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A074140
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Sum of least integers of prime signatures over all partitions of n.
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9
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1, 2, 10, 50, 346, 3182, 38770, 609290, 11226106, 250148582, 7057182250, 216512001950, 7903965900226, 321552174623162, 13779150603234010, 644574260638821590, 33968684108427733426, 1994885097404292104942, 121496572792097514728530, 8114030083731371137603190
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history;
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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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
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EXAMPLE
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a(6) = 64+96+144+216+240+360+900+840+1260+4620+30030 = 38770.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy, Aug 28 2002
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EXTENSIONS
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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
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STATUS
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approved
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