A063647 -id:A063647 - cp's OEIS frontend

A339107 a(1) = 3; for n>1, a(n) = the smallest positive number not occurring earlier such that a(n-1)*a(n) is divisible by a(n-1)+a(n), where a(n) is not a prime or 4.

3, 6, 12, 24, 8, 56, 42, 21, 28, 70, 30, 15, 10, 40, 60, 20, 80, 48, 16, 112, 84, 14, 35, 140, 105, 120, 72, 9, 18, 36, 45, 90, 135, 108, 54, 27, 216, 270, 180, 144, 240, 160, 96, 32, 224, 168, 126, 63, 378, 189, 252, 315, 210, 231, 132, 44, 77, 462, 22, 99, 198, 165, 110, 374, 204, 68, 221, 2652

Offset: 1

Scott R. Shannon, Nov 23 2020

nonn

Given a(n-1) the candidates for a(n) are k*a(n-1)/(a(n-1)-k), where 1<=kA063647(n). The values of a(n-1)*a(n)/(a(n-1)+a(n)) are given by the companion sequence A339133.
The first term is 3 as one can easily show 1 and 2 cannot occur in the sequence; if a(n-1)=1 then 1*a(n)/(1+a(n)) has no integer solution while if a(n-1)=2 then 2*a(n)/(2+a(n)) has the one integer solution a(n)=2, but that is a(n-1).
One can also show that a(n) can never be a prime. The only way a(n)=p can be produced is if a(n-1) = p*(p-1). However the only candidate for a(n+1) if a(n)=p is the number p*(p-1), but that is a(n-1). Thus allowing a(n) to be a prime will halt the sequence; if a number of the form p*(p-1) occurs in the sequence the smallest candidate other than p must be chosen for the next term.
Likewise to avoid halting the sequence the number 4 cannot be chosen; if a(n-1)=4 the only candidates for a(n) would be 4 and 12, but a(3)=12 and 4 cannot occur again, thus a(n-1)=4 would halt the sequence.
It is likely all numbers other than 1,2,4 and the primes appear in the sequence, although this is unknown. The smallest composite not to have appeared after 100 thousand terms is 794. The one fixed point in the first 100 thousand terms is a(20572) = 20572.

			a(2) = 6. The only candidate for a(2) that satisfies 3*a(2) being divisible by 3+a(2) is a(2) = 6.
a(3) = 12. The candidates for a(3) given a(2) = 6 are 3,6,12,30, all of which satisfy 6*a(3) being divisible by 6+a(3). 3 and 6 have already appeared so the next smallest candidate is chosen, being 12.
a(4) = 24. The candidates for a(4) given a(3) = 12 are 4,6,12,24,36,60,132, all of which satisfy 12*a(4) being divisible by 12+a(4). 4 is not allowed as there would be no candidates for a(5), and 6 and 12 have already appeared, so the next smallest candidate is chosen, being 24.
a(5) = 8. There are 10 candidates for a(5) given a(4) = 24, the smallest that has not appeared is 8.

Robert Israel, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Line graph of the first 100000 terms.

Cf. A339133, A063647, A063427, A027750, A000040, A098550, A336957, A064413.

R:= 3: a:= 3: S:= {4,3}:
for i from 2 to 100 do
  Cands:= remove (t -> t < 1 or isprime(t), map(`-`,numtheory:-divisors(a^2),a) minus S);
  a:= min(Cands); R:= R, a; S:= S union {a};
od:
R; # Robert Israel, Mar 23 2023

A343074 Number of partitions of 1/n into n distinct reciprocals of positive integers.

Original entry on oeis.org

1, 1, 15, 586, 112535, 131223239

Offset: 1

Ilya Gutkovskiy, Apr 04 2021

			a(2) = 1 because we have 1/2 = 1/3 + 1/6.

