A367090 -id:A367090 - cp's OEIS frontend

A367091 Length of runs of consecutive numbers in A367090, i.e., size of gaps in the set of sums of distinct powers of 3 and distinct powers of 4.

2, 2, 36, 36, 2, 2, 23, 2, 2, 36, 36, 2, 2, 23, 2, 2, 36, 2, 2, 36, 2, 2, 23, 2, 2, 36, 36, 2, 2, 23, 2, 2, 2, 2, 2, 2, 23, 2, 2, 36, 36, 2, 2, 23, 2, 2, 36, 2, 2, 14, 14, 2, 2, 36, 2, 2, 23, 2, 2, 36, 36, 2, 2, 23, 2, 2, 2, 2, 2, 2, 23, 2, 2, 36, 36, 2, 2, 23, 2, 2, 36, 2, 2, 36

Offset: 1

M. F. Hasler, Nov 08 2023

nonn

The numbers that occur in this sequence are, in order of first appearance: 2, 36, 23, 14, 1081, 20, ... It is not known which numbers will eventually appear and which numbers will never occur in this sequence.
The first 1's (which correspond to isolated numbers in A367090, or gaps that are a singleton) appear as a(131) = a(132) = 1.
This set exhibits an interesting self-similar, pseudo-symmetric structure. This is due to the Proposition given in A367090.

			Sequence A367090 (= numbers that are not the sum of distinct powers of 3 or 4) starts (62, 63, 143, 144, 207, 208, 209, 210, ...), so the first two runs of consecutive terms are 2 = #{62, 63} and 2 = #{143, 144}, the next run is of length 36.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A367090; A005836 and A000695 (sums of distinct powers of 3 resp. 4).

D(v)=v[^1]-v[^-1] \\ first differences
A367091_upto(N, DA=D(A367090_upto(N)))= D([ k | k<-[0..#DA], !k|| DA[k]-1 ])

A367092 Starting values of runs of consecutive numbers in A367090, i.e., minima of gaps in the set of sums of distinct powers of 3 and distinct powers of 4.

Original entry on oeis.org

62, 143, 207, 463, 561, 642, 706, 791, 872, 936, 1487, 1585, 1666, 1730, 1815, 1896, 1960, 2249, 2395, 2650, 2748, 2829, 2893, 2978, 3059, 3123, 3674, 3772, 3853, 3917, 4002, 4083, 4158, 4582, 4657, 4738, 4802, 4887, 4968, 5032, 5583, 5681, 5762, 5826, 5911, 5992, 6056, 6345, 6491

Offset: 1

M. F. Hasler, Nov 08 2023

nonn

Also: terms a(n) of A367090 such that a(n)-1 is not in A367090.
("Consecutive" includes the possibility of having a gap of just one single isolated missing number.)
This sequence together with A367091 (run lengths), provide a "compressed version" of A367090.

			Sequence A367090 (= numbers that are not the sum of distinct powers of 3 or 4) starts (62, 63, 143, 144, 207, 208, 209, 210, ...), so the first three runs of consecutive terms start at a(1) = 62, a(2) = 143, and a(3) = 207.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A367090, A367091; A005836 and A000695 (sums of distinct powers of 3 resp. 4).

A367092_upto(N, A=A367090_upto(N))=[ A[k] | k<-[1..#A], A[k-(k>1)]!=A[k]-1 ]

{ x in A367090 | x-1 is not in A367090 }

A327621 Sums of distinct powers of 3 and powers of 4 (greater than 1).

Original entry on oeis.org

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

Offset: 1

Giuseppe Melfi, Sep 19 2019

nonn

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)

			40 is in the sequence because 40 = 27 + 9 + 4.

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.

Cf. A000244 (powers of 3), A000302 (powers of 4).

Cf. A005836 and A000695 (sums of distinct powers of 3 and of 4).

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 *)

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

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

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.

More terms from Giovanni Resta, Sep 19 2019

cp's OEIS Frontend

A367091 Length of runs of consecutive numbers in A367090, i.e., size of gaps in the set of sums of distinct powers of 3 and distinct powers of 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A367092 Starting values of runs of consecutive numbers in A367090, i.e., minima of gaps in the set of sums of distinct powers of 3 and distinct powers of 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A327621 Sums of distinct powers of 3 and powers of 4 (greater than 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions