A332626 -id:A332626 - cp's OEIS frontend

A074140 Sum of least integers of prime signatures over all partitions of n.

1, 2, 10, 50, 346, 3182, 38770, 609290, 11226106, 250148582, 7057182250, 216512001950, 7903965900226, 321552174623162, 13779150603234010, 644574260638821590, 33968684108427733426, 1994885097404292104942, 121496572792097514728530, 8114030083731371137603190

Offset: 0

Amarnath Murthy, Aug 28 2002

nonn

Old name was: Sum of terms in n-th group in A036035.

a(n) is also the sum of terms in n-th row of A063008, A087443 or A227955.

			a(6) = 64+96+144+216+240+360+900+840+1260+4620+30030 = 38770.

Peter Luschny and Alois P. Heinz, Table of n, a(n) for n = 0..350
Eric Weisstein's World of Mathematics, Prime Signature
Wikipedia, Partition (number theory)
Wikipedia, Prime signature
Index entries for sequences related to prime signature

Cf. A036035, A063008, A074139, A074141, A025487, A087443, A227955, A332626.

b:= proc(n, i, j) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, j)+
      `if`(i>n, 0, ithprime(j)^i*b(n-i, i, j+1))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 03 2013

b[n_, i_, j_] := b[n, i, j] = If[n == 0, 1, If[i<1, 0, b[n, i-1, j]+If[i>n, 0, Prime[j]^i*b[n-i, i, j+1]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)

def A074140(n):
    L = []
    P = primes_first_n(n)
    for p in Partitions(n):
        m = mul(P[i]^pi for i, pi in enumerate(p))
        L.append(m)
    return add(L)
[A074140(n) for n in (0..20)]  # Peter Luschny, Aug 02 2013

More terms from Alford Arnold, Sep 10 2002
a(10)-a(12) from Thomas A. Rockwell (LlewkcoRAT(AT)aol.com), Sep 30 2004
a(12) corrected by Peter Luschny, Aug 03 2013
New name from Alois P. Heinz, Aug 03 2013

A328524 T(n,k) is the k-th smallest least integer of prime signatures for partitions of n into distinct parts; triangle T(n,k), n>=0, 1<=k<=A000009(n), read by rows.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 72, 64, 96, 144, 360, 128, 192, 288, 432, 720, 256, 384, 576, 864, 1440, 2160, 512, 768, 1152, 1728, 2592, 2880, 4320, 10800, 1024, 1536, 2304, 3456, 5184, 5760, 8640, 12960, 21600, 75600, 2048, 3072, 4608, 6912, 10368, 11520

Offset: 0

Alois P. Heinz, Feb 18 2020

			Triangle T(n,k) begins:
     1;
     2;
     4;
     8,   12;
    16,   24;
    32,   48,   72;
    64,   96,  144,  360;
   128,  192,  288,  432,  720;
   256,  384,  576,  864, 1440, 2160;
   512,  768, 1152, 1728, 2592, 2880, 4320, 10800;
  1024, 1536, 2304, 3456, 5184, 5760, 8640, 12960, 21600, 75600;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Eric Weisstein's World of Mathematics, Prime Signature
Wikipedia, Partition (number theory)
Wikipedia, Prime signature
Index entries for sequences related to prime signature

Column k=1-3 give: A000079, A003945 for n>2, A116453 for n>4.
Row sums give A332626.
Last elements of rows give A332644.
Cf. A000009, A087443 (for all partitions), A087980 (as sorted sequence).

b:= proc(n, i, j) option remember; `if`(i*(i+1)/2 x*ithprime(j)^i,
       b(n-i, min(n-i, i-1), j+1))[], b(n, i-1, j)[]]))
    end:
T:= n-> sort(b(n$2, 1))[]:
seq(T(n), n=0..12);

b[n_, i_, j_] := b[n, i, j] = If[i(i+1)/2 < n, {}, If[n == 0, {1}, Join[# * Prime[j]^i& /@ b[n - i, Min[n - i, i - 1], j + 1], b[n, i - 1, j]]]];
T[n_] := Sort[b[n, n, 1]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 07 2020, after Maple *)

cp's OEIS Frontend

A074140 Sum of least integers of prime signatures over all partitions of n.

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