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A370521 The smallest number k that can be partitioned in n ways as the sum of two Blum numbers (A016105). 1

%I #15 Mar 19 2024 13:52:48

%S 1,42,90,162,234,474,270,378,558,594,774,846,970,810,1050,630,1370,

%T 1134,990,1170,1470,1730,1530,2054,1970,1386,1638,1710,2178,2070,2630,

%U 2250,1890,2730,2394,2310,3234,3230,3530,2790,2898,3650,3470,4010,3570,3654,2970,3150

%N The smallest number k that can be partitioned in n ways as the sum of two Blum numbers (A016105).

%H Michael S. Branicky, <a href="/A370521/b370521.txt">Table of n, a(n) for n = 0..11549</a>

%e 1 cannot be written as the sum of two Blum numbers, so a(0) = 1.

%e Since A016105(k) >= 21, for k >= 1, the numbers 2 through 41 cannot be written as the sum of two Blum numbers. 42 = 21 + 21 = A016105(1) + A016105(1), so a(1) = 42.

%e 90 = 21 + 69 = A016105(1) + A016105(4), 90 = 33 + 57 = A016105(2) + A016105(3), and the numbers 1 to 89 cannot be written in two ways as the sum of two Blum numbers. Thus a(2) = 90.

%o (Magma) pp:=PrimeDivisors; blum:=func<n|#Divisors(n) eq 4 and #pp(n) eq 2 and pp(n)[1] mod 4 eq 3 and pp(n)[2] mod 4 eq 3>;b:=[n: n in [1..5000]|blum(n)]; a:=[]; for n in [0..47] do k:=1; while #RestrictedPartitions(k,2,Set(b)) ne n do k:=k+1; end while; Append(~a,k); end for; a;

%o (Python)

%o from sympy import factorint

%o from itertools import takewhile

%o from collections import Counter

%o def okA016105(n):

%o f = factorint(n)

%o return len(f)==2 and sum(f.values())==2 and all(p%4==3 for p in f)

%o def aupto(N): # N is limit of terms considered; use 2*10**6 for b-file

%o s = [k for k in range(1, N+1) if okA016105(k)]

%o c = Counter(x+y for i, x in enumerate(s) if 2*i<=N for y in s[i:] if x+y<=N)

%o adict = {0: 1}

%o for k in sorted(c):

%o v = c[k]

%o if v not in adict: adict[v] = k

%o adict_rev = (adict.get(i) for i in range(max(adict)+1))

%o return list(takewhile(lambda v:v!=None, adict_rev))

%o print(aupto(4010)) # _Michael S. Branicky_, Feb 28 2024

%Y Cf. A016105.

%K nonn

%O 0,2

%A _Marius A. Burtea_, Feb 27 2024

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Last modified July 13 11:10 EDT 2024. Contains 374282 sequences. (Running on oeis4.)