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

%I A074140 #44 Feb 16 2025 08:32:47
%S A074140 1,2,10,50,346,3182,38770,609290,11226106,250148582,7057182250,
%T A074140 216512001950,7903965900226,321552174623162,13779150603234010,
%U A074140 644574260638821590,33968684108427733426,1994885097404292104942,121496572792097514728530,8114030083731371137603190
%N A074140 Sum of least integers of prime signatures over all partitions of n.
%C A074140 Old name was: Sum of terms in n-th group in A036035.
%C A074140 a(n) is also the sum of terms in n-th row of A063008, A087443 or A227955.
%H A074140 Peter Luschny and Alois P. Heinz, <a href="/A074140/b074140.txt">Table of n, a(n) for n = 0..350</a>
%H A074140 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSignature.html">Prime Signature</a>
%H A074140 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A074140 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_signature">Prime signature</a>
%H A074140 <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>
%e A074140 a(6) = 64+96+144+216+240+360+900+840+1260+4620+30030 = 38770.
%p A074140 b:= proc(n, i, j) option remember;
%p A074140       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, j)+
%p A074140       `if`(i>n, 0, ithprime(j)^i*b(n-i, i, j+1))))
%p A074140     end:
%p A074140 a:= n-> b(n$2, 1):
%p A074140 seq(a(n), n=0..40);  # _Alois P. Heinz_, Aug 03 2013
%t A074140 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_ *)
%o A074140 (Sage)
%o A074140 def A074140(n):
%o A074140     L = []
%o A074140     P = primes_first_n(n)
%o A074140     for p in Partitions(n):
%o A074140         m = mul(P[i]^pi for i, pi in enumerate(p))
%o A074140         L.append(m)
%o A074140     return add(L)
%o A074140 [A074140(n) for n in (0..20)]  # _Peter Luschny_, Aug 02 2013
%Y A074140 Cf. A036035, A063008, A074139, A074141, A025487, A087443, A227955, A332626.
%K A074140 nonn
%O A074140 0,2
%A A074140 _Amarnath Murthy_, Aug 28 2002
%E A074140 More terms from _Alford Arnold_, Sep 10 2002
%E A074140 a(10)-a(12) from Thomas A. Rockwell (LlewkcoRAT(AT)aol.com), Sep 30 2004
%E A074140 a(12) corrected by _Peter Luschny_, Aug 03 2013
%E A074140 New name from _Alois P. Heinz_, Aug 03 2013