A286219 -id:A286219 - cp's OEIS frontend

A286531 Restricted growth sequence of A278531 (prime-signature of A163511).

1, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 7, 5, 5, 2, 8, 6, 9, 4, 10, 7, 7, 3, 9, 7, 11, 5, 7, 5, 5, 2, 12, 8, 13, 6, 14, 9, 9, 4, 14, 10, 15, 7, 10, 7, 7, 3, 13, 9, 15, 7, 15, 11, 11, 5, 9, 7, 11, 5, 7, 5, 5, 2, 16, 12, 17, 8, 18, 13, 13, 6, 19, 14, 20, 9, 14, 9, 9, 4, 18, 14, 21, 10, 21, 15, 15, 7, 14, 10, 15, 7, 10, 7, 7, 3, 17, 13, 20, 9, 21, 15, 15, 7, 20, 15

Offset: 0

Antti Karttunen, May 17 2017

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A046523, A163511, A286533, A286535, A286219, A286622, A305433, A305434.

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A054429(n) = ((3<<#binary(n\2))-n-1); \\ After M. F. Hasler, Aug 18 2014
A278531(n) = if(!n,1,A278222(A054429(n)));
write_to_bfile(0,rgs_transform(vector(65538,n,A278531(n-1))),"b286531.txt");

A286218 Number of partitions of n into parts with an odd number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 4, 4, 6, 7, 9, 11, 13, 16, 19, 23, 28, 33, 40, 46, 55, 65, 76, 89, 104, 121, 141, 163, 190, 219, 253, 290, 334, 383, 439, 502, 573, 653, 744, 845, 961, 1089, 1234, 1395, 1576, 1780, 2007, 2259, 2544, 2856, 3209, 3598, 4033, 4516, 5051, 5644, 6304, 7033, 7843

Offset: 0

Ilya Gutkovskiy, May 04 2017

nonn

			a(8) = 4 because we have [8], [5, 3], [3, 3, 2] and [2, 2, 2, 2].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prime Factor
Index entries for related partition-counting sequences

Cf. A001156, A026424, A285799, A286219.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      `if`(bigomega(d)::odd, d, 0), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, May 04 2017

nmax = 60; CoefficientList[Series[Product[1/(1 - Boole[OddQ[PrimeOmega[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^A026424(k)).

A286223 Number of partitions of n into distinct parts with an even number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 3, 2, 0, 1, 3, 4, 4, 2, 1, 4, 6, 5, 4, 4, 6, 10, 10, 6, 6, 10, 13, 14, 11, 9, 14, 21, 21, 17, 17, 23, 31, 31, 25, 25, 33, 41, 43, 39, 38, 50, 61, 60, 56, 58, 68, 83, 87, 79, 82, 99, 115, 121, 118, 118, 139, 163, 164, 157, 165, 189, 216, 228, 221, 229, 265, 296

Offset: 0

Ilya Gutkovskiy, May 04 2017

nonn

			a(10) = 3 because we have [10], [9, 1] and [6, 4].

Amiram Eldar, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prime Factor
Index entries for related partition-counting sequences

Cf. A028260, A286219, A286221, A286222.

nmax = 75; CoefficientList[Series[Product[1 + Boole[EvenQ[PrimeOmega[k]]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A028260(k)).

A286227 Number of compositions (ordered partitions) of n into parts with an even number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 7, 10, 15, 24, 36, 53, 78, 118, 179, 271, 405, 605, 907, 1366, 2055, 3086, 4628, 6948, 10440, 15689, 23560, 35371, 53110, 79771, 119821, 179958, 270243, 405833, 609495, 915394, 1374780, 2064647, 3100680, 4656676, 6993575, 10503180, 15773877, 23689467, 35577360

Offset: 0

Ilya Gutkovskiy, May 04 2017

nonn

			a(6) = 5 because we have [6], [4, 1, 1], [1, 4, 1], [1, 1, 4] and [1, 1, 1, 1, 1, 1].

Amiram Eldar, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Prime Factor
Index entries for sequences related to compositions

Cf. A028260, A286219, A286223, A286225, A286226.

nmax = 45; CoefficientList[Series[1/(1 - Sum[Boole[EvenQ[PrimeOmega[k]]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^A028260(k)).

cp's OEIS Frontend

A286531 Restricted growth sequence of A278531 (prime-signature of A163511).

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A286218 Number of partitions of n into parts with an odd number of prime divisors (counted with multiplicity).

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Maple

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A286223 Number of partitions of n into distinct parts with an even number of prime divisors (counted with multiplicity).

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Mathematica

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A286227 Number of compositions (ordered partitions) of n into parts with an even number of prime divisors (counted with multiplicity).

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Mathematica

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