Index entries for sequences related to Egyptian fractions

Cf. A006585, A063647, A227610, A241883, A342267.

a(5) from Bert Dobbelaere, Apr 05 2021

a(6) from Jud McCranie, Sep 03 2021

A063648 Smallest c such that 1/n=1/c+1/b has integer solutions with c>b.

Original entry on oeis.org

6, 12, 12, 30, 15, 56, 24, 36, 30, 132, 28, 182, 42, 40, 48, 306, 45, 380, 45, 70, 66, 552, 56, 150, 78, 108, 77, 870, 66, 992, 96, 132, 102, 84, 84, 1406, 114, 156, 90, 1722, 91, 1892, 132, 120, 138, 2256, 112, 392, 150, 204, 156, 2862, 135, 176, 120, 228, 174

Offset: 2

Henry Bottomley, Jul 23 2001

nonn

Largest c is n(n+1) since 1/n=1/(n(n+1))+1/(n+1)

			a(10)=30 since 1/10=1/20+1/20=1/30+1/15=1/35+1/14=1/60+1/12=1/110+1/11, but the first sum does not have c>b, leaving the second sum to provide the value.

Cf. A018892, A063427, A063428, A063647, A063649.

a(n) =n*A063649(n)/(A063649(n)-n) =A063427(n)+2n.

A339133 a(n) = A339107(n)*A339107(n+1)/(A339107(n)+A339107(n+1)).

Original entry on oeis.org

2, 4, 8, 6, 7, 24, 14, 12, 20, 21, 10, 6, 8, 24, 15, 16, 30, 12, 14, 48, 12, 10, 28, 60, 56, 45, 8, 6, 12, 20, 30, 54, 60, 36, 18, 24, 120, 108, 80, 90, 96, 60, 24, 28, 96, 72, 42, 54, 126, 108, 140, 126, 110, 84, 33, 28, 66, 21, 18, 66, 90, 66, 85, 132, 51, 52, 204, 51, 36, 78, 72, 40, 52, 60, 26

Offset: 1

Scott R. Shannon, Nov 24 2020

nonn

This is the ratio of the multiplication and addition of successive terms in A339107. See that sequence for further details.

			a(1) = 2 as A339107(1) = 3 and A339107(2) = 6, and 3*6/(3+6) = 2.

Scott R. Shannon, Line graph of the first 100000 terms.

Cf. A339107, A063647, A027750, A000040, A098550, A336957, A064413.

A127731 Triangle read by rows, where row n consists of the r's where r = (nm)/(n+m) and the m's are positive integers such that (n+m) divides (nm).

Original entry on oeis.org

1, 2, 2, 3, 4, 2, 3, 4, 5, 6, 4, 6, 7, 6, 8, 5, 6, 8, 9, 10, 3, 4, 6, 8, 9, 10, 11, 12, 7, 10, 12, 13, 6, 10, 12, 14, 8, 12, 14, 15, 16, 6, 9, 12, 14, 15, 16, 17, 18, 4, 10, 12, 15, 16, 18, 19, 12, 14, 18, 20, 11, 18, 20, 21, 22, 6, 8, 12, 15, 16, 18, 20, 21, 22, 23, 20, 24, 13, 22, 24, 25

Offset: 2

Leroy Quet, Jan 26 2007

The maximum term of the n-th row, for n >= 2, is (n-1). The minimum term of row n is A063428(n), for n >= 3. Row n contains A063647(n) terms (according to a comment by Benoit Cloitre). For p prime, row p^k has k terms. (Each term in row p^k is of the form p^(k-j)*(p^j -1), 1<=j<=k.)

			Row 6 is (2,3,4,5) because row 6 of irregular array A127730 is (3,6,12,30); and (6*3)/(6+3) = 2, (6*6)/(6+6) = 3, (6*12)/(6+12) = 4 and (6*30)/(6+30) = 5.

