A100555
Smallest square that is equal to the sum of n not-necessarily-distinct primes plus 1.
Original entry on oeis.org
1, 4, 9, 9, 9, 16, 16, 16, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81, 81, 81, 81, 81, 81, 81, 100, 100, 100, 100, 100, 100, 100, 100, 100, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 144, 144
Offset: 0
a(1)=4 because 2^2=1+3.
a(2)=9 because 3^2=1+3+5.
a(3)=9 because 3^2=1+2+3+3.
a(4)=9 because 3^2=1+2+2+2+2.
A101778
Last term in each row of triangle referenced in A101777.
Original entry on oeis.org
3, 5, 3, 2, 7, 5, 3, 7, 7, 5, 3, 2, 11, 7, 7, 5, 3, 13, 11, 7, 7, 5, 3, 2, 13, 13, 11, 7, 7, 5, 3, 17, 13, 13, 11, 7, 7, 5, 3, 2, 19, 17, 13, 13, 11, 7, 7, 5, 3, 19, 19, 17, 13, 13, 11, 7, 7, 5, 3, 2, 23, 19, 19, 17, 13, 13, 11, 7, 7, 5, 3, 23, 23, 19, 19, 17, 13, 13, 11, 7, 7, 5, 3, 2, 23
Offset: 1
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A020482(k) = forprime(q=2, k, if(isprime(2*k-q), return(2*k-q)));
a(n) = {my(r=(ceil(sqrt(2*n+1)))^2-2*n+3); if(r%2==0, r=A020482(r/2), if(isprime(r-2), r-=2, r=A020482(r\2))); r; } \\ Jinyuan Wang, Jan 29 2020
A101777
Triangle read by rows where n-th row is the lexicographically least set of n not-necessarily-distinct primes summing to A100555(n)-1.
Original entry on oeis.org
3, 3, 5, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 5, 7, 2, 2, 2, 2, 2, 2, 2, 3, 7, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 7
Offset: 1
Triangle begins:
{3}
{3,5}
{2,3,3}
{2,2,2,2}
{2,2,2,2,7}
{2,2,2,2,2,5}
{2,2,2,2,2,2,3}
{2,2,2,2,2,2,5,7}
{2,2,2,2,2,2,2,3,7}
{2,2,2,2,2,2,2,2,3,5}
{2,2,2,2,2,2,2,2,2,3,3}
{2,2,2,2,2,2,2,2,2,2,2,2}
{2,2,2,2,2,2,2,2,2,2,2,2,11}
{2,2,2,2,2,2,2,2,2,2,2,3,3,7}
{2,2,2,2,2,2,2,2,2,2,2,2,2,2,7}
A333615
a(n) is the number of ways to express 2*n+1 as a sum of parts x such that x+2 is an odd prime.
Original entry on oeis.org
1, 2, 3, 4, 7, 10, 13, 20, 26, 34, 48, 61, 78, 103, 129, 162, 206, 256, 314, 391, 479, 579, 711, 859, 1028, 1243, 1485, 1764, 2107, 2497, 2941, 3477, 4092, 4783, 5610, 6557, 7615, 8872, 10303, 11901, 13781, 15910, 18292, 21062, 24196, 27697, 31726, 36287
Offset: 0
For n = 3, 2*n + 1 = 7. There are 4 partitions of 7 into parts with sizes 1, 3, 5, 9, 11 ... (the odd primes minus 2):
7 = 5 + 1 + 1
7 = 3 + 3 + 1
7 = 3 + 1 + 1 + 1 + 1
7 = 1 + 1 + 1 + 1 + 1 + 1 + 1
So, a(3) = 4.
Cf.
A069259 (partitions of 2*n, instead of 2*n+1).
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a[n_] := Module[{p},
p = Table[Prime[i] - 2, {i, 2, PrimePi[2*n + 3]}];
Length[IntegerPartitions[2*n + 1, {0, Infinity}, p]]]
Table[a[n], {n, 0, 60}]
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\\ Slowish:
partitions_into(n,parts,from=1) = if(!n,1, if(#parts==from, (0==(n%parts[from])), my(s=0); for(i=from,#parts,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s)));
odd_primes_minus2_upto_n_reversed(n) = { my(lista=List([])); forprime(p=3,n+2,listput(lista,p-2)); Vecrev(Vec(lista)); };
A333615(n) = partitions_into(n+n+1, odd_primes_minus2_upto_n_reversed(n+n+1)); \\ Antti Karttunen, May 09 2020
Showing 1-4 of 4 results.