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 A327621 Sums of distinct powers of 3 and powers of 4 (greater than 1). 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS From M. F. Hasler, Nov 16 2023: (Start) 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) LINKS M. F. Hasler, Table of n, a(n) for n = 1..10000, Nov 02 2023 S. A. Burr, P. Erdős, R. L. Graham, and W. Wen-Ching Li, Complete sequences of sets of integer powers, Acta Arithmetica 77(2) (1996), 133-138. P. Erdős, Conjecture about the enumerating function A(x) G. Melfi, An additive problem about powers of fixed integers, Rend. Circ. Mat. Palermo 50 (2001), 239-246. FORMULA 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. EXAMPLE 40 is in the sequence because 40 = 27 + 9 + 4. MATHEMATICA 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 *) PROG (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) def A327621_upto(N): "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, ... # M. F. Hasler, Nov 09 2023 CROSSREFS Cf. A000244 (powers of 3), A000302 (powers of 4). Cf. A005836 and A000695 (sums of distinct powers of 3 and of 4). Sequence in context: A032729 A035270 A120451 * A060428 A352274 A035238 Adjacent sequences: A327618 A327619 A327620 * A327622 A327623 A327624 KEYWORD nonn AUTHOR Giuseppe Melfi, Sep 19 2019 EXTENSIONS More terms from Giovanni Resta, Sep 19 2019 STATUS approved

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Last modified September 13 04:25 EDT 2024. Contains 375859 sequences. (Running on oeis4.)