A325044 - OEIS

A325044

Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 60, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 88, 89, 92, 94, 96, 97, 101

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OFFSET

1,1

COMMENTS

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is less than or equal to their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A096276.

LINKS

Table of n, a(n) for n=1..67.

FORMULA

A003963(a(n)) <= A056239(a(n)).

a(n) = A325038(n)/2.

Union of A301987 and A325038.

EXAMPLE

The sequence of terms together with their prime indices begins:

2: {1}

3: {2}

4: {1,1}

5: {3}

6: {1,2}

7: {4}

8: {1,1,1}

9: {2,2}

10: {1,3}

11: {5}

12: {1,1,2}

13: {6}

14: {1,4}

16: {1,1,1,1}

17: {7}

18: {1,2,2}

19: {8}

20: {1,1,3}

22: {1,5}

23: {9}

24: {1,1,1,2}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], Times@@primeMS[#]<=Plus@@primeMS[#]&]

CROSSREFS

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325042.

Sequence in context: A070776 A320230 A076564 * A343822 A303550 A164563

Adjacent sequences: A325041 A325042 A325043 * A325045 A325046 A325047

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 25 2019

STATUS

approved