 11;
22, 15, 20, 18, 20, 14, 11, 15;
...
From _Gus Wiseman_, Oct 23 2023: (Start)
Row n = 9 counts the following partitions:
  (9)          (72)        (54)       (63)      (81)
  (711)        (5211)      (522)      (6111)    (621)
  (531)        (3321)      (4311)     (432)     (441)
  (51111)      (321111)    (411111)   (42111)   (4221)
  (333)        (21111111)  (32211)    (3222)    (22221)
  (33111)                  (2211111)  (222111)
  (3111111)
  (111111111)
(End)

Cf. A000009, A173305, A174712, A174714, A174715.
Row sums are A000041.
The odd version is A365067.
The corresponding rank statistic is A366531, odd version A366528.
A000009 counts partitions into odd parts, ranks A066208.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A035363, A066967, A130780, A171966, A182616, A366527, A366533.

Table[Length[Select[IntegerPartitions[n],Total[Select[#,EvenQ]]==k&]],{n,0,15},{k,0,n,2}] (* Gus Wiseman, Oct 23 2023 *)

As infinite lower triangular matrices, A173305 * A174712.

T(n,k) = A000009(n-2k) * A000041(k). - Gus Wiseman, Oct 23 2023

A233307 a(n) = |{0 < k < n: p(k)^2 + q(n-k)^2 is prime}|, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 07 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Also, for any integer n > 4, p(k)*q(n-k) - 1 is prime for some 0 < k < n/2.
(ii) If n > 9, then prime(k)*p(n-k) + 1 is prime for some 0 < k < n. If n > 2, then prime(k)*q(n-k) - 1 is prime for some 0 < k < n, and also prime(k)*q(n-k) + 1 is prime for some 0 < k < n.
(iii) If n > 11, then prime(k) + p(n-k) is prime for some 0 < k < n. If n > 4, then prime(k) + q(n-k) is prime for some 0 < k < n, and also prime(k)^2 + q(n-k)^2 is prime for some 0 < k < n.

			a(5) = 1 since 5 = 1 + 4 with p(1)^2 + q(4)^2 = 1^2 + 2^2 = 5 prime.
a(6) = 1 since 6 = 3 + 3 with p(3)^2 + q(3)^2 = 3^2 + 2^2 = 13 prime.
a(19) = 1 since 19 = 3 + 16 with p(3)^2 + q(16)^2 = 3^2 + 32^2 = 1033 prime.
a(28) = 1 since 28 = 3 + 25 with p(3)^2 + q(25)^2 = 3^2 + 142^2 = 20173 prime.
a(29) = 1 since 29 = 6 + 23 with p(6)^2 + q(23)^2 = 11^2 + 104^2 = 10937 prime.
a(38) = 1 since 38 = 1 + 37 with p(1)^2 + q(37)^2 = 1^2 + 760^2 = 577601 prime.
a(75) = 1 since 75 = 13 + 62 with p(13)^2 + q(62)^2 = 101^2 + 13394^2 = 179409437 prime.
a(160) = 1 since 160 = 48 + 112 with p(48)^2 + q(112)^2 = 147273^2 + 1177438^2 = 1408049580373 prime.
a(210) = 1 since 210 = 71 + 139 with p(71)^2 + q(139)^2 = 4697205^2 + 8953856^2 = 102235272080761 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, A000041, A231201, A232504.

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

A285229 Expansion of g.f. Product_{j>=1} 1/(1-y*x^j)^A000009(j), triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 3, 1, 1, 0, 3, 4, 3, 1, 1, 0, 4, 8, 5, 3, 1, 1, 0, 5, 11, 10, 5, 3, 1, 1, 0, 6, 18, 16, 11, 5, 3, 1, 1, 0, 8, 25, 29, 18, 11, 5, 3, 1, 1, 0, 10, 38, 44, 34, 19, 11, 5, 3, 1, 1, 0, 12, 52, 72, 55, 36, 19, 11, 5, 3, 1, 1