Cf. A063428, A063647, A127730.

f[n_] := Select[Table[n*m/(n + m), {m, n^2}], IntegerQ];Table[f[n], {n, 2, 26}] // Flatten (* Ray Chandler, Feb 13 2007 *)

Extended by Ray Chandler, Feb 13 2007

A146747 Numbers k such that sigma_1(k)*sigma_0(k)/(sigma_1(k)-sigma_0(k)) is an integer.

Original entry on oeis.org

2, 3, 5, 6, 7, 20, 30

Offset: 1

Ctibor O. Zizka, Nov 01 2008

Numbers k such that A000203(k)*A000005(k)/(A000203(k)-A000005(k)) is an integer.

Cf. A000005, A000203, A063647, A083795.

q[n_] := Module[{s = DivisorSigma[1, n], d = DivisorSigma[0, n]}, Divisible[s*d, s - d]]; Select[Range[2, 100], q] (* Amiram Eldar, Apr 07 2024 *)

is(n) = {my(f = factor(n), s = sigma(f), d = numdiv(f)); n > 1 && !((s*d) % (s-d));} \\ Amiram Eldar, Apr 07 2024

A153193 a(n) is the number of integers of the form n(n+1)k / (k - n*(n+1)) where k is an integer >= 1.

Original entry on oeis.org

4, 13, 22, 22, 40, 40, 31, 52, 67, 40, 67, 67, 40, 121, 121, 40, 67, 67, 67, 202, 121, 40, 94, 157, 67, 94, 157, 67, 121, 121, 49, 148, 121, 121, 337, 112, 40, 121, 283, 94, 121, 121, 67, 337, 202, 40, 121, 202, 112, 202, 202, 67, 94, 283, 283, 283, 121, 40

Offset: 1

Ctibor O. Zizka, Dec 20 2008

nonn

1/n - 1/(n+1) - 1/k = 1/c where c is an integer, k >= 1.

			The a(1)=4 integers of the form n*(n+1)*k/(k - n*(n+1)) = 1*(1+1)*k/(k - 1*(1+1)) = 2*k/(k-2) occur at
  k=1: 2*1/(1-2) = -2,
  k=3: 2*3/(3-2) =  6,
  k=4: 2*4/(4-2) =  4, and
  k=6: 2*6/(6-2) =  3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A146564, A152492, A063647, A000005, A146566.

f:= proc(n) local D;
   D:= numtheory:-divisors((n*(n+1))^2);
   nops(D) + nops(select(`<=`,D,n*(n+1)-1))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 21 2024

a(13)-a(58) from Jon E. Schoenfield, Mar 15 2022

A243016 Number of solutions for k*n/(k+n) = p for integer k > 0 and prime p.

Original entry on oeis.org

0, 0, 1, 2, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Derek Orr, May 29 2014

nonn

It is unknown whether a(6) = 3 is the highest number in this sequence.
No terms higher than 3 among the first 10000 terms. - Antti Karttunen, Jan 20 2025
a(n) is the number of primes among n-1, n/2 and q, where q satisfies q*(q+1)=n. So a(n) <= 2 for n > 6, and a(n) = 2 iff n != 6 is in A053185 + 1 or A077068. - Jinyuan Wang, Jan 20 2025

			4*k/(4+k) has two solutions: k=4, p=2 and k=12, p=3. Thus a(4) = 2.
From _Antti Karttunen_, Jan 18 2025: (Start)
For n=3, the ratio (k*n)/(k+n) obtains for k=1..3*(3-1) the values 3/4, 6/5, 3/2, 12/7, 15/8, 2, and only the last one of these is prime, therefore a(3) = 1.
For n=26, the only k such that (k*n)/(k+n) is a prime, is k=26, with (26^2)/(2*26) = 13, therefore a(26) = 1. (End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A053185, A063647, A077068.

