A073610 -id:A073610 - cp's OEIS frontend

A340961 Number of ways to write n as an ordered sum of 5 primes.

1, 5, 10, 15, 25, 36, 50, 65, 70, 90, 110, 125, 155, 170, 200, 241, 270, 300, 350, 375, 435, 500, 530, 600, 640, 696, 760, 850, 840, 985, 990, 1170, 1160, 1370, 1250, 1570, 1445, 1760, 1600, 2000, 1710, 2340, 1950, 2555, 2165, 2876, 2320, 3340, 2560, 3595, 2880, 3985, 3050

Offset: 10

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 10..10000

Column k=5 of A121303.

Cf. A000040, A010051, A073610, A098238, A259195, A340960, A340962, A340963, A340964, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 5):
seq(a(n), n=10..62);  # Alois P. Heinz, Jan 31 2021

nmax = 62; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^5, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (Sum_{k>=1} x^prime(k))^5.

A340962 Number of ways to write n as an ordered sum of 6 primes.

Original entry on oeis.org

1, 6, 15, 26, 45, 72, 106, 150, 186, 236, 306, 366, 455, 540, 636, 782, 912, 1056, 1236, 1410, 1617, 1896, 2106, 2400, 2696, 2976, 3348, 3716, 4026, 4446, 4917, 5340, 5982, 6380, 7017, 7476, 8377, 8640, 9765, 9936, 11202, 11496, 13132, 12930, 15117, 14672, 17178, 16800, 19696

Offset: 12

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 12..10000

Column k=6 of A121303.

Cf. A000040, A010051, A073610, A098238, A259196, A340960, A340961, A340963, A340964, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 6):
seq(a(n), n=12..60);  # Alois P. Heinz, Jan 31 2021

nmax = 60; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 12] &

G.f.: (Sum_{k>=1} x^prime(k))^6.

A340963 Number of ways to write n as an ordered sum of 7 primes.

Original entry on oeis.org

1, 7, 21, 42, 77, 133, 210, 316, 434, 574, 770, 980, 1239, 1547, 1876, 2331, 2828, 3367, 4032, 4746, 5565, 6574, 7602, 8757, 10136, 11480, 13132, 14882, 16646, 18662, 20951, 23268, 26082, 28861, 31787, 35218, 38745, 42532, 46403, 50883, 54810, 60613, 65016, 71302, 76069

Offset: 14

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 14..10000

Column k=7 of A121303.

Cf. A000040, A010051, A073610, A098238, A259197, A340960, A340961, A340962, A340964, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 7):
seq(a(n), n=14..58);  # Alois P. Heinz, Jan 31 2021

nmax = 58; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^7, {x, 0, nmax}], x] // Drop[#, 14] &

G.f.: (Sum_{k>=1} x^prime(k))^7.

A340964 Number of ways to write n as an ordered sum of 8 primes.

Original entry on oeis.org

1, 8, 28, 64, 126, 232, 392, 624, 925, 1296, 1800, 2416, 3158, 4088, 5152, 6504, 8142, 9976, 12216, 14784, 17738, 21296, 25272, 29736, 35023, 40768, 47328, 54832, 62728, 71744, 81796, 92736, 105078, 118664, 132924, 149424, 167002, 186144, 206852, 229272, 253023

Offset: 16

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 16..10000

Column k=8 of A121303.

Cf. A000040, A010051, A073610, A098238, A259198, A340960, A340961, A340962, A340963, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 8):
seq(a(n), n=16..56);  # Alois P. Heinz, Jan 31 2021

nmax = 56; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^8, {x, 0, nmax}], x] // Drop[#, 16] &

G.f.: (Sum_{k>=1} x^prime(k))^8.

A340965 Number of ways to write n as an ordered sum of 9 primes.

Original entry on oeis.org

1, 9, 36, 93, 198, 387, 696, 1170, 1845, 2740, 3960, 5562, 7566, 10125, 13248, 17133, 22014, 27774, 34776, 43173, 53010, 64869, 78696, 94617, 113415, 134946, 159552, 188164, 219960, 256041, 297180, 342846, 394614, 452595, 516276, 587997, 667938, 755109, 852444

Offset: 18

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 18..10000

Column k=9 of A121303.

Cf. A000040, A010051, A073610, A098238, A259200, A340960, A340961, A340962, A340963, A340964, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 9):
seq(a(n), n=18..56);  # Alois P. Heinz, Jan 31 2021

nmax = 56; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^9, {x, 0, nmax}], x] // Drop[#, 18] &

G.f.: (Sum_{k>=1} x^prime(k))^9.

A340966 Number of ways to write n as an ordered sum of 10 primes.

Original entry on oeis.org

1, 10, 45, 130, 300, 622, 1185, 2100, 3495, 5480, 8266, 12100, 17140, 23730, 32155, 42802, 56400, 73180, 93820, 119250, 149872, 187090, 231765, 284490, 347335, 421332, 507580, 608840, 725500, 859450, 1014473, 1190700, 1392100, 1621710, 1879950, 2172610, 2503580

Offset: 20

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 20..10000

Column k=10 of A121303.

Cf. A000040, A010051, A073610, A098238, A259201, A340960, A340961, A340962, A340963, A340964, A340965.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 10):
seq(a(n), n=20..56);  # Alois P. Heinz, Jan 31 2021

nmax = 56; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^10, {x, 0, nmax}], x] // Drop[#, 20] &

G.f.: (Sum_{k>=1} x^prime(k))^10.