Offset: 0

Alois P. Heinz, Apr 14 2017

			T(n,k) is the number of multisets of exactly k partitions of positive integers into distinct parts with total sum of parts equal to n.
T(4,1) = 2: {4}, {31}.
T(4,2) = 3: {3,1}, {21,1}, {2,2}.
T(4,3) = 1: {2,1,1}.
T(4,4) = 1: {1,1,1,1}.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,   1;
  0,  2,  3,   1,  1;
  0,  3,  4,   3,  1,  1;
  0,  4,  8,   5,  3,  1,  1;
  0,  5, 11,  10,  5,  3,  1,  1;
  0,  6, 18,  16, 11,  5,  3,  1,  1;
  0,  8, 25,  29, 18, 11,  5,  3,  1, 1;
  0, 10, 38,  44, 34, 19, 11,  5,  3, 1, 1;
  0, 12, 52,  72, 55, 36, 19, 11,  5, 3, 1, 1;
  0, 15, 75, 110, 96, 60, 37, 19, 11, 5, 3, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Index entries for triangles generated by the Multiset Transformation

Columns k=0..10 give: A000007, A000009 (for n>0), A320787, A320788, A320789, A320790, A320791, A320792, A320793, A320794, A320795.
Row sums give A089259.
T(2n,n) give A285230.
Cf. A061260, A360763.

with(numtheory):
g:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
       x^j*binomial(g(i)+j-1, j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..16);

L[n_] := QPochhammer[x^2]/QPochhammer[x] + O[x]^n;
A[n_] := Module[{c = L[n]}, CoefficientList[#, y]& /@ CoefficientList[ 1/Product[(1 - x^k*y + O[x]^n)^SeriesCoefficient[c, {x, 0, k}], {k, 1, n}], x]];
A[12] // Flatten (* Jean-François Alcover, Jan 19 2020, after Andrew Howroyd *)
g[n_] := g[n] = If[n==0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n];
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[b[n - i*j, i - 1]*x^j* Binomial[g[i] + j - 1, j], {j, 0, n/i}]]];
T[n_] := CoefficientList[b[n, n] + O[x]^(n+1), x];
T /@ Range[0, 16] // Flatten (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
A(n)={my(c=L(n), v=Vec(1/prod(k=1, n, (1 - x^k*y + O(x*x^n))^polcoef(c,k)))); vector(#v, n, Vecrev(v[n],n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

G.f.: Product_{j>=1} 1/(1-y*x^j)^A000009(j).

A234475 Number of ways to write n = k + m with 2 < k <= m such that q(phi(k)*phi(m)/4) + 1 is prime, 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, 1, 1, 2, 2, 3, 3, 4, 4, 3, 4, 5, 5, 4, 7, 7, 6, 5, 5, 7, 3, 6, 7, 7, 5, 7, 4, 8, 4, 7, 7, 8, 7, 4, 5, 5, 4, 4, 5, 5, 6, 5, 4, 5, 3, 5, 4, 6, 6, 4, 6, 5, 4, 3, 6, 4, 9, 4, 8, 6, 7, 6, 8, 4, 7, 4, 7, 8, 9, 2, 3, 1, 8, 6, 9, 6, 6, 6, 6, 4, 7, 5, 8, 8, 4, 5, 5, 9, 7, 10, 4, 10, 3, 7, 8, 6

Offset: 1

Zhi-Wei Sun, Dec 26 2013

nonn

Conjecture: a(n) > 0 for all n > 5.

This implies that there are infinitely many primes p with p - 1 a term of A000009.

			a(6) = 1 since 6 = 3 + 3 with q(phi(3)*phi(3)/4) + 1 = q(1) + 1 = 2 prime.
a(76) = 1 since 76 = 18 + 58 with q(phi(18)*phi(58)/4) + 1 = q(42) + 1 = 1427 prime.
a(197) = 1 since 197 = 4 + 193 with q(phi(4)*phi(193)/4) + 1 = q(96) + 1 = 317789.
a(356) = 1 since 356 = 88 + 268 with q(phi(88)*phi(268)/4) + 1 = q(1320) + 1 = 35940172290335689735986241 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2525
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000010, A000040, A232504, A233307, A233346, A233547, A233390, A233393, A234309, A234310, A234337, A234344, A234347, A234359, A234360, A234361, A234451, A234470, A234514, A234530, A234567, A234569

f[n_,k_]:=PartitionsQ[EulerPhi[k]*EulerPhi[n-k]/4]+1
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,3,n/2}]
Table[a[n],{n,1,100}]

