A208614 Number of partitions of n into distinct primes where all of the prime factors of n are represented in the partition.
1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 3, 1, 4, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 2, 3, 1, 2, 1, 3, 4, 2, 1, 5, 9, 6, 5, 4, 1, 6, 6, 7, 4, 3, 1, 4, 1, 4, 9, 20, 7, 3, 1, 7, 8, 6, 1, 15, 1, 5, 19, 11, 13, 9, 1, 21, 52, 7, 1
Offset: 0
Keywords
Examples
a(p) = 1 for any prime p. a(n) = 0 for 1, 4, 6, 8, 9, 22. a(25) = 3 because 25 = 3 + 5 + 17 = 5 + 7 + 13 = 2 + 5 + 7 + 11.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..5000
- Rajesh Bhowmick, A bit different from Goldbach's conjecture (February 27-28, 2012)
Programs
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Maple
with(numtheory): a:= proc(n) local b, l, f; b:= proc(h, j) option remember; `if`(h=0, 1, `if`(j<1, 0, `if`(l[j]>h, 0, b(h-l[j], j-1)) +b(h, j-1))) end; forget(b); f:= factorset(n); l:= sort([({seq(ithprime(i), i=1..pi(n))} minus f)[]]); b(n-add(i, i=f), nops(l)) end: seq(a(n), n=0..300); # Alois P. Heinz, Mar 20 2012 -
Mathematica
restrictedIntegerPartition[ n_Integer, list_List ] := 1 /; n == 0 restrictedIntegerPartition[ n_Integer, list_List ] := 0 /; n < 0 || Total[list] < n || n < Min[list] restrictedIntegerPartition[ n_Integer, list_List ] := restrictedIntegerPartition[n - First[list], Rest[list]] + restrictedIntegerPartition[n, Rest[list]] distinctPrimeFactors[ n_Integer ] := distinctPrimeFactors[n] = Map[First, FactorInteger[n]] oeisA076694[ n_Integer ] := oeisA076694[n] = n - Total[distinctPrimeFactors[n]] oeisA208614[ n_Integer ] := restrictedIntegerPartition[oeisA076694[n], Sort[Complement[Prime @ Range @ PrimePi @ oeisA076694 @ n, distinctPrimeFactors[n]] , Greater ]] Table[oeisA208614[n], {n,1,100}]
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Maxima
countRestrictedIntegerPartitions(n, L) := if ( n = 0 ) then 1 else if ( ( n < 0 ) or ( lsum(k, k, L) < n ) or ( n < lmin( L ) ) ) then 0 else block( [ m, R ], m : first(L), R : rest(L), countRestrictedIntegerPartitions(n, R) + countRestrictedIntegerPartitions(n - m, R)); distinctPrimeFactors(n) := map(first,ifactors(n)); oeisA076694(n) := n - lsum(k,k,distinctPrimeFactors(n)); listOfPrimesLessThanOrEqualTo (n) := block( [ list : [] , i], for i : 2 step 0 while i <= n do ( list : cons(i, list) , i : next_prime(i) ) , list ); oeisA208614(n) := block([ m, list ], m : oeisA076694(n), list : sort(listify(setdifference(setify(listOfPrimesLessThanOrEqualTo(m)), setify(distinctPrimeFactors(n)))), ordergreatp), countRestrictedIntegerPartitions(m, list)); makelist(oeisA208614(j), j, 1, 100);
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