A330394 -id:A330394 - cp's OEIS frontend

A307539 Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)...prime(y_k).

1, 2, 9, 125, 2401, 161051, 4826809, 410338673, 16983563041, 1801152661463, 420707233300201, 25408476896404831, 6582952005840035281, 925103102315013629321, 73885357344138503765449, 12063348350820368238715343, 3876269050118516845397872321

Offset: 0

Gus Wiseman, Apr 13 2019

nonn

			The square partition (4,4,4,4) has Heinz number prime(4)^4 = 7^4 = 2401.

Alois P. Heinz, Table of n, a(n) for n = 0..303

After a(0) = 1, same as A062457.
Cf. A002024, A047993, A056239, A096771, A106529, A112798, A115720, A174090, A257990, A263297.
Cf. A088218, A330394.

a:= n-> mul(ithprime(i), i=[n$n]):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 03 2020

Mathematica
```
Table[If[n==0,1,Prime[n]]^n,{n,0,10}]
```

a(n) = A330394(A088218(n)). - Alois P. Heinz, Mar 03 2020

A332967 Sum of all integers m satisfying Omega(m) = n and pi(p) <= n for all prime factors p of m.

Original entry on oeis.org

1, 2, 19, 410, 14343, 1139166, 89131918, 10861230692, 1271028562739, 203393524967230, 52274418436233714, 11160490802017899420, 3415612116240107778630, 1088775430914588654276060, 311608007930071575510930780, 99738699420765496000734958440

Offset: 0

Alois P. Heinz, Mar 04 2020

nonn

			a(2) = 4 + 6 + 9 = 2*2 + 2*3 + 3*3 = 19.

Alois P. Heinz, Table of n, a(n) for n = 0..279

Row sums of A330394.
Main diagonal of A343751.
Cf. A000040, A000720, A001222, A027748, A088218, A124960.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(ithprime(j)*b(n-1, j), j=1..i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);

a(n) = [x^n] Product_{i=1..n} 1/(1-prime(i)*x).
a(n) = A124960(2n,n).
a(n) = Sum_{k=1..A088218(n)} A330394(n,k).
a(n) = A343751(n,n).

cp's OEIS Frontend

A307539 Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)...prime(y_k).

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A332967 Sum of all integers m satisfying Omega(m) = n and pi(p) <= n for all prime factors p of m.

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cp's OEIS Frontend

A307539 Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Views

Author

Keywords

Examples

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Crossrefs

Programs

Maple

Mathematica

Formula

A332967 Sum of all integers m satisfying Omega(m) = n and pi(p) <= n for all prime factors p of m.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A307539 Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)...prime(y_k).