A064814 -id:A064814 - cp's OEIS frontend

A073258 Numbers n such that gcd(c(n),n) = gcd(A002808(n),n) = A064814(n)=1 where c(n) is the n-th composite number.

1, 3, 4, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 53, 57, 58, 59, 61, 66, 67, 68, 69, 71, 73, 79, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 97, 100, 101, 103, 106, 107, 108, 109, 111, 113, 115, 116, 117, 118, 119

Offset: 1

Labos Elemer, Jul 22 2002

nonn

			n=256: composite(256) = 323 = 17*19, gcd(323,256)=1, so 256 is a term.

Cf. A064814, A002808, A073257.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x]; t=Table[0, {256}]; s=0; k=0; Do[s=GCD[c[n], n]; If[Equal[s, 1], k=k+1; t[[k]]=n; Print[{k, n}]], {n, 1, 256}] t

lista(nn) = {my(n=0, list=List()); forcomposite (c=1, nn, n++; if (gcd(n, c) == 1, listput(list, c))); Vec(list); } \\ Michel Marcus, Jul 19 2020

A073257 Smallest k such that gcd(c(k),k) = gcd(A002808(k),k) = A064814(k) = n.

Original entry on oeis.org

1, 2, 12, 20, 5, 6, 7, 64, 234, 50, 55, 24, 26, 28, 30, 32, 629, 1008, 209, 220, 231, 1012, 506, 168, 425, 182, 189, 2716, 2204, 1080, 93, 96, 99, 2176, 105, 4428, 1369, 5586, 1755, 1800, 6109, 2478, 2279, 3916, 5760, 644, 4606, 1920, 1960, 10250, 2040, 2444

Offset: 1

Labos Elemer, Jul 22 2002

nonn

			50th composite is 70, gcd(50,70)=10 appears first here, a(10)=50.

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A064814, A002808, A073258.

f[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] t=Table[0, {100}]; Do[s=GCD[f[n], n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 100000}]; t

a(n) = min{x: gcd(c(x), x)=n}, where c(x) is the x-th composite number.

A073259 Number of iterations of f(n,k) = n+pi(k)+1 starting from f(n,n) until a fixed point is reached.

Original entry on oeis.org

4, 4, 4, 3, 3, 4, 4, 3, 3, 4, 4, 4, 3, 4, 4, 3, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 3, 4, 5, 4, 3, 4, 4, 4, 3, 3, 4, 4, 4, 3, 3, 4, 5, 4, 4, 3, 3, 4, 4, 4, 4, 5, 4, 4, 4, 3, 4, 4, 4, 5, 4, 4, 4, 3, 4, 4, 4, 4, 3, 3, 3, 4, 4, 4, 4, 5, 4, 4, 5, 5, 4, 4, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 4, 4, 4, 3, 4, 5

Offset: 1

Labos Elemer, Jul 22 2002

nonn

Original name: Length of FixedPointList leading to value of n-th composite number.

			n=1000000:the list={1000000,1078499,1084157,1084577,1084604,1084605}, its length including initial term is 6, while composite[1000000]=1084605.

Cf. A000720, A064814, A002808, A073255-A073264.

Table[Length[FixedPointList[w+PrimePi[ # ]+1&, w]]-1, {w, 1, 128}]

See program below.

Name clarified by Sean A. Irvine, Nov 21 2024

A337490 a(0)=1; for n > 0, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

1, 2, 4, 7, 11, 5, 6, 13, 21, 30, 10, 21, 33, 46, 14, 29, 45, 62, 18, 37, 57, 78, 22, 45, 69, 94, 26, 53, 81, 110, 140, 171, 203, 236, 270, 305, 341, 378, 416, 39, 79, 120, 162, 205, 249, 294, 340, 387, 3, 52, 102, 17, 69, 122, 176, 231, 287, 344, 402, 461, 521, 582, 644, 707, 771, 836, 902, 969

Offset: 0

Scott R. Shannon, Aug 29 2020

nonn

The sequence displays the unusual behavior of decreasing 53 times in the first 1975 terms, due to the existence of a GCD which has not previously appeared in the sequence, but then not decreasing again for n up to at least 100 million. In this period there are 37 repeated terms, the first being 21 at n=11 and the last 161202 at n=2054. In the same range many values do not appear, for example 16,23,28,32,36. It is unknown when the sequence decreases again, or if all values eventually appear. The 100 millionth term is 4999999948050717.

