A000009 -id:A000009 - cp's OEIS frontend

A233390 a(n) = |{0 < k < n: 2^k - 1 + q(n-k) is prime}|, where q(.) is the strict partition function (A000009).

0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 4, 4, 2, 3, 2, 3, 3, 7, 4, 4, 5, 3, 4, 5, 5, 5, 6, 7, 6, 5, 4, 4, 9, 3, 6, 6, 5, 4, 7, 1, 4, 5, 6, 9, 6, 8, 6, 8, 4, 5, 8, 7, 4, 3, 4, 7, 6, 6, 3, 6, 5, 6, 4, 6, 8, 7, 8, 4, 5, 3, 6, 7, 7, 3, 10, 7, 5, 6, 10, 4, 8, 4, 6, 7, 6, 8, 10, 4, 6, 8, 9, 5, 6, 5, 7, 13, 5, 5, 6

Offset: 1

Zhi-Wei Sun, Dec 08 2013

nonn

Conjecture: a(n) > 0 for all n > 1.

We have verified this for n up to 150000. For n = 124669, the least positive integer k with 2^k - 1 + q(n-k) prime is 13413.

			a(6) = 1 since 2^2 - 1 + q(4) = 3 + 2 = 5 is prime.
a(10) = 1 since 2^4 - 1 + q(6) = 15 + 4 = 19 is prime.
a(41) = 1 since 2^{16} - 1 + q(25) = 65535 + 142 = 65677 is prime.
a(127) = 1 since 2^{21} - 1 + q(106) = 2097151 + 728260 = 2825411 is prime.
a(153) = 1 since 2^{70} - 1 + q(83) = 1180591620717411303423 + 101698 = 1180591620717411405121 is prime.
a(164) = 1 since 2^{26} - 1 + q(138) = 67108863 + 8334326 = 75443189 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..8000
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2017.

Cf. A000009, A000040, A000225, A233307, A233346, A233359, A233360, A233393, A233417, A233439.

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

A378971 Antidiagonal-sums of absolute value of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

Original entry on oeis.org

1, 1, 1, 5, 8, 18, 30, 47, 70, 110, 177, 309, 574, 1063, 1892, 3107, 4598, 6166, 8737, 20603, 62457, 149132, 314116, 614093, 1155968, 2176048, 4244322, 8753864, 19006756, 42472117, 95235017, 210396059, 453414950, 949510166, 1931941261, 3826650257, 7400745917

Offset: 0

Gus Wiseman, Dec 14 2024

nonn

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = 8.

For primes we have A376681 or A376684, signed version A140119 or A376683.
For composites we have A377035, signed version A377034.
For squarefree numbers we have A377040, signed version A377039.
For nonsquarefree numbers we have A377048, signed version A377049.
For prime powers we have A377053, signed version A377052.
For partition numbers we have A378621, signed version A377056.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The signed version is A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244, A325245, A325268.

nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A118303 Even values of the PartitionsQ function A000009.

Original entry on oeis.org

2, 2, 4, 6, 8, 10, 12, 18, 22, 32, 38, 46, 54, 64, 76, 104, 122, 142, 192, 222, 256, 296, 340, 390, 448, 512, 668, 760, 864, 982, 1260, 1426, 1610, 1816, 2048, 2304, 2590, 2910, 3264, 3658, 4582, 5120, 5718, 6378, 7108, 8808, 9792, 10880, 12076, 13394, 14848

Offset: 1

Reinhard Zumkeller, Apr 22 2006

nonn

a(n) = A000009(A090864(n)).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000009, A051044, A090864.

Select[PartitionsQ[Range[100]], EvenQ] (* Jean-François Alcover, Nov 10 2021 *)

A233393 Primes of the form 2^k - 1 + q(m) with k > 0 and m > 0, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 61, 67, 71, 73, 79, 83, 101, 107, 109, 127, 131, 137, 139, 149, 157, 167, 173, 181, 191, 193, 199, 223, 229, 257, 263, 269, 271, 277, 293, 311, 331, 347, 349, 359, 383, 397, 421, 449, 463, 467, 479, 521, 523, 557, 587

Offset: 1

Zhi-Wei Sun, Dec 08 2013

nonn

Conjecture: The sequence has infinitely many terms.

This follows from the conjecture in A233390.

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..542
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.

Cf. A000040, A000225, A232504, A233307, A233346, A233359, A233360, A233390.

