Search: a066874 -id:a066874
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A014652
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Number of partitions of n in its prime divisors with at least one part of size 1.
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+10
6
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1, 1, 1, 2, 1, 5, 1, 4, 3, 8, 1, 16, 1, 11, 11, 8, 1, 33, 1, 26, 15, 17, 1, 56, 5, 20, 9, 36, 1, 226, 1, 16, 23, 26, 23, 120, 1, 29, 27, 92, 1, 422, 1, 56, 78, 35, 1, 208, 7, 140, 35, 66, 1, 261, 35, 128, 39, 44, 1, 1487, 1, 47, 108, 32, 41, 996, 1, 86, 47, 1062, 1, 456, 1, 56
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OFFSET
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1,4
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LINKS
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FORMULA
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Coefficient of x^(n-1) in expansion of (1/(1-x))*1/Product_{d is prime divisor of n} (1-x^d). - Vladeta Jovovic, Apr 11 2004
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PROG
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(PARI)
\\ This is for computing just a moderate number of terms:
prime_factors_with1_reversed(n) = vecsort(setunion([1], factor(n)[, 1]~), , 4);
partitions_into_with_trailing_ones(n, parts, from=1) = if(!n, 1, if(#parts<=(from+1), if(#parts == from, 1, (1+(n\parts[from]))), my(s=0); for(i=from, #parts, if(parts[i]<=n, s += partitions_into_with_trailing_ones(n-parts[i], parts, i))); (s)));
A014652(n) = partitions_into_with_trailing_ones(n-1, prime_factors_with1_reversed(n)); \\ Antti Karttunen, Sep 10 2018
(PARI) \\ For an efficient program to compute large numbers of terms, see David A. Corneth's PARI program included in the Links section. - Antti Karttunen, Sep 12 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A317624
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Number of integer partitions of n where all parts are > 1 and whose LCM is n.
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+10
6
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0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 17, 1, 1, 1, 7, 1, 60, 1, 1, 1, 1, 1, 76, 1, 1, 1, 55, 1, 105, 1, 11, 10, 1, 1, 187, 1, 6, 1, 13, 1, 30, 1, 111, 1, 1, 1, 5043, 1, 1, 15, 1, 1, 230, 1, 17, 1, 242, 1, 4173, 1, 1, 12, 19, 1
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OFFSET
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0,13
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LINKS
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EXAMPLE
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The a(20) = 5 partitions are (20), (10,4,4,2), (10,4,2,2,2), (5,5,4,4,2), (5,5,4,2,2,2).
The a(45) = 10 partitions:
(45),
(15,15,9,3,3), (15,9,9,9,3),
(15,9,9,3,3,3,3), (15,9,5,5,5,3,3), (9,9,9,5,5,5,3),
(15,9,3,3,3,3,3,3,3), (9,9,5,5,5,3,3,3,3), (9,5,5,5,5,5,5,3,3),
(9,5,5,5,3,3,3,3,3,3,3).
Let sum(t) denote the sum of elements of a tuple t. The tuples t with distinct divisors of 45 that have lcm(t) = 45 and sum(t) <= 45 are {(45) and (3, 9, 15), (3, 5, 9, 15), (3, 5, 9), (5, 9), (9, 15), (5, 9, 15)}. For each such tuple t, find the number of partitions of 45 - s(t) into distinct parts of t.
For the tuple (45), there is 1 partition of 45 - 45 = 0 into parts with 45. That is: {()}.
For the tuple (3, 9, 15), there are 4 partitions of 45 - (3 + 9 + 15) = 18 into parts with 3, 9 and 15. They are {(3, 15), (9, 9), (3, 3, 3, 9), (3, 3, 3, 3, 3, 3)}.
For the tuple (3, 5, 9), there are 4 partitions of 45 - (3 + 5 + 9) = 28 into parts with 3, 5 and 9; they are {(5, 5, 9, 9), (3, 3, 3, 5, 5, 9), (3, 5, 5, 5, 5, 5), (3, 3, 3, 3, 3, 3, 5, 5)}.
For the tuple (3, 5, 9, 15), there is 1 partition of 45 - (3 + 5 + 9 + 15) = 13 into parts with 3, 5, 9 and 15. That is (3, 5, 5).
The other tuples, (5, 9), (9, 15), and (5, 9, 15); they give no extra tuples. That's because there is no solution to the Diophantine equation for 5x + 9y = 45 - (5 + 9), corresponding to the tuple (5, 9) with nonnegative x, y.
That also excludes (9, 15); if there is a solution for that, there would also be a solution for (5, 9). This could whittle down the number of seeds even further. Similarly, (5, 9, 15) gives no solution.
Therefore a(45) = 1 + 4 + 4 + 1 = 10.
