A091114 -id:A091114 - cp's OEIS frontend

A091094 Number of partitions of n-th composite number not containing the smallest prime factor.

3, 6, 11, 19, 20, 35, 58, 99, 96, 154, 242, 407, 375, 573, 1331, 861, 1435, 1282, 1886, 2745, 4539, 3961, 9279, 5667, 8038, 13208, 11323, 15836, 22001, 35960, 30383, 41715, 120351, 56953, 92670, 77363, 104566, 247050, 140668, 227999, 188397

Offset: 1

Reinhard Zumkeller, Feb 22 2004

nonn

a(n) = A000041(A002808(n)) - A091114(n).

a(n) = A000041(A002808(n)) - A000041(A085271(n)). - Charlie Neder, Jan 11 2019

Cf. A002808, A000041, A020639, A056608.

Incorrect formula removed by Charlie Neder, Jan 11 2019

A091109 Number of occurrences of smallest prime factor in all partitions of n-th composite number: a(n)=A066633(A002808(n), A056608(n)).

Original entry on oeis.org

3, 8, 19, 15, 41, 83, 160, 122, 295, 526, 911, 683, 1538, 2540, 853, 4115, 3050, 6551, 10269, 15873, 11664, 24222, 8415, 36532, 54509, 39784, 80524, 117862, 171036, 124143, 246211, 351769, 72718, 499042, 360550, 703268, 984857, 353996

Offset: 1

Reinhard Zumkeller, Feb 22 2004

nonn

			n=2: A002808(2)=6=2*3 has A000041(6)=11 partitions: 6 = 5+1 = 4+2 = 4+1+1 = 3+3 = 3+2+1 = 3+1+1+1 = 2+2+2 = 2+2+1+1 = 2+1+1+1+1 = 1+1+1+1+1+1, 2 occurs 8 times, therefore a(2)=8.

Cf. A002808, A000041, A056608, A020639, A091114.

Count[Flatten[IntegerPartitions[#]],FactorInteger[#][[1,1]]]&/@ Select[ Range[ 60],CompositeQ] (* Harvey P. Dale, Sep 08 2018 *)

cp's OEIS Frontend

A091094 Number of partitions of n-th composite number not containing the smallest prime factor.

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A091109 Number of occurrences of smallest prime factor in all partitions of n-th composite number: a(n)=A066633(A002808(n), A056608(n)).

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