Pow[n_]:=Pow[n]=Mod[n,2]==0&&2^(IntegerExponent[n,2])==n
n=0
Do[Do[If[Pow[Prime[m]-PartitionsQ[k]+1],
n=n+1;Print[n," ",Prime[m]];Goto[aa]];If[PartitionsQ[k]>=Prime[m],Goto[aa]];Continue,{k,1,2*Prime[m]}];
Label[aa];Continue,{m,1,110}]

A235344 Numbers m with m - 1, m + 1 and q(m) + 1 all prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

4, 6, 18, 42, 72, 102, 270, 282, 312, 618, 1032, 1062, 1320, 1950, 2082, 3528, 7350, 7488, 10332, 15138, 17388, 21600, 40038, 44700, 134922, 156258, 187908, 243708, 339138, 389568, 495360, 610920, 761712, 911292, 916218, 943800, 1013532, 1217472, 1312602

Offset: 1

Zhi-Wei Sun, Jan 06 2014

Clearly, any term after the first term 4 is a multiple of 6. By part (i) of the conjecture in A235343, this sequence should have infinitely many terms. The prime q(a(51)) + 1 = q(3235368) + 1 has 1412 decimal digits.
See A235356 for primes of the form q(m) + 1 with m - 1 and m + 1 both prime.
See also A235346 for a similar sequence.

			a(1) = 4 since 4 - 1, 4 + 1 and q(4) + 1 = 3 are all prime.
a(2) = 6 since 6 - 1, 6 + 1 and q(6) + 1 = 5 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..51
Zhi-Wei Sun, Twin primes and the strict partition function, a message to Number Theory List, Jan. 15, 2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000040, A014574, A234530, A234569, A234644, A235343, A235346, A235356, A235357, A235358.

f[k_]:=PartitionsQ[Prime[k]+1]+1
n=0;Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[f[k]],n=n+1;Print[n," ",Prime[k]+1]],{k,1,10000}]

A235346 Numbers m with m - 1, m + 1 and q(m) - 1 all prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

6, 240, 420, 1032, 1062, 1278, 2238, 4020, 12612, 15972, 19890, 22110, 34500, 44772, 134370, 141768, 145602, 191142, 217368, 290658, 436482, 454578, 464382, 618030, 668202, 849348, 888870, 964260, 1179150, 1364970, 1446900, 1593498, 1737102, 1866438, 2291802, 3237432

Offset: 1

Zhi-Wei Sun, Jan 06 2014

Clearly, each term is a multiple of 6. By the conjecture in A235358 (which is part (ii) of the conjecture in A235343), this sequence should have infinitely many terms. q(a(36)) - 1 = q(3237432) - 1 is a prime having 1412 decimal digits.
See A235357 for primes of the form q(m) - 1 with m - 1 and m + 1 both prime.
See also A235344 for a similar sequence.

			a(1) = 6 since q(4) - 1 = 1 is not a prime, and 6 - 1, 6 + 1 and q(6) - 1 = 3 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..36
Zhi-Wei Sun, Twin primes and the strict partition function, a message to Number Theory List, Jan. 15, 2014.

Cf. A000009, A000040, A014574, A234530, A234569, A234644, A235343, A235344, A235356, A235357, A235358.

f[k_]:=PartitionsQ[Prime[k]+1]-1
n=0;Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[f[k]],n=n+1;Print[n," ",Prime[k]+1]],{k,1,10000}]
Select[Mean/@Select[Partition[Prime[Range[10000]],2,1],#[[2]]-#[[1]] == 2&],PrimeQ[PartitionsQ[#]-1]&] (* The program generates the first 14 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* Harvey P. Dale, Feb 01 2022 *)

A284908 a(n) = A000009(A000009(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 10, 15, 27, 46, 89, 192, 390, 864, 2304, 5718, 16444, 53250, 173682, 618784, 2556284, 11086968, 53466624, 299016608, 1780751883, 11784471548, 94036004868, 795888123110, 7723778471936, 91117574462854, 1168225267521350

Offset: 0

Alois P. Heinz, Apr 05 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..70

Cf. A000009, A000041, A058699, A284909, A284910.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> b(b(n)):
seq(a(n), n=0..35);

Table[PartitionsQ@ PartitionsQ@ n, {n, 0, 50}] (* Indranil Ghosh, Apr 07 2017 *)

A304877 G.f.: Sum_{k>=0} q(k)^2 * x^k / Sum_{k>=0} q(k)*x^k, where q(n) is A000009(n).

