A125688 Number of partitions of n into three distinct primes.
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 4, 3, 4, 2, 5, 3, 5, 4, 6, 1, 6, 3, 6, 4, 6, 3, 9, 3, 8, 5, 8, 4, 11, 3, 11, 5, 10, 3, 13, 3, 13, 6, 12, 2, 14, 5, 15, 6, 13, 2, 18, 5, 17, 6, 14, 4, 21, 5, 19, 7, 17, 4, 25, 4, 20, 8, 21, 4, 26, 4, 25, 9, 22, 4
Offset: 1
Examples
a(42) = #{2+3+37, 2+11+29, 2+17+23} = 3.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
Crossrefs
Programs
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Maple
b:= proc(n, i) option remember; `if`(n=0, [1,0$3], `if`(i<1, [0$4], zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$3], b(n-ithprime(i), i-1)[1..3])[]], 0))) end: a:= n-> b(n, numtheory[pi](n))[4]: seq(a(n), n=1..100); # Alois P. Heinz, Nov 15 2012 -
Mathematica
b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i<1, {0, 0, 0, 0}, Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, {0, 0, 0}, Take[b[n-Prime[i], i-1], 3]]]}]]]; a[n_] := b[n, PrimePi[n]][[4]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *) dp3Q[{a_,b_,c_}]:=Length[Union[{a,b,c}]]==3&&AllTrue[{a,b,c},PrimeQ]; Table[ Count[IntegerPartitions[n,{3}],?dp3Q],{n,100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Jan 30 2019 *) -
PARI
a(n)=my(s);forprime(p=n\3,n-4,forprime(q=(n-p)\2+1,min(n-p,p-1),if(isprime(n-p-q),s++)));s \\ Charles R Greathouse IV, Aug 27 2012
Formula
From Alois P. Heinz, Nov 22 2012: (Start)
G.f.: Sum_{0
a(n) = [x^n*y^3] Product_{i>=1} (1+x^prime(i)*y). (End)
a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/2)} A010051(i) * A010051(k) * A010051(n-i-k). - Wesley Ivan Hurt, Mar 29 2019
A283762 Expansion of (x + Sum_{k>=1} x^prime(k))^3.
0, 0, 0, 1, 3, 6, 7, 9, 9, 13, 12, 15, 9, 15, 12, 22, 15, 24, 12, 27, 18, 34, 18, 36, 15, 42, 24, 45, 15, 42, 12, 51, 24, 52, 18, 60, 21, 66, 24, 58, 15, 69, 18, 75, 30, 75, 24, 87, 21, 93, 36, 91, 24, 99, 18, 108, 36, 97, 18, 108, 21, 126, 42, 111, 21, 135, 30, 141, 36, 112, 18, 150, 30, 153, 42, 138, 33, 177, 30, 171, 42
Offset: 0
Keywords
Comments
Number of ways to write n as an ordered sum of 3 noncomposite numbers (1 together with the primes) (A008578).
a(2k+1) > 0 for all k > 0 (from the ternary Goldbach's conjecture, proved by H. A. Helfgott).
Examples
a(6) = 7 because we have [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 2, 2], [2, 1, 3], [1, 3, 2] and [1, 2, 3].
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10000
- Ilya Gutkovskiy, Extended graphical example
Programs
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Mathematica
nmax = 80; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^3, {x, 0, nmax}], x] -
PARI
concat([0, 0, 0], Vec((x + sum(k=1, 80, x^prime(k)))^3 + O(x^81))) \\ Indranil Ghosh, Mar 16 2017
Formula
G.f.: (x + Sum_{k>=1} x^prime(k))^3.
A068873 Smallest prime which is a sum of n distinct primes.