A065577 Number of Goldbach partitions of 10^n.

Original entry on oeis.org

2, 6, 28, 127, 810, 5402, 38807, 291400, 2274205, 18200488, 149091160, 1243722370, 10533150855, 90350630388

Offset: 1

Robert G. Wilson v, Dec 01 2001

Number of ways of writing 10^n as the sum of two odd primes, when the order does not matter.

			a(1)=2 because 10 = 3+7 = 5+5;
a(2)=6 because 100 = 3+97 = 11+89 = 17+83 = 29+71 = 41+59 = 47+53; ...

Ivars Peterson's MathTrek, Goldbach's Prime Pairs
Science News Online, Goldbach's Prime Pairs, week of Aug. 19, 2000; Vol. 158, No. 8.

Cf. A001031, A073610, A107318.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[n_] := Block[{c = 0, lmt = n/2, p = 3}, While[p <= lmt, If[ PrimeQ[n - p], c++ ]; p = NextPrim@p]; c]; Array[f, 10] (* Robert G. Wilson v, Nov 01 2006 *)
a[n]:=Length[Select[n - Prime[Range[PrimePi[n/2]]], PrimeQ]]; Table[a[n],{n, 10^3, 10^3}] (* Luciano Ancora, Mar 16 2015 *)

a(n) = A061358(10^n).

a(9) from Zak Seidov Nov 01 2006
a(10) from R. J. Mathar and David W. Wilson, Nov 02 2006
a(11) from David W. Wilson and Russ Cox, Nov 03 2006
a(12) from Russ Cox, Nov 04 2006
a(13) from Donovan Johnson, Nov 16 2009
a(14) from Huang Yuanbing (bailuzhou(AT)163.com), Dec 24 2009

A076608 Number of nonprimes k < n such that also n-k is not a prime.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 2, 1, 2, 4, 2, 3, 4, 4, 4, 7, 4, 7, 6, 7, 6, 10, 6, 11, 8, 12, 8, 13, 10, 15, 12, 13, 12, 18, 12, 21, 14, 16, 16, 21, 16, 23, 18, 21, 18, 24, 18, 27, 20, 27, 20, 27, 22, 31, 24, 29, 24, 32, 26, 37, 28, 30, 28, 37, 28, 41, 30, 33, 32, 41, 32, 43, 34, 40, 34, 43, 34

Offset: 1

Reinhard Zumkeller, Oct 21 2002

nonn

a(n) is odd iff n is even and n/2 is not prime.

Convolution of A005171 with itself. - R. J. Mathar, Sep 10 2021

			5=1+4=2+3=3+2=4+1, hence a(5)=2;
6=1+5=2+4=3+3=4+2=5+1, hence a(6)=0.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A073610, A005171.

Table[With[{nn=m},Total[Table[If[NoneTrue[{n,nn-n},PrimeQ],1,0],{n,nn-1}]]],{m,80}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 15 2020 *)

A341982 Number of ways to write n as an ordered sum of 2 primes (counting 1 as a prime).

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 4, 2, 3, 0, 4, 2, 5, 2, 4, 0, 6, 2, 6, 2, 5, 0, 8, 2, 5, 0, 4, 0, 8, 2, 6, 2, 7, 0, 8, 0, 5, 2, 6, 0, 10, 2, 8, 2, 7, 0, 12, 2, 8, 0, 6, 0, 12, 2, 6, 0, 7, 0, 14, 2, 7, 2, 10, 0, 12, 0, 6, 2, 10, 0, 14, 2, 11, 2, 10, 0, 14, 0, 10, 2, 9, 0, 18, 2, 9, 0, 8

Offset: 2

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A008578, A073610, A096139, A280917, A283762, A341945, A341973, A341983, A341984.

b:= proc(n) option remember; series(`if`(n=0, 1, x*add(
      `if`(j=1 or isprime(j), b(n-j), 0), j=1..n)), x, 3)
    end:
a:= n-> coeff(b(n), x, 2):
seq(a(n), n=2..88);  # Alois P. Heinz, Feb 24 2021

nmax = 88; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &

G.f.: ( x + Sum_{k>=1} x^prime(k) )^2.

A347739 Number of compositions (ordered partitions) of n into at most 2 prime parts.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 11 2021

nonn

Cf. A000040, A023360, A073610, A121303, A347740, A347741, A347742, A347743.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,2,Prime@Range@PrimePi@n],1],{n,0,100}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

a(n) = Sum_{k=1..2} A121303(n,k) for n >= 2. - Alois P. Heinz, Sep 11 2021

cp's OEIS Frontend

A340961 Number of ways to write n as an ordered sum of 5 primes.

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A340962 Number of ways to write n as an ordered sum of 6 primes.

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A340963 Number of ways to write n as an ordered sum of 7 primes.

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A340964 Number of ways to write n as an ordered sum of 8 primes.

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A340965 Number of ways to write n as an ordered sum of 9 primes.

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A340966 Number of ways to write n as an ordered sum of 10 primes.

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A065577 Number of Goldbach partitions of 10^n.

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A076608 Number of nonprimes k < n such that also n-k is not a prime.

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