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A327621
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Sums of distinct powers of 3 and powers of 4 (greater than 1).
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5
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3, 4, 7, 9, 12, 13, 16, 19, 20, 23, 25, 27, 28, 29, 30, 31, 32, 34, 36, 39, 40, 43, 46, 47, 50, 52, 55, 56, 59, 64, 67, 68, 71, 73, 76, 77, 80, 81, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 101, 103, 104, 106, 107, 108, 109, 110, 111, 112
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OFFSET
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1,1
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COMMENTS
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Record gaps in this sequence are : a(2) - a(1) = 1, a(3) - a(2) = 3, a(30) - a(29) = 5, a(112) - a(111) = 39, a(9863) - a(9862) = 1084, a(34096) - a(34095) = 7682, ...
These gaps are closely related to the gaps in the set where 3^0 and 4^0 are (both) also allowed to be in the sum, in which case the first missing numbers are A367090 = (62, 63, 143, 144, 207, ...), see also Melfi's paper. It is obvious that the study of these gaps is crucial for the proof of Erdös conjecture.
The record gap a(9863) - a(9862) = 1084 explains the discontinuity seen in the graph of a(1..10^4). (End)
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LINKS
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FORMULA
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For A(x) the enumerating function, Erdős conjectured that A(x) > c*x.
G. Melfi proved that A(x) > x^0.965 for sufficiently large x.
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EXAMPLE
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40 is in the sequence because 40 = 27 + 9 + 4.
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MATHEMATICA
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f[b_, m_] := Select[b Range[0, m/b], Max@ IntegerDigits[#, b] < 2 &]; mx=200; Union@ Select[Total /@ Tuples[{f[3, mx], f[4, mx]}], 0 < # < mx &] (* Giovanni Resta, Sep 19 2019 *)
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PROG
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(PARI) A327621_upto(N, S=[0])={for(b=3, 4, for(k=1, logint(N, b), my(p=b^k); S=setunion(S, [x+p|x<-S, x+p<=N]))); S[^1]} \\ M. F. Hasler, Nov 02 2023
(Python)
"list(x < N | x = sum(3^j, j in J) + sum(4^k, k in K); J, K subset N*)."
S = {0} # empty sum
for b in (3, 4):
p = b
while p < N: S |= {k+p for k in S if k+p < N} ; p *= b
return sorted(S) # includes a(0) = 0, so a(1, 2, 3, ...) = 3, 4, 9, ...
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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