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

Author

Amarnath Murthy, Aug 28 2002

Keywords

• nonn

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.

Examples

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

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

Crossrefs

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

Programs

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 *)

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

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