A316557 -id:A316557 - cp's OEIS frontend

A316556 Number of distinct LCMs of nonempty submultisets of the integer partition with Heinz number n.

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

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).

Number of distinct values obtained when A290103 is applied to all divisors of n larger than one. - Antti Karttunen, Sep 25 2018

			462 is the Heinz number of (5,4,2,1) which has possible LCMs of nonempty submultisets {1,2,4,5,10,20} so a(462) = 6.

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

Cf. A056239, A074761, A108917, A122768, A275972, A290103, A296150, A301957, A316313, A316314, A316430, A316555, A316557.

Cf. also A304793, A305611, A319685, A319695 for other similarly constructed sequences.

Table[Length[Union[LCM@@@Rest[Subsets[If[n==1,{},Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]]]]],{n,100}]

A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A316556(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s=A290103(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 25 2018

More terms from Antti Karttunen, Sep 25 2018

A316555 Number of distinct GCDs of nonempty submultisets of the integer partition with Heinz number n.

Original entry on oeis.org

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

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).

Number of distinct values obtained when A289508 is applied to all divisors of n larger than one. - Antti Karttunen, Sep 28 2018

			455 is the Heinz number of (6,4,3) which has possible GCDs of nonempty submultisets {1,2,3,4,6} so a(455) = 5.

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

Cf. A056239, A108917, A122768, A275972, A289508, A289509, A296150, A316313, A316314, A316430, A316556, A316557.

Cf. also A304793, A305611, A319685, A319695 for other similarly constructed sequences.

Table[Length[Union[GCD@@@Rest[Subsets[If[n==1,{},Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]]]]],{n,100}]

A289508(n) = gcd(apply(p->primepi(p),factor(n)[,1]));
A316555(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s=A289508(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 28 2018

More terms from Antti Karttunen, Sep 28 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}]]]]]]]&]

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A316556 Number of distinct LCMs of nonempty submultisets of the integer partition with Heinz number n.

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A316555 Number of distinct GCDs of nonempty submultisets of the integer partition with Heinz number n.

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

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