2, 5, 19, 17, 43, 41, 79, 83, 127, 131, 199, 197, 283, 281, 379, 389, 499, 509, 643, 641, 809, 809, 983, 971, 1171, 1163, 1381, 1373, 1609, 1607, 1861, 1861, 2137, 2137, 2437, 2441, 2749, 2767, 3109, 3109, 3457, 3457, 3833, 3847, 4243, 4241, 4663, 4679, 5119
Offset: 1
Keywords
Comments
Conjectured terms a(50)-a(76): 5147, 5623, 5591, 6079, 6101, 6599, 6607, 7151, 7151, 7699, 7699, 8273, 8293, 8893, 8893, 9521, 9547, 10211, 10223, 10889, 10891, 11597, 11617, 12343, 12373, 13099, 13127. - Jean-François Alcover, Apr 22 2020
Examples
a(3) = 19 as 19 is the smallest prime which can be expressed as the sum of three primes as 19 = 3 + 5 + 11. a(4) = 17= 2+3+5+7. a(2)=A038609(1). a(3)=A124867(7). Further examples in A102330.
References
- Shantanu Dey & Moloy De, Two conjectures on prime numbers, Journal of Recreational Mathematics, Vol. 36 (3), pp 205-206. Baywood Publ. Co, Amityville NY 2011.
Links
- David A. Corneth, Table of n, a(n) for n = 1..10000 (first 200 terms from Jean-François Alcover)
- Jean-François Alcover, Conjectured terms up to a(200).
Programs
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Maple
# Number of ways to write n as a sum of k distinct primes, the smallest # being smalp sumkprims := proc(n,k,smalp) option remember; local a,res,pn; res := n-smalp ; if res < 0 then return 0; elif res > 0 and k <=0 then return 0; elif res = 0 and k = 1 then return 1; else pn := nextprime(smalp) ; a := 0 ; while pn <= res do a := a+procname(res,k-1,pn) ; pn := nextprime(pn) ; end do: a ; end if; end proc: # Number of ways of writing n as a sum of k distinct primes A000586k := proc(n,k) local a,i,smalp ; a := 0 ; for i from 1 do smalp := ithprime(i) ; if k*smalp > n then return a; end if; a := a+sumkprims(n,k,smalp) ; end do: end proc: # Smallest prime which is a sum of n distinct primes A068873 := proc(n) local a,i; a := A007504(n) ; a := nextprime(a-1) ; for i from 1 do if A000586k(a,n) > 0 then return a; end if; a := nextprime(a) ; end do: end proc: # R. J. Mathar, May 04 2014 -
PARI
a(n)= { my(P=primes(n), k=n, t, res = oo); while(1, forvec(v=vector(n-1, i, [1, k-1]), t=sum(i=1, n-1, P[v[i]])+P[k]; if(isprime(t), res = min(res, t); ) , 2 \\ flag: only strictly increasing vectors v ); P=concat(P, nextprime(P[k]+1)); k++; if(P[k] + sum(i = 1+bitand(n,1), n-1+bitand(n,1), P[i]) > res, return(res) ) ); } \\ Charles R Greathouse IV, Sep 19 2015; corrected by David A. Corneth, May 12 2025
Formula
Extensions
More terms from Sascha Kurz, Feb 03 2003
Corrected by Ray Chandler, Feb 02 2005
A124868 Natural numbers that are not the sum of 3 distinct primes.
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 17
Offset: 1
Comments
(Conjectured) Every number n > 17 is the sum of 3 distinct primes. a(n) is the complement of A124867(n) = {10, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, ...}, numbers that are the sum of 3 distinct primes.
A125688(a(n)) = 0. - Reinhard Zumkeller, Nov 30 2006
A224534 Prime numbers that are the sum of three distinct prime numbers.
19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307
Offset: 1
Keywords
Comments
Similar to Goldbach's weak conjecture.
