A233360 -id:A233360 - 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}]

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}]

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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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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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).

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Mathematica