See the companion sequence A333980 for the sum of the terms from a(0) to a(n).

			a(2) = 4 as the sum of all previous terms is a(0)+a(1) = 3, and the GCD of 3 and 2 is 1, which has already appeared in the sequence. Therefore a(2) = a(1) + n = 2 + 2 = 4.
a(4) = 11 as the sum of all previous terms is a(0)+...+a(3) = 14, and the GCD of 14 and 4 is 2. However 2 has already appeared so a(4) = a(3) + n = 7 + 4 = 11.
a(5) = 5 as the sum of all previous terms is a(0)+...+a(4) = 25, and the GCD of 25 and 5 is 5, and as 5 has not previous appeared a(5) = 5.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Graph of the terms for n=0..2500. This includes the last known decrease in the sequence, n(1974) = 42.
Scott R. Shannon, Graph of the terms for n=0..10000000.

Cf. A333980, A333826 (same rules but starting a(1)=1), A165430, A064814, A082299, A005132, A336957.

lista(nn) = {my(va = vector(nn), s=0); va[1] = 1; s += va[1]; for (n=2, nn, my(g = gcd(n-1, s)); if (#select(x->(x==g), va), va[n] = va[n-1]+n-1, va[n] = g); s += va[n];); va;} \\ Michel Marcus, Sep 05 2020

A333980 Partial sums of A337490.

Original entry on oeis.org

1, 3, 7, 14, 25, 30, 36, 49, 70, 100, 110, 131, 164, 210, 224, 253, 298, 360, 378, 415, 472, 550, 572, 617, 686, 780, 806, 859, 940, 1050, 1190, 1361, 1564, 1800, 2070, 2375, 2716, 3094, 3510, 3549, 3628, 3748, 3910, 4115, 4364, 4658, 4998, 5385, 5388, 5440, 5542, 5559, 5628, 5750

Offset: 0

Scott R. Shannon, Sep 04 2020

nonn

See A337490 for an explanation of the sequence.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A337490, A165430, A064814, A082299, A005132, A336957.

A071224 LCM of n and n-th composite number.

Original entry on oeis.org

4, 6, 24, 36, 10, 12, 14, 120, 144, 90, 220, 84, 286, 168, 75, 208, 459, 252, 570, 160, 231, 374, 805, 72, 950, 78, 1080, 84, 1276, 90, 1426, 96, 1617, 850, 1785, 468, 1998, 2090, 2184, 2280, 2378, 420, 2666, 2772, 2880, 2990, 3102, 816, 3381

Offset: 1

Amarnath Murthy, May 17 2002

nonn

			a(10)=90 as 18 is the 10th composite number and 10*18/gcd(10,18)=10*18/2=90.

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Module[{nn=100,cmps},cmps=Select[Range[nn],CompositeQ];LCM@@@Thread[ {Range[ Length[ cmps]],cmps}]] (* Harvey P. Dale, Jul 21 2024 *)

a(n) = n*A002808(n)/A064814(n). - Lior Manor, Jun 02 2002

More terms from Reinhard Zumkeller, May 28 2002

More terms from Lior Manor, Jun 02 2002

A333826 a(1)=1; for n>1, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

1, 3, 6, 2, 7, 13, 20, 4, 13, 23, 34, 46, 59, 73, 88, 8, 25, 43, 62, 10, 31, 53, 76, 100, 125, 151, 178, 206, 235, 15, 46, 78, 111, 145, 5, 41, 78, 116, 155, 195, 236, 278, 321, 365, 410, 456, 503, 551, 600, 50, 101, 153, 206, 260, 315, 371, 428, 486, 545, 605, 666, 728, 791, 855, 920, 986, 1053

Offset: 1

Scott R. Shannon, Sep 03 2020

nonn

This is a variation of A337490; here we start with an offset of 1, so a(1) = 1. See that sequence for further details.