Primes in A124867, and by the comment in A124867 also the set of all primes >=19. - R. J. Mathar, Apr 19 2013
"Goldbach's original conjecture (sometimes called the 'ternary' Goldbach conjecture), written in a June 7, 1742 letter to Euler, states 'at least it seems that every number that is greater than 2 is the sum of three primes' (Goldbach 1742; Dickson 2005, p. 421). Note that here Goldbach considered the number 1 to be a prime, a convention that is no longer followed." [Weisstein] - Jonathan Vos Post, May 15 2013
Examples
19 = 3 + 5 + 11.
Links
- H. A. Helfgott and David J. Platt, Numerical Verification of the Ternary Goldbach Conjecture up to 8.875e30, arXiv:1305.3062 [math.NT], 2013-2014.
- H. A. Helfgott and David J. Platt, Numerical verification of the Ternary Goldbach Conjecture up to 8.875*10^30, Exp. Math. 22 (4) (2013) 406-409.
- Eric W. Weisstein, Goldbach conjecture
- Wikipedia, Goldbach's conjecture
- Wikipedia, Goldbach's weak conjecture
Programs
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Mathematica
Union[Select[Total /@ Subsets[Prime[Range[2, 30]], {3}], PrimeQ]]
A238397 Numbers of the form pq + qr + rp where p, q and r are distinct primes (sorted sequence without duplicates).
31, 41, 59, 61, 71, 87, 91, 101, 103, 113, 119, 121, 129, 131, 143, 151, 161, 167, 171, 185, 191, 199, 211, 213, 215, 221, 227, 239, 241, 243, 247, 251, 263, 269, 271, 275, 281, 293, 297, 299, 301, 311, 321, 327, 331, 339, 341, 343, 347, 355
Offset: 1
Keywords
Comments
Numbers of the form e2(p, q, r) for distinct primes p, q, r, where e2 is the elementary symmetric polynomial of degree 2. Other sequences are obtained with different numbers of distinct primes and degrees: A000040 for 1 prime, A038609 and A006881 for 2 primes, A124867, this sequence, and A007304 for 3 primes. The 4-prime sequences are not presently in the OEIS with the exception of A046386. - Charles R Greathouse IV, Feb 26 2014
Examples
71 = 3*5 + 5*7 + 7*3 = 2*3 + 3*13 + 13*2 is in the sequence (only once, though 2 solutions exist).
Links
- Jean-François Alcover, Table of n, a(n) for n = 1..1000
Programs
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Mathematica
terms = 50; dm (* initial number of primes *) = 10; f[p_, q_, r_] := p*q + q*r + r*p; Clear[A238397]; A238397[m_] := A238397[m] = Take[u = Union[f @@@ Subsets[Prime /@ Range[m], {3}]], Min[Length[u], terms]]; A238397[dm]; A238397[m = 2*dm]; While[Print["m = ", m]; A238397[m] != A238397[m - dm], m = m + dm]; A238397[m]
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PARI
is(n)=forprime(r=(sqrtint(3*n-3)+5)\3, (n-6)\5, forprime(q= sqrtint(r^2+n)-r+1, min((n-2*r)\(r+2), r-2), if((n-q*r)%(q+r)==0 && isprime((n-q*r)/(q+r)), return(1)))); 0 \\ Charles R Greathouse IV, Feb 26 2014
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PARI
list(n)=my(v=List()); forprime(r=5, (n-6)\5, forprime(q=3, min((n-2*r)\(r+2), r-2), my(S=q+r, P=q*r); forprime(p=2, min((n-P)\S, q-1), listput(v, p*S+P)))); Set(v) \\ Charles R Greathouse IV, Feb 26 2014
A177697 Sums of 3 distinct primorials.
9, 33, 37, 38, 213, 217, 218, 241, 242, 246, 2313, 2317, 2318, 2341, 2342, 2346, 2521, 2522, 2526, 2550, 30033, 30037, 30038, 30061, 30062, 30066, 30241, 30242, 30246, 30270, 32341, 32342, 32346, 32370, 32550, 510513, 510517, 510518, 510541, 510542
Offset: 1
Comments
Examples
9 = 6+2+1 33 = 30+2+1 37 = 30+6+1 38 = 30+6+2 213 = 210+2+1
Programs
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Mathematica
Take[Total/@Subsets[Join[{1},FoldList[Times,Prime[Range[10]]]],{3}]// Union,40] (* Harvey P. Dale, Nov 07 2017 *)
A224535 Odd numbers that are the sum of three distinct prime numbers.
15, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135
Offset: 1
Keywords
Comments
Odd terms in A124867. - R. J. Mathar, Jun 09 2014
Examples
15 = 3 + 5 + 7.
Crossrefs
Cf. A224534 (primes in this sequence).
Programs
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Mathematica
Union[Total /@ Subsets[Prime[Range[2, 30]], {3}]]
Extensions
"Odd" added by Irina Gerasimova, Apr 15 2013
A177709 Sums of 4 distinct primorials.
39, 219, 243, 247, 248, 2319, 2343, 2347, 2348, 2523, 2527, 2528, 2551, 2552, 2556, 30039, 30063, 30067, 30068, 30243, 30247, 30248, 30271, 30272, 30276, 32343, 32347, 32348, 32371, 32372, 32376, 32551, 32552, 32556, 32580, 510519, 510543
Offset: 1
Comments
This is to numbers that are the sum of 4 different primes (A177708) as primorials (A002110) are to primes (A000040). The subsequence of primes among these sums of 4 distinct primorials begins: 2347, 2551, 30271, 32371, 510751. The subsequence of nontrivial powers a^b with b>1 begin: a(3) = 243, a(24) = 30276 = 30030+210+30+6 = 2^2 x 3^2 x 29^2.
Examples
a(1) = 39 = 30+6+2+1 a(2) = 219 = 210+6+2+1 a(3) = 243 = 210+30+2+1 = 3^5 a(4) = 247 = 210+30+6+1 a(5) = 248 = 210+30+6+2.
Extensions
Corrected (2348 inserted) by R. J. Mathar, May 15 2010
A178041 Number of ways to represent the n-th prime (which has a nonzero number of such representations) as the sum of 4 distinct primes.
1, 2, 3, 3, 5, 6, 6, 6, 8, 10, 13, 14, 13, 18, 21, 17, 21, 30, 21, 32, 23, 37, 27, 45, 35, 34, 54, 43, 60, 61, 67, 44, 52, 55, 79, 58, 89, 57, 92, 100, 111, 69, 119, 76, 83, 122, 91, 89, 94, 102, 147, 146, 106, 159, 116, 176, 125, 190, 119, 195, 202, 136, 230, 148, 154, 222
Offset: 1
Examples
a(1) = 1 because 17 = 2+3+5+7 is the unique solution for the smallest such prime. a(2) = 2 because 23 = 2+3+5+13 = 2+3+7+11 are the only two solutions for the 2nd smallest such prime. a(3) = 3 because 29 = 2+3+5+19 = 2+3+7+17 = 2+3+11+13 are the only 3 solutions for the 3rd smallest such prime. a(4) = 3 because 31 = 2+3+7+19 = 2+5+7+17 = 2+5+11+13 are the only 3 solutions for the 4th smallest such prime. a(5) = 5 because 37 = 2+3+13+19 = 2+5+7+23 = 2+5+11+19 = 2+5+13+17 = 2+7+11+17 are the only 5 solutions for the 5th smallest such prime.
Links
- Eric W. Weisstein, Goldbach Conjecture,
Crossrefs
Programs
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Mathematica
max=367;lim=PrimePi[max];p4=Sort[Total/@Subsets[Prime[Range[lim]],{4}]];p4p=Select[p4,PrimeQ[#]&&#<=max&]; s={};Do[c=Count[p4p,Prime[p]];If[c>0,AppendTo[s,c]],{p,lim}];s (* James C. McMahon, Jan 11 2025 *)
Extensions
Extended by Zak Seidov
Comments