A234530 Primes p with q(p) + 1 also prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

2, 3, 11, 13, 29, 37, 47, 71, 79, 89, 103, 127, 131, 179, 181, 197, 233, 271, 331, 379, 499, 677, 691, 757, 887, 911, 1019, 1063, 1123, 1279, 1429, 1531, 1559, 1637, 2251, 2719, 3571, 4007, 4201, 4211, 4297, 4447, 4651, 4967, 5953, 6131, 7937, 8233, 8599, 8819, 9013, 11003, 11093, 11813, 12251, 12889, 12953, 13487, 13687, 15259

Offset: 1

Zhi-Wei Sun, Dec 27 2013

nonn

By the conjecture in A234514, this sequence should have infinitely many terms.
It seems that a(n+1) < a(n) + a(n-1) for all n > 4.
See A234366 for primes of the form q(p) + 1 with p prime.
See also A234644 for a similar sequence.

			a(1) = 2 since 2 and q(2) + 1 = 2 are both prime.
a(2) = 3 since 3 and q(3) + 1 = 3 are both prime.
a(3) = 11 since 11 and q(11) + 1 = 13 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..150

Cf. A000009, A000040, A233346, A233393, A234366, A234470, A234475, A234514, A234567, A234569, A234572, A234615, A234644, A234647

n=0;Do[If[PrimeQ[PartitionsQ[Prime[k]]+1],n=n+1;Print[n," ",Prime[k]]],{k,1,10^5}]
Select[Prime[Range[2000]],PrimeQ[PartitionsQ[#]+1]&] (* Harvey P. Dale, Apr 23 2017 *)

A056848 Numbers k that divide the number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, 10, 16, 65, 160, 180, 366, 406, 896, 1436, 3904, 5464, 6312, 7168, 12558, 17957, 36960, 48097, 48256, 61952, 88646, 94400, 107340, 112240, 114863, 127540, 171856, 270336, 383360, 392736, 459012, 623639, 960484, 1222656, 1312768, 1463990, 1480704, 2244736, 2380968, 3183563, 4161888, 4787280, 5107455, 5606400, 6826556, 7878400, 9188414, 9533238, 10219520, 10356472, 12981760, 15162808, 22062080, 25240360, 28313472, 32215040, 41284864, 72160576, 79563520, 91164167

Offset: 1

Robert G. Wilson v, Sep 02 2000

nonn

No other terms below 10^8. - Max Alekseyev, Jul 10 2015

Cf. A000009, A051177, A162468, A259943.

Do[ If[ Mod[ PartitionsQ[n], n] == 0, Print[n]], {n, 1, 48000}]

Extended by Max Alekseyev, Jul 04 2009

a(49)-a(60) from Max Alekseyev, Jul 10 2015

cp's OEIS Frontend

A266232 Binomial transform of the number of partitions into distinct parts (A000009).

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A129519 First differences of the binomial transform of the distinct partition numbers (A000009).

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A233346 Primes of the form p(k)^2 + q(m)^2 with k > 0 and m > 0, where p(.) is the partition function (A000041), and q(.) is the strict partition function (A000009).

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A234514 Number of ways to write n = k + m with k > 0 and m > 0 such that p = k + phi(m)/2 and q(p) + 1 are both prime, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).

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A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).

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A233307 a(n) = |{0 < k < n: p(k)^2 + q(n-k)^2 is prime}|, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

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A285229 Expansion of g.f. Product_{j>=1} 1/(1-y*x^j)^A000009(j), triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A234475 Number of ways to write n = k + m with 2 < k <= m such that q(phi(k)*phi(m)/4) + 1 is prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

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