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

0,2

Comments

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.

Links

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

Example

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

Maple

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

Mathematica

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 *)

Prog

(Sage)
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

Crossrefs

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

Sequence in context: A020088 A205772 A306336 * A128449 A037514 A037717

Adjacent sequences: A074137 A074138 A074139 * A074141 A074142 A074143

Keyword

nonn

Author

Amarnath Murthy, Aug 28 2002

Extensions

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

Status

approved