A316440 -id:A316440 - cp's OEIS frontend

A316557 Number of distinct integer averages of subsets of the integer partition with Heinz number n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 2, 1, 3, 3, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 2, 3, 2, 2, 1, 2, 3, 3, 1, 4, 1, 3, 2, 3, 1, 2, 1, 3, 2, 2, 1, 2, 3, 3, 3, 2, 1, 3, 1, 3, 3, 1, 2, 4, 1, 4, 2, 4, 1, 2, 1, 2, 2, 2, 2, 5, 1, 3, 1, 3, 1, 4, 3, 2, 3, 4, 1, 3, 3, 3, 2, 3, 2, 2, 1, 3, 3, 3, 1, 4, 1, 2, 3

Offset: 1

Gus Wiseman, Jul 06 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(78) = 5 distinct integer averages of subsets of (6,2,1) are {1, 2, 3, 4, 6}.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A056239, A067538, A122768, A237984, A296150, A316313, A316314, A316440, A316555, A316556.

Table[Length[Select[Union[Mean/@Subsets[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],IntegerQ]],{n,100}]

up_to = 65537;
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
v056239 = vector(up_to,n,A056239(n));
A316557(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&(1==denominator(s = v056239[d]/bigomega(d)))&&!mapisdefined(m,s), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 25 2018

a(n) <= A316314(n). - Antti Karttunen, Sep 25 2018

More terms from Antti Karttunen, Sep 25 2018

A319335 Numerator of the average of the averages of all integer partitions of n.

Original entry on oeis.org

1, 3, 11, 31, 187, 131, 247, 1993, 4463, 3635, 395077, 24441, 81149, 10414421, 12868591, 10764151, 61170133, 419426561, 353495183, 3429826973, 29219934899, 5110021867, 142319532929, 606916707064, 87086496509, 4426308633083, 15954910019953, 38414031851849

Offset: 1

Gus Wiseman, Sep 17 2018

			The sequence of average averages begins: 1, 3/2, 11/6, 31/15, 187/84, 131/55, 247/100, 1993/770, 4463/1680, 3635/1323.

Andrew Howroyd, Table of n, a(n) for n = 1..500

Denominators are in A319336.

Cf. A056239, A067538, A237984, A296150, A316313, A316314, A316440.

Table[Numerator[Mean[Mean/@IntegerPartitions[n]]],{n,20}]

seq(n)={[numerator(poldegree(p)*subst(intformal(p/y)/p, y, 1)) | p <- Vec(-1+1/prod(k=1, n, 1 - x^k*y + O(x*x^n)))]} \\ Andrew Howroyd, Sep 19 2018

A319336 Denominator of the average of the averages of all integer partitions of n.

Original entry on oeis.org

1, 2, 6, 15, 84, 55, 100, 770, 1680, 1323, 141120, 8470, 27720, 3474900, 4228224, 3468465, 19459440, 131030900, 109156320, 1042578108, 8779605120, 1514663280, 41736380400, 175685635125, 24960905112, 1254125149200, 4476730258000, 10664476594200, 73326164511600

Offset: 1

Gus Wiseman, Sep 17 2018

			The sequence of average averages begins: 1, 3/2, 11/6, 31/15, 187/84, 131/55, 247/100, 1993/770, 4463/1680, 3635/1323.

Andrew Howroyd, Table of n, a(n) for n = 1..500

Numerators are in A319335.

Cf. A056239, A067538, A237984, A296150, A316313, A316314, A316440.

Table[Denominator[Mean[Mean/@IntegerPartitions[n]]],{n,20}]

seq(n)={[denominator(poldegree(p)*subst(intformal(p/y)/p, y, 1)) | p <- Vec(-1+1/prod(k=1, n, 1 - x^k*y + O(x*x^n)))]} \\ Andrew Howroyd, Sep 19 2018

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

Gus Wiseman, Jul 06 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Supersequence of A000961. - David A. Corneth, Jul 06 2018

			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).

Cf. A056239, A067538, A122768, A237984, A296150, A316313, A316314, A316440, A316557.

Select[Range[100],And@@IntegerQ/@Mean/@Union[Rest[Subsets[If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]&]

cp's OEIS Frontend

A316557 Number of distinct integer averages of subsets of the integer partition with Heinz number n.

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A319335 Numerator of the average of the averages of all integer partitions of n.

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A319336 Denominator of the average of the averages of all integer partitions of n.

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A316465 Heinz numbers of integer partitions such that every nonempty submultiset has an integer average.

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