A316465
Heinz numbers of integer partitions such that every nonempty submultiset has an integer average.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 29, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 79, 81, 82, 83, 85, 87, 89, 91, 94, 97, 101, 103, 107, 109, 110, 111, 113, 115, 118, 121, 125, 127, 128
Offset: 1
Sequence of partitions begins (), (1), (2), (1,1), (3), (4), (1,1,1), (2,2), (3,1), (5), (6), (1,1,1,1), (7), (8), (4,2), (5,1), (9), (3,3), (2,2,2).
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Select[Range[100],And@@IntegerQ/@Mean/@Union[Rest[Subsets[If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]&]
A316525
Numbers whose average of prime factors is prime.
Original entry on oeis.org
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 21, 23, 25, 27, 29, 31, 32, 33, 37, 41, 43, 44, 47, 49, 53, 57, 59, 60, 61, 64, 67, 68, 69, 71, 73, 79, 81, 83, 85, 89, 93, 97, 101, 103, 105, 107, 109, 112, 113, 116, 121, 125, 127, 128, 129, 131, 133, 137, 139
Offset: 1
60 = 2*2*3*5 has average of prime factors (2+2+3+5)/4 = 3, which is prime, so 60 belongs to the sequence.
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Select[Range[100],PrimeQ[Mean[If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[p,{k}]]]]]]&]
Select[Range[200],PrimeQ[Mean[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ #]]]]&] (* Harvey P. Dale, Sep 28 2018 *)
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isok(n) = {my(f=factor(n)); iferr(isprime(sum(k=1, #f~, f[k,1]*f[k,2])/sum(k=1, #f~, f[k,2])), E, 0);} \\ Michel Marcus, Jul 06 2018
A360070
Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.
Original entry on oeis.org
1, 4, 8, 9, 12, 16, 18, 20, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 121, 125, 128, 144, 147, 150, 160, 162, 169, 175, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 245, 250, 252, 256, 272
Offset: 1
A partition of 20 with the same mean as its multiplicities is (5,4,3,2,1,1,1,1,1,1), so 20 is in the sequence.
A360670
Number of integer partitions of n whose parts have the same mean as their negated first differences.
Original entry on oeis.org
1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 2, 0, 2, 0, 2, 2, 0, 0, 5, 1, 0, 3, 3, 0, 4, 0, 5, 3, 0, 2, 10, 0, 0, 4, 10, 0, 5, 0, 7, 9, 0, 0, 17, 1, 6, 5, 10, 0, 9, 8, 14, 6, 0, 0, 34, 0, 0, 9, 18, 13, 13, 0, 17, 7, 19, 0, 40, 0, 0, 28
Offset: 0
The a(n) partitions for n = 4, 12, 24, 27, 30, 44:
(3,1) (9,3) (18,6) (19,7,1) (21,8,1) (33,11)
(9,2,1) (17,6,1) (20,5,2) (22,6,2) (34,5,4,1)
(18,4,2) (21,3,3) (23,4,3) (34,6,3,1)
(19,2,2,1) (25,2,1,1,1) (34,7,2,1)
(19,3,1,1) (34,8,1,1)
(35,4,3,2)
(35,5,2,2)
For example, the partition y = (28,4,3,1), with mean 9, has negated first differences (24,2,1), with mean 9, so y is counted under a(36).
These partitions have ranks
A360669.
A360614/
A360615 = mean of first differences of 0-prepended prime indices.
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Table[Length[Select[IntegerPartitions[n], Mean[#]==Mean[Differences[Reverse[#]]]&]],{n,0,30}]
A316520
Heinz numbers of integer partitions whose average is a prime number.
Original entry on oeis.org
3, 5, 9, 10, 11, 17, 21, 22, 25, 27, 28, 30, 31, 41, 46, 57, 59, 67, 68, 78, 81, 82, 83, 84, 85, 88, 90, 91, 98, 99, 100, 105, 109, 110, 111, 121, 125, 127, 146, 155, 157, 164, 179, 184, 191, 194, 203, 208, 211, 213, 222, 228, 241, 243, 247, 252, 253, 264, 267
Offset: 1
60 = 2*2*3*5 has prime indices {1,1,2,3} with average 7/4, so 60 does not belong to the sequence.
90 = 2*3*3*5 has prime indices {1,2,2,3} with average 8/2 = 4, so 90 belongs to the sequence.
Cf.
A000040,
A000041,
A000607,
A056239,
A067538,
A078175,
A100118,
A237984,
A296150,
A316313,
A316413.
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Select[Range[100],PrimeQ[Mean[If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]&]
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