A074140 - OEIS

%I #44 Feb 16 2025 08:32:47

%S 1,2,10,50,346,3182,38770,609290,11226106,250148582,7057182250,

%T 216512001950,7903965900226,321552174623162,13779150603234010,

%U 644574260638821590,33968684108427733426,1994885097404292104942,121496572792097514728530,8114030083731371137603190

%N Sum of least integers of prime signatures over all partitions of n.

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

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

%H Peter Luschny and Alois P. Heinz, <a href="/A074140/b074140.txt">Table of n, a(n) for n = 0..350</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSignature.html">Prime Signature</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_signature">Prime signature</a>

%H <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>

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

%p b:= proc(n, i, j) option remember;

%p       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, j)+

%p       `if`(i>n, 0, ithprime(j)^i*b(n-i, i, j+1))))

%p     end:

%p a:= n-> b(n$2, 1):

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Aug 03 2013

%t 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 (Sage)

%o def A074140(n):

%o     L = []

%o     P = primes_first_n(n)

%o     for p in Partitions(n):

%o         m = mul(P[i]^pi for i, pi in enumerate(p))

%o         L.append(m)

%o     return add(L)

%o [A074140(n) for n in (0..20)]  # _Peter Luschny_, Aug 02 2013

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

%K nonn,changed

%O 0,2

%A _Amarnath Murthy_, Aug 28 2002

%E More terms from _Alford Arnold_, Sep 10 2002

%E a(10)-a(12) from Thomas A. Rockwell (LlewkcoRAT(AT)aol.com), Sep 30 2004

%E a(12) corrected by _Peter Luschny_, Aug 03 2013

%E New name from _Alois P. Heinz_, Aug 03 2013