A035026 -id:A035026 - cp's OEIS frontend

A238711 Product of all primes p such that 2n - p is also prime.

2, 3, 15, 105, 35, 231, 2145, 5005, 4641, 53295, 1616615, 119301, 21505, 7436429, 21489, 57998985, 3038795305, 4123, 13844919, 10393190665, 12838371, 299859855, 7292509103495, 12023917269, 70691995, 37198413949697, 62483343, 2769282065, 98755025688454681

Offset: 2

Reinhard Zumkeller, Mar 06 2014

nonn

Product of n-th row in triangle A171637;
All terms greater than 3 are odd, composite and squarefree numbers, cf. A024556.
n is prime iff n is a factor of a(n).
Product of the distinct primes in the Goldbach partitions of 2n. - Wesley Ivan Hurt, Sep 29 2020

Reinhard Zumkeller, Table of n, a(n) for n = 2..2000

Cf. A000040, A010051, A238778, subsequence of A056911.

a238711 n = product $ filter ((== 1) . a010051') $
   map (2 * n -) $ takeWhile (<= 2 * n) a000040_list

Table[Times@@Select[Select[Prime[Range[2 n]], # < 2 n &], PrimeQ[2 n - #] &], {n, 2, 30}] (* Robert Price, Apr 26 2025 *)

A020639(a(n)) = A020481(n); A006530(a(n)) = A020482(n);
A001221(a(n)) = A035026(n); A008472(a(n)) = A238778(n);
A027748(a(n),k) + A027748(a(n),l+1-k) = 2*n for k=1..l, with l=A001221(a(n)); particulary A020639(a(n))+A006530(a(n)) = 2*n;
a(n) = n^c(n) * Product_{i=1..n-1} (i*(2*n-i))^(c(i)*c(2*n-i)), where c is the prime characteristic (A010051). - Wesley Ivan Hurt, Sep 29 2020

A238778 Sum of all primes p such that 2n - p is also a prime.

Original entry on oeis.org

2, 3, 8, 15, 12, 21, 32, 36, 40, 55, 72, 65, 56, 90, 64, 119, 144, 57, 120, 168, 132, 161, 240, 200, 156, 270, 168, 203, 360, 155, 320, 396, 136, 350, 432, 333, 380, 546, 320, 369, 672, 387, 352, 810, 368, 423, 672, 294, 600, 816, 520, 583, 864, 660, 784

Offset: 2

Reinhard Zumkeller, Mar 06 2014

nonn

Sum of n-th row in triangle A171637.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A008472, A238711.

Cf. A010051, A035026, A171637

a238778 n = sum $ filter ((== 1) . a010051') $
   map (2 * n -) $ takeWhile (<= 2 * n) a000040_list

Table[Total@Select[Select[Prime[Range[2 n]], # < 2 n &], PrimeQ[2 n - #] &], {n, 2, 56}] (* Robert Price, Apr 26 2025 *)

a(n) = my(s=0); forprime(p=2, 2*n, if(isprime(2*n-p), s+=p)); s; \\ Michel Marcus, Jan 24 2022

a(n) = A008472(A238711(n)).
a(n) mod 2 = A010051(n).
a(n) = n*A035026(n). - Robert G. Wilson v, Apr 28 2018

A199331 Number of ordered ways of writing n as the sum of two semiprimes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 2, 2, 2, 2, 0, 3, 4, 3, 2, 0, 2, 4, 4, 2, 2, 3, 4, 5, 6, 4, 0, 2, 6, 6, 4, 2, 6, 6, 4, 5, 8, 7, 4, 2, 8, 10, 6, 5, 2, 5, 6, 4, 10, 6, 4, 4, 10, 12, 12, 2, 6, 10, 6, 7, 8, 9, 6, 5, 12, 14, 10, 6, 6, 7, 8, 7, 10, 10, 6, 4, 14, 14

Offset: 1

Robert G. Wilson v, Nov 05 2011

Conjecture: Only the integers 1, 2, 3, 4, 5, 6, 7, 9, 11, 17, 22, 33 (A072966) cannot be partitioned into a set of two semiprimes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A035026, A064911, A072966, A073610.

sp:= select(t -> numtheory:-bigomega(t)=2, [$1..100]):
G:= expand(add(x^t,t=sp)^2):
seq(coeff(G,x,i),i=1..100); # Robert Israel, Nov 24 2020

mx=200; semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2; t = Select[ Range@ mx, semiPrimeQ]; s = Sort[Plus @@@ Tuples[t, 2]]; Transpose[ Tally@ Join[ Range@ mx, s]][[2]] - 1

a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c = A064911. - Wesley Ivan Hurt, Jan 07 2024

Definition clarified by Robert Israel, Nov 24 2020

A341459 Number of compositions of n^2 into n prime parts.