Original entry on oeis.org

1, 0, 0, 2, 0, 4, 4, 8, 4, 20, 20, 28, 38, 52, 80, 128, 128, 176, 300, 316, 476, 648, 832, 972, 1428, 1720, 2340, 3014, 3844, 4588, 6556, 7476, 9760, 12588, 15596, 19480, 25140, 29796, 37728, 47604, 58140, 70856, 90148, 107692, 133228, 167284, 198692, 242728

Offset: 0

Vaclav Kotesovec, May 20 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000009, A054440, A304873, A304878.

nmax = 50; CoefficientList[Series[Sum[PartitionsQ[k]^2*x^k, {k, 0, nmax}] / Sum[PartitionsQ[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ sqrt(3) * exp(Pi*sqrt(n)) / (2^(11/2) * n^(3/2)).

A089254 G.f.: Product_{m>=1} 1/(1+x^m)^A000009(m).

Original entry on oeis.org

1, -1, 0, -2, 1, -2, 2, -2, 5, -3, 8, -3, 13, -6, 16, -15, 21, -28, 21, -52, 22, -90, 23, -136, 40, -190, 79, -237, 185, -254, 385, -219, 740, -116, 1279, 49, 2039, 198, 2923, 130, 3785, -541, 4215, -2464, 3638, -6581, 1204, -14062, -3889, -26110, -12267, -43399, -23264, -65206, -34085, -87955, -37666

Offset: 0

N. J. A. Sloane, Dec 23 2003

sign

Cf. A050342, A000009, A001970, A089259.

A233359 a(n) = |{0 < k < n: L(k) + q(n-k) is prime}|, where L(k) is the k-th Lucas number (A000204), and q(.) is the strict partition function (A000009).

Original entry on oeis.org

0, 1, 1, 2, 3, 1, 2, 4, 2, 2, 3, 3, 2, 4, 3, 5, 1, 4, 5, 3, 1, 3, 3, 7, 3, 3, 4, 5, 2, 2, 9, 2, 4, 4, 9, 2, 6, 6, 6, 3, 3, 1, 5, 7, 4, 4, 5, 7, 4, 9, 5, 6, 4, 1, 5, 6, 11, 9, 4, 2, 5, 5, 4, 6, 8, 9, 12, 3, 7, 5, 4, 10, 6, 7, 6, 3, 5, 8, 4, 4, 4, 4, 7, 7, 5, 1, 4, 9, 7, 4, 8, 7, 6, 5, 2, 3, 7, 11, 5, 5

Offset: 1

Zhi-Wei Sun, Dec 08 2013

nonn

Conjecture: a(n) > 0 for all n > 1.
We have verified this for n up to 60000.
Note that for n = 19976 there is no k = 0,...,n such that F(k) + q(n-k) is prime, where F(0), F(1), ... are the Fibonacci numbers.

			a(7) = 2 since L(1) + q(6) = 1 + 4 = 5 and L(6) + q(1) = 18 + 1 = 19 are both prime.
a(17) = 1 since L(13) + q(4) = 521 + 2 = 523 is prime.
a(21) = 1 since L(5) + q(16) = 11 + 32 = 43 is prime.
a(42) = 1 since L(22) + q(20) = 39603 + 64 = 39667 is prime.
a(54) = 1 since L(8) + q(46) = 47 + 2304 = 2351 is prime.
a(86) = 1 since L(67) + q(19) = 100501350283429 + 54 = 100501350283483 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Cf. A000009, A000032, A000040, A000204, A233307, A233346, A233360.

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

cp's OEIS Frontend

A233390 a(n) = |{0 < k < n: 2^k - 1 + q(n-k) is prime}|, where q(.) is the strict partition function (A000009).

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A378971 Antidiagonal-sums of absolute value of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

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A118303 Even values of the PartitionsQ function A000009.

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A233393 Primes of the form 2^k - 1 + q(m) with k > 0 and m > 0, where q(.) is the strict partition function (A000009).

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A235344 Numbers m with m - 1, m + 1 and q(m) + 1 all prime, where q(.) is the strict partition function (A000009).

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A235346 Numbers m with m - 1, m + 1 and q(m) - 1 all prime, where q(.) is the strict partition function (A000009).

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A284908 a(n) = A000009(A000009(n)).

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A304877 G.f.: Sum_{k>=0} q(k)^2 * x^k / Sum_{k>=0} q(k)*x^k, where q(n) is A000009(n).

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Formula

A089254 G.f.: Product_{m>=1} 1/(1+x^m)^A000009(m).

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