A243016(n) = { my(s); sum(k=1, n*(n-1), s = (k*n)/(k+n); (1==denominator(s) && isprime(s))); }; \\ Edited by Antti Karttunen, Jan 18 2025

a(n) <= A063647(n). - Antti Karttunen, Jan 18 2025

Data section extended up to a(105) and incorrect terms, that were caused by dropping of a(26) and a(27) (first discrepancies at n=26, 28, 30, 34, etc.) corrected by Antti Karttunen, Jan 18 2025

A309149 Number of solutions of the Diophantine equation 1/n + 1/x = 1/y + 1/z, where n >= 1, x >= n, y > n and z >= y.

Original entry on oeis.org

0, 3, 9, 14, 22, 33, 29, 44, 56, 65, 50, 108, 53, 101, 141, 97, 64, 150, 73, 216, 184, 119, 81, 274, 138, 141, 182, 263, 101, 378, 90, 222, 270, 183, 375, 419, 102, 185, 319, 479, 121, 486, 115, 394, 520, 187, 118, 565, 227, 362

Offset: 1

S. Nazardonyavi, Jul 14 2019

nonn

			n=2: 1/2 + 1/6 = 1/3 + 1/3, 1/2 + 1/12 = 1/3 + 1/4, 1/2 + 1/30 = 1/3 + 1/5.

Cf. A309150, A063647, A018892.

a[n_]:=Length@Solve[1/(n)+1/(x)==1/y+1/z&&x>=n&&z>=y&&y>n,{x,y,z},Integers];
Array[a,50]

A309150 Number of solutions of the Diophantine equation 1/n + 1/x = 1/y + 1/z, where n >= 1, x > n, y > n and z > y.

Original entry on oeis.org

0, 2, 7, 12, 20, 29, 27, 41, 52, 60, 48, 101, 51, 96, 134, 93, 62, 142, 71, 209, 176, 114, 79, 264, 134, 136, 176, 256, 99, 363, 88, 217, 262, 178, 368, 406, 100, 180, 311, 469, 119, 471, 113, 386, 508, 182, 116, 552, 223, 353

Offset: 1

S. Nazardonyavi, Jul 14 2019

nonn

			n=2:  1/2 + 1/12 = 1/3 + 1/4, 1/2 + 1/30 = 1/3 + 1/5.

Cf. A309149, A063647, A018892.

a[n_]:=Length@Solve[1/(n)+1/(x)==1/y+1/z&&x>n&&z>y&&y>n,{x,y,z},Integers];
Array[a,50]

cp's OEIS Frontend

A339107 a(1) = 3; for n>1, a(n) = the smallest positive number not occurring earlier such that a(n-1)*a(n) is divisible by a(n-1)+a(n), where a(n) is not a prime or 4.

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A343074 Number of partitions of 1/n into n distinct reciprocals of positive integers.

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Extensions

A063648 Smallest c such that 1/n=1/c+1/b has integer solutions with c>b.

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A339133 a(n) = A339107(n)*A339107(n+1)/(A339107(n)+A339107(n+1)).

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A127731 Triangle read by rows, where row n consists of the r's where r = (n*m)/(n+m) and the m's are positive integers such that (n+m) divides (n*m).

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A146747 Numbers k such that sigma_1(k)*sigma_0(k)/(sigma_1(k)-sigma_0(k)) is an integer.

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PARI

A153193 a(n) is the number of integers of the form n*(n+1)*k / (k - n*(n+1)) where k is an integer >= 1.

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Maple

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A243016 Number of solutions for k*n/(k+n) = p for integer k > 0 and prime p.

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PARI

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Extensions

A309149 Number of solutions of the Diophantine equation 1/n + 1/x = 1/y + 1/z, where n >= 1, x >= n, y > n and z >= y.

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Author

A127731 Triangle read by rows, where row n consists of the r's where r = (nm)/(n+m) and the m's are positive integers such that (n+m) divides (nm).

A153193 a(n) is the number of integers of the form n(n+1)k / (k - n*(n+1)) where k is an integer >= 1.