Original entry on oeis.org

1, 0, 1, 4, 22, 241, 2696, 35218, 529888, 8998419, 169486964, 3496417024, 78344008779, 1891733424205, 48923563968087, 1347813311456319, 39371345548420060, 1214570579814316742, 39430967625404799740, 1343040950675651131103, 47862610677098010505554

Offset: 0

Alois P. Heinz, Feb 12 2021

nonn

			a(3) = 4: 333, 225, 252, 522.

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A023360, A035026, A121303, A299168.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(isprime(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..22);

b[n_, t_] := b[n, t] =
     If[n == 0, If[t == 0, 1, 0], If[t < 1, 0, Sum[
     If[PrimeQ[j], b[n-j, t-1], 0], {j, 1, n}]]];
a[n_] := b[n^2, n];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)

a(n) = A121303(n^2,n).

A094113 Total area of all 1-histograms of length n.

Original entry on oeis.org

1, 7, 44, 268, 1609, 9583, 56792, 335448, 1976689, 11627735, 68308580, 400870468, 2350563097, 13773547487, 80663415344, 472175746096, 2762854639585, 16160861104423, 94502471413916, 552472329537660

Offset: 1

Ralf Stephan, May 03 2004

nonn

Arises in analysis of first-come-first-served (FCFS) printer policy.

Fung Lam, Table of n, a(n) for n = 1..1300
D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, FUN with algorithm'01, Isola d'Elba, 2001.
D. Merlini, R. Sprugnoli and M.C. Verri, Waiting patterns for a printer, Discrete Applied Mathematics, Volume 144, 359-373, 2004

Cf. A001003, A035026.

Rest[CoefficientList[Series[(1+x-Sqrt[1-6*x+x^2])/(4*(1-6*x+x^2)), {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 23 2014 *)

G.f.: (1+x-sqrt(1-6*x+x^2))/(4*(1-6*x+x^2)).
Recurrence: (n+1)*a(n) = (8-n)*a(n-10) + 3*(10*n-71)*a(n-9) + (2263-365*n)*a(n-8) + 4*(570*n-3021)*a(n-7) + 2*(16654-3785*n)*a(n-6) + 6138*(2*n-7)*a(n-5) + 2*(9841-3785*n)*a(n-4) + 4*(570*n-969)*a(n-3) + (292-365*n)*a(n-2) + 3*(10*n+1)*a(n-1), n>=10. - Fung Lam, Feb 07 2014
Recurrence (of order 4): n*a(n) = 3*(4*n-3)*a(n-1) - 19*(2*n-3)*a(n-2) + 3*(4*n-9)*a(n-3) - (n-3)*a(n-4). - Vaclav Kotesovec, Feb 23 2014
a(n) ~ (sqrt(2)-1)/8 * (3+2*sqrt(2))^(n+1). - Vaclav Kotesovec, Feb 23 2014

A254713 All numbers k such that the number of distinct parts of all A045917(k) Goldbach partitions of 2k is prime.

Original entry on oeis.org

4, 5, 6, 7, 11, 13, 17, 19, 23, 29, 31, 53, 59, 61, 67, 73, 83, 89, 97, 101, 103, 109, 113, 127, 131, 139, 151, 157, 163, 173, 179, 191, 193, 199, 223, 227, 229, 251, 263, 271, 307, 313, 337, 347, 353, 359, 367, 379, 389, 401, 449, 479, 521, 523, 577, 587, 599, 601, 607, 613, 631, 643

Offset: 1

Ivan N. Ianakiev, Feb 06 2015

Conjecture: a(k) is prime for k > 3. Tested for k up to 3*10^4.

			For k = 4, 2k = 8. The number of the distinct Goldbach parts of 8 (3 and 5) is prime, therefore 4 is in the sequence.
5 is in the sequence because the 2 = A045917(5) Goldbach partitions of 10 are 5 + 5 and 3 + 7, and there are 3 distinct parts, namely 3, 5 and 7. - _Wolfdieter Lang_, Feb 23 2015

Eric Weisstein's World of Mathematics, Goldbach Partition.