In the first 4212 terms the sequence decreases 69 times while 45 terms are repeated, the first being 13 at n=9 and the last 399876 at n=4212. After n(4166)=84 the sequence does not decrease again for n up to at least 100 million. The lowest numbers that have not appeared in that range are 30,37,47,48,49,51. The 100 millionth term is 4999999941527298.

			a(2) = 3 as the sum of all previous terms is a(1) = 1, and the GCD of 1 and 2 is 1. However 1 has already appeared so a(2) = a(1) + n = 1 + 2 = 3.
a(4) = 2 as the sum of all previous terms is a(1)+a(2)+a(3) = 10, and the GCD of 10 and 4 is 2, and as 2 has not previous appeared a(4) = 2.
a(8) = 4 as the sum of all previous terms is a(1)+...+a(7) = 52, and the GCD of 52 and 8 is 4, and as 4 has not previous appeared a(8) = 4.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Graph of the terms for n=1..4500. This includes the last known decrease in the sequence, n(4166)=84.

Cf. A337490 (same sequence rules but starting a(0)=1), A333980, A165430, A064814, A082299, A005132, A336957.

lista(nn) = {my(va = vector(nn), s=0); va[1] = 1; s += va[1]; for (n=2, nn, my(g = gcd(n, s)); if (#select(x->(x==g), va), va[n] = va[n-1]+n, va[n] = g); s += va[n];); va;} \\ Michel Marcus, Sep 05 2020

A334880 Numbers k such that gcd(k, k-th composite number) > 1.

Original entry on oeis.org

2, 5, 6, 7, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 34, 36, 42, 48, 50, 51, 52, 54, 55, 56, 60, 62, 63, 64, 65, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 93, 96, 98, 99, 102, 104, 105, 110, 112, 114, 120, 122, 124, 126, 128, 130, 132, 138, 148

Offset: 1

Clark Kimberling, Jul 17 2020

			In the following table, c(k) = A002808(k) = k-th composite number.
  k     c(k)   gcd(k, c(k))
  1      4         1
  2      6         2
  3      8         1
  4      9         1
  5     10         5
  6     12         6
2, 5, 6 are in this sequence, and 1,3,4 are in A073258.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002808, A064814, A073258 (complement), A336323.

c = Select[Range[2, 150], ! PrimeQ[#] &]; (* A002808 *)
Select[Range[Length[c]], GCD[c[[#]], #] > 1 &]  (* A334880 *)
Module[{nn=200,cmps,len},cmps=Select[Range[nn],CompositeQ];len=Length[ cmps];Select[Thread[{Range[len],cmps}],GCD@@#>1&]][[All,1]] (* Harvey P. Dale, Sep 20 2020 *)

lista(nn) = {my(n=0, list=List()); forcomposite (c=1, nn, n++; if (gcd(n, c) > 1, listput(list, n))); Vec(list);} \\ Michel Marcus, Jul 19 2020

A073260 Length of FixedPointList leading to value of [10^n]-th composite number.

Original entry on oeis.org

4, 4, 4, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9

Offset: 1

Labos Elemer, Jul 22 2002

One plus the number of iterations necessary to reach the composite number using the formula in the program. - Robert G. Wilson v, Jul 23 2002

			n=10^11: the list= {100000000000, 104118054814, 104280509328, 104286914053, 104287166025, 104287176027, 104287176414, 104287176419}, its length including initial term is 8, so a(11)=8.

Cf. A064814, A065857, A002808, A073255-A073264.

Table[Length[FixedPointList[10^w+PrimePi[ # ]+1&, 10^w]]-1, {w, 1, 11}]

See program below.

More terms from Robert G. Wilson v, Jul 23 2002

A118310 a(n) = gcd(n,m(n)), where m(n) is the n-th nonprime positive integer (1 or composite).

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 1, 2, 3, 2, 1, 4, 1, 2, 3, 1, 1, 9, 1, 10, 1, 11, 1, 1, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 5, 3, 1, 2, 1, 8, 1, 2, 1, 2, 9, 2, 1, 6, 1, 1, 1, 4, 1, 3, 1, 7, 3, 2, 1, 2, 1, 1, 1, 1, 1, 6, 1, 4, 3, 2, 1, 24, 1, 1, 25, 2, 1, 3, 1, 4, 1, 1, 1, 6, 5, 2, 3, 2, 1, 30, 1, 2, 3, 2, 5, 6, 1, 1, 1, 4, 1, 2, 1

Offset: 1

Leroy Quet, May 14