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A316291 Common denominators of all Egyptian fraction representations of unity (EFROUs) such that replacing two terms with their sum never results in another EFROU.

6, 20, 28, 30, 40, 48, 60, 66, 72, 80, 84, 88, 90, 96, 104, 120, 126, 132, 140, 144, 150, 156, 160, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 260, 264

Offset: 1

David V. Feldman, Jun 28 2018

nonn

The relevant EFROUs serve as generators, general EFROUs arising by repeatedly replacing terms 1/a with 1/b + 1/c. a(b+c)=bc requires taking b=D(B+C)B and c=D(B+C)C, where B,C|a, gcd(B,C)=1 and D=a/BC.

			For 6, 1 = 1/2 + 1/3 + 1/6 (combining 1/3 + 1/6 would duplicate 1/2). For 20, 1 =  1/2 + 1/4 + 1/5 + 1/20.  Observe that 1 = 1/2 + 1/3 + 1/12 + 1/20 + 1/30 has common denominator 60 even though 1/60 itself does not appear as a summand; since also 1 =  1/3 + 1/4 + 1/5 + 1/10 + 1/12 + 1/30, uniqueness fails for the relevant EFROU, the first such example.

A316262 Numbers k such that gcd(k, floor(phi*k)) > 1, where phi is the golden ratio.

Original entry on oeis.org

4, 6, 8, 10, 14, 15, 20, 21, 24, 25, 26, 30, 35, 36, 39, 40, 45, 46, 50, 52, 54, 55, 56, 62, 65, 66, 68, 69, 72, 76, 78, 82, 84, 88, 90, 91, 92, 93, 94, 98, 102, 104, 108, 114, 117, 118, 120, 124, 126, 130, 132, 134, 136, 140, 141, 143, 144, 146, 147, 150

Offset: 1

David V. Feldman, Jun 27 2018

nonn

			2 divides both 4 and floor(phi*4)=6, so 4 is a term.

Cf. A000201, A001622.

select(n->gcd(n,floor(((sqrt(5)-1)/2)*n))>1,[$1..160]); # Muniru A Asiru, Jun 28 2018

Select[Range[150], GCD[#, Floor[GoldenRatio #]] > 1 &] (* Giovanni Resta, Jun 28 2018 *)

is(n) = gcd(n, floor((sqrt(5)-1)/2*n)) > 1 \\ Felix Fröhlich, Jun 29 2018

from math import gcd, isqrt
from itertools import count, islice
def A316262_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:gcd(n,n+isqrt(5*n**2)>>1)>1,count(max(startvalue,1)))
A316262_list = list(islice(A316262_gen(),30)) # Chai Wah Wu, Aug 10 2022

A316221 Let S(n) = set of divisors of n, excluding n; sequence gives n such that there is a unique relatively prime subset of S(n) that sums to n.

Original entry on oeis.org

6, 12, 18, 20, 28, 42, 54, 56, 66, 88, 100, 104, 162, 176, 196, 208, 272, 304, 368, 414, 460, 464, 486, 490, 496, 500, 558, 572, 580, 650, 666, 726, 736, 748, 812, 820, 868, 928, 968, 992, 1036, 1148, 1184, 1204, 1312, 1316, 1352, 1372, 1376, 1458, 1484, 1504

Offset: 1

David V. Feldman, Jun 27 2018

nonn

The relatively prime condition arises naturally from the perspective of Egyptian fractions representations of unity which in turn arise upon dividing the elements of such a subset all by n. In particular the condition guarantees that the Egyptian fraction representation of unity doesn't arise already from any smaller n.

			6=1+2+3, 12=1+2+3+6, 18=1+2+6+9, 20=1+4+5+10, 28=1+2+4+7+14, 42=1+6+14+21.

Subsequence of A005835. Related to A064771.

ric[r_, g_, p_] := Block[{v}, If[r==0, If[g==1, c++], If[c<2 && Total@p >= r, ric[r, g, Rest@ p]; v = p[[1]]; If[r>=v, ric[r-v, GCD[g, v], Rest@ p]]]]]; ok[n_] := DivisorSigma[1, n] >= 2 n && (c = 0; ric[n, n, Reverse@ Most@ Divisors@ n]; c == 1); Select[ Range[2000], ok] (* Giovanni Resta, Jun 27 2018 *)

More terms from Giovanni Resta, Jun 27 2018

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User: David V. Feldman

A316291 Common denominators of all Egyptian fraction representations of unity (EFROUs) such that replacing two terms with their sum never results in another EFROU.

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A316262 Numbers k such that gcd(k, floor(phi*k)) > 1, where phi is the golden ratio.

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A316221 Let S(n) = set of divisors of n, excluding n; sequence gives n such that there is a unique relatively prime subset of S(n) that sums to n.

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Mathematica

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