(End)
In general, there are A318670(n) (<= A069626(n)) such seed sets of divisors where to start extending the partition from. (See the second PARI program which uses subroutine toplevel_starting_sets.) - Antti Karttunen, Sep 08 2018
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], And[Min@@#>=2, LCM@@#==n]&]], {n, 30}]
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PROG
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(PARI)
strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
partitions_into_lcm(orgn, n, parts, from=1, m=1) = if(!n, (m==orgn), my(k = #parts, s=0); for(i=from, k, if(parts[i]<=n, s += partitions_into_lcm(orgn, n-parts[i], parts, i, lcm(m, parts[i])))); (s));
A317624(n) = if(n<=1, 0, partitions_into_lcm(n, n, strong_divisors_reversed(n))); \\ Antti Karttunen, Sep 07 2018
(PARI)
strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
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)));
toplevel_starting_sets(orgn, n, parts, from=1, ss=List([])) = { my(k = #parts, s=0, newss); if(lcm(Vec(ss))==orgn, s += partitions_into(n, ss)); for(i=from, k, if(parts[i]<=n, newss = List(ss); listput(newss, parts[i]); s += toplevel_starting_sets(orgn, n-parts[i], parts, i+1, newss))); (s) };
A317624(n) = if(n<=1, 0, toplevel_starting_sets(n, n, strong_divisors_reversed(n))); \\ Antti Karttunen, Sep 08-10 2018
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CROSSREFS
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Cf. A000837, A018818, A066874, A067538, A069626, A074761, A074971, A143773, A259936, A281116, A285572, A290103, A305566, A316429, A316431, A316432, A316433, A318670.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A097798
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Number of partitions of n into abundant numbers.
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+10
4
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1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 0, 2, 0, 4, 0, 2, 0, 0, 0, 7, 0, 2, 0, 2, 0, 8, 0, 5, 0, 2, 0, 14, 0, 4, 0, 4, 0, 14, 0, 8, 0, 5, 0, 23, 0, 9, 0, 9, 0, 26, 0, 18, 0, 9, 0, 38, 0, 16, 0, 17, 0, 46, 0, 29, 0, 19, 0, 65, 0, 32, 0
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OFFSET
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0,25
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COMMENTS
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n = 977 = 945 + 32 is the first prime for which sequence obtains a nonzero value, as a(977) = a(32) = 1. 945 is the first term in A005231. - Antti Karttunen, Sep 06 2018
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LINKS
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Eric Weisstein's World of Mathematics, Partition
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MATHEMATICA
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n = 100; d = Select[Range[n], DivisorSigma[1, #] > 2 # &]; CoefficientList[ Series[1/Product[1 - x^d[[i]], {i, 1, Length[d]} ], {x, 0, n}], x] (* Amiram Eldar, Aug 02 2019 *)
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PROG
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(PARI)
abundants_up_to_reversed(n) = { my(s = Set([])); for(k=1, n, if(sigma(k)>(2*k), s = setunion([k], s))); vecsort(s, , 4); };
partitions_into(n, parts, from=1) = if(!n, 1, my(k = #parts, s=0); for(i=from, k, if(parts[i]<=n, s += partitions_into(n-parts[i], parts, i))); (s));
(PARI) \\ see Corneth link
(Magma) v:=[n:n in [1..100]| SumOfDivisors(n) gt 2*n]; [#RestrictedPartitions(n, Set(v)): n in [0..100]]; // Marius A. Burtea, Aug 02 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A014649
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Number of partitions of n into its nonprime power divisors with at least one part of size 1.
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+10
3
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 15, 1, 1, 1, 1, 1, 16, 1, 1, 1, 6, 1, 21, 1, 2, 3, 1, 1, 26, 1, 5, 1, 2, 1, 18, 1, 6, 1, 1, 1, 238, 1, 1, 3, 1, 1, 31, 1, 2, 1, 31, 1, 139, 1, 1, 5, 2, 1, 37, 1, 26, 1, 1, 1, 414, 1, 1, 1, 6, 1, 612, 1, 2, 1, 1
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OFFSET
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1,12
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LINKS
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PROG
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(PARI)
\\ This is for computing a small number of terms:
nonprimepower_divisors_with1_reversed(n) = vecsort(select(d -> ((1==d) || !isprimepower(d)), divisors(n)), , 4);
partitions_into_with_trailing_ones(n, parts, from=1) = if(!n, 1, if(#parts<=(from+1), if(#parts == from, 1, (1+(n\parts[from]))), my(s=0); for(i=from, #parts, if(parts[i]<=n, s += partitions_into_with_trailing_ones(n-parts[i], parts, i))); (s)));
A014649(n) = partitions_into_with_trailing_ones(n-1, nonprimepower_divisors_with1_reversed(n)); \\ Antti Karttunen, Aug 23 2019
(PARI) \\ For an efficient program to compute large numbers of terms, see PARI program included in the Links-section.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A014650
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Number of partitions of n into its divisors that are powers of primes (A000961) with at least one part of size 1.