Cf. A035026, A002372, A045917.

lstIn={};lstFin={};
goldPart[x_]:=Module[{h=x/2},While[h>1,If[And[PrimeQ[h],PrimeQ[x-h]],AppendTo[lstIn,{h,x-h}]];h--];
lstFin=Length[Union[Flatten[lstIn]]];lstIn={};lstFin];
a254713=Flatten[Position[PrimeQ[goldPart/@Range[2,2002,2]],True]]

Edited. Wolfdieter Lang, Feb 23 2015

A280667 a(n) = number of primes of the form 4k + 1 such that 2n - 4k - 1 is prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, 2, 2, 2, 2, 3, 1, 2, 2, 2, 3, 4, 0, 3, 4, 3, 4, 5, 2, 3, 4, 3, 5, 6, 0, 5, 6, 2, 4, 6, 3, 5, 6, 4, 3, 8, 2, 4, 6, 4, 4, 7, 2, 6, 8, 5, 5, 8, 4, 7, 10, 6, 6, 12, 3, 5, 10, 3, 6, 9, 4, 5, 6, 7, 8, 11, 3, 5, 10, 4, 8, 11, 2, 8, 10, 5, 6, 13, 6, 6, 8, 7, 7, 14, 2, 8, 12, 5

Offset: 1

Juri-Stepan Gerasimov, Jan 07 2017

nonn

Primes p such that a(p) = 0: 2, 3, 7, 19, 31.

			a(8) = 2 because 2*8 - 4*1 - 1 = 11 is prime where 4*1 + 1 = 5 is prime of the form 4k+1 and 2*8 - 4*3 - 1 = 3 is prime where 4*3 + 1 = 13 is prime of the form 4k+1.

Cf. A002144, A035026, A278287, A279027.

A280667 := func; [A280667(n):n in[1..100]];

a(n) = sum(k=1, 2*n, isprime(k) && isprime(2*n-k) && ((2*n-k) % 4 == 1)); \\ Michel Marcus, Jan 07 2017

A306196 Irregular triangle read by rows where row n lists the primes 2n - k, with 1 < k < 2n-1, and if k is composite also 2n - p has to be prime for some prime divisor p of k.

Original entry on oeis.org

2, 3, 2, 3, 5, 3, 5, 7, 2, 5, 7, 2, 3, 5, 7, 11, 3, 5, 7, 11, 13, 3, 5, 7, 11, 13, 2, 3, 5, 7, 11, 13, 17, 2, 3, 5, 7, 11, 13, 17, 19, 2, 3, 5, 7, 11, 13, 17, 19, 2, 3, 5, 7, 11, 13, 17, 19, 23, 3, 5, 11, 13, 17, 23, 2, 7, 11, 13, 17, 19, 23, 2, 3, 5, 11, 13, 17, 19, 23, 29

Offset: 2

Juri-Stepan Gerasimov, Jan 28 2019

Conjectures:
(i) 1 <= A035026(n) <= (n-th row length of this triangle) for n >= 2;
(ii) a(n,1) < A171637(n,1) for n >= 4.
Numbers m such that m-th row length of this triangle is equal to A000720(m): 1, 2, 11, 13, 25, 56, 60, ...

			Row 2 = [2] because 2*2 = 2 + 2;
Row 3 = [3] because 2*3 = 3 + 3;
Row 4 = [2,3,5] because 2*4 - 2 = 6 = 2*3 and 2*4 = 3 + 5;
Row 5 = [3,5,7] because 2*5 = 3 + 7 = 5 + 5.
The table starts:
  2;
  3;
  2,  3,  5;
  3,  5,  7;
  2,  5,  7;
  2,  3,  5,  7, 11;
  3,  5,  7, 11, 13;
  3,  5,  7, 11, 13;
  2,  3,  5,  7, 11, 13, 17;
  2,  3,  5,  7, 11, 13, 17, 19;
  2,  3,  5,  7, 11, 13, 17, 19;
  2,  3,  5,  7, 11, 13, 17, 19, 23;
  3,  5, 11, 13, 17, 23;
  2,  7, 11, 13, 17, 19, 23;
  2,  3,  5, 11, 13, 17, 19, 23, 29;

Supersequence of A171637.

Cf. A000720, A020481, A035026, A306247, A306261.

isok(k,n) = {if (isprime(2*n-k), pf = factor(k)[,1]; for (j=1, #pf, if (isprime(2*n-pf[j]), return (1));););}
row(n) = {my(v = []); for (k=1, 2*n, if (isok(k,n), v = concat(v, 2*n-k))); vecsort(v);} \\ Michel Marcus, Mar 02 2019

cp's OEIS Frontend

A238711 Product of all primes p such that 2n - p is also prime.

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A238778 Sum of all primes p such that 2n - p is also a prime.

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Haskell

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A199331 Number of ordered ways of writing n as the sum of two semiprimes.

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Maple

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A341459 Number of compositions of n^2 into n prime parts.

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Maple

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A094113 Total area of all 1-histograms of length n.

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A254713 All numbers k such that the number of distinct parts of all A045917(k) Goldbach partitions of 2k is prime.

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A280667 a(n) = number of primes of the form 4k + 1 such that 2n - 4k - 1 is prime.

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Magma

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A306196 Irregular triangle read by rows where row n lists the primes 2n - k, with 1 < k < 2n-1, and if k is composite also 2n - p has to be prime for some prime divisor p of k.

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