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+10
3
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1, 1, 1, 2, 1, 5, 1, 6, 3, 8, 1, 27, 1, 11, 11, 26, 1, 43, 1, 63, 15, 17, 1, 215, 5, 20, 18, 114, 1, 226, 1, 166, 23, 26, 23, 734, 1, 29, 27, 728, 1, 422, 1, 261, 181, 35, 1, 2697, 7, 179, 35, 357, 1, 791, 35, 1729, 39, 44, 1, 6747, 1, 47, 325, 1626, 41, 996, 1, 594, 47, 1062, 1, 20345, 1, 56, 327, 735, 47, 1374, 1, 13485, 216, 62, 1
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OFFSET
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1,4
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LINKS
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PROG
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(PARI)
\\ This is for computing a small number of terms:
primepower_divisors_with1_reversed(n) = vecsort(select(d -> ((1==d) || isprimepower(d)), divisors(n)), , 4);
partitions_into_with_trailing_ones(n, parts, from=1) = if(!n, 1, if(#parts<=(from+1), if(#parts == from, 1, (1+(n\parts[from]))), my(s=0); for(i=from, #parts, if(parts[i]<=n, s += partitions_into_with_trailing_ones(n-parts[i], parts, i))); (s)));
A014650(n) = partitions_into_with_trailing_ones(n-1, primepower_divisors_with1_reversed(n)); \\ Antti Karttunen, Sep 10 2018
(PARI) \\ For an efficient program to compute large numbers of terms, see David A. Corneth's PARI program included in the Links-section. - Antti Karttunen, Sep 12 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A286852
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Number of partitions of n into unitary prime divisors of n.
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+10
3
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1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 2, 0, 1, 1, 21, 1, 0, 2, 2, 2, 0, 1, 2, 2, 1, 1, 28, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 5, 1, 2, 1, 0, 2, 42, 1, 1, 2, 43, 1, 0, 1, 2, 1, 1, 2, 49, 1, 1, 0, 2, 1, 5, 2, 2, 2, 1, 1, 10, 2, 1, 2, 2, 2
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OFFSET
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0,7
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LINKS
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FORMULA
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a(n) = [x^n] Product_{p|n, p prime, gcd(p, n/p) = 1} 1/(1 - x^p).
a(n) = 0 if n is a powerful number (A001694).
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EXAMPLE
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a(6) = 2 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are unitary prime divisors {2, 3} therefore we have [3, 3] and [2, 2, 2].
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MATHEMATICA
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Join[{1}, Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[GCD[n/d[[k]], d[[k]]] == 1 && PrimeQ[d[[k]]]] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 1, 95}]]
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PROG
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(PARI)
A055231(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); } \\ From A055231
unitary_prime_factors(n) = { my(ufs = factor(A055231(n))); ufs[, 1]~; };
partitions_into(n, parts, from=1) = if(!n, 1, my(k = #parts, s=0); for(i=from, k, if(parts[i]<=n, s += partitions_into(n-parts[i], parts, i))); (s));
A286852(n) = if(n<2, 1-n, partitions_into(n, vecsort(unitary_prime_factors(n), , 4))); \\ Antti Karttunen, Jul 02 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A286851
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Number of compositions (ordered partitions) of n into unitary divisors of n.
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+10
2
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1, 1, 2, 2, 2, 2, 25, 2, 2, 2, 129, 2, 170, 2, 742, 450, 2, 2, 4603, 2, 1503, 3321, 29967, 2, 9278, 2, 200390, 2, 13460, 2, 154004511, 2, 2, 226020, 9262157, 51886, 127654, 2, 63346598, 2044895, 170354, 2, 185493291001, 2, 1304512, 567124, 2972038875, 2, 59489916, 2, 20367343494, 184947044, 14324735, 2
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = [x^n] 1/(1 - Sum_{d|n, gcd(d, n/d) = 1} x^d).
a(n) = 2 if n is a prime power (A246655).
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EXAMPLE
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a(8) = 2 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are unitary divisors {1, 8} therefore we have [8] and [1, 1, 1, 1, 1, 1, 1, 1].
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MAPLE
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a:= proc(n) option remember; local b, l; l, b:=
select(x-> igcd(x, n/x)=1, numtheory[divisors](n)),
proc(m) option remember; `if`(m=0, 1,
add(`if`(j>m, 0, b(m-j)), j=l))
end; b(n)
end:
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MATHEMATICA
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Join[{1}, Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[GCD[n/d[[k]], d[[k]]] == 1] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 1, 53}]]
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PROG
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(Python)
from sympy import divisors, gcd
from sympy.core.cache import cacheit
@cacheit
def a(n):
l=[x for x in divisors(n) if gcd(x, n//x)==1]
@cacheit
def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)
return b(n)
print([a(n) for n in range(61)]) # Indranil Ghosh, Aug 01 2017, after Maple code
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A349542
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Number of partitions of n into distinct unitary divisors of n.
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+10
1
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1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2
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OFFSET
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0,7
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LINKS
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FORMULA
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a(n) = [x^n] Product_{d|n, gcd(d,n/d) = 1} (1 + x^d).
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MATHEMATICA
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a[n_] := SeriesCoefficient[Product[(1 + Boole[GCD[n/d, d] == 1] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 114}]
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PROG
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(PARI) A349542(n) = if(!n, 1, my(p=1); fordiv(n, d, if(1==gcd(d, n/d), p *= (1 + 'x^d))); polcoeff(p, n)); \\ Antti Karttunen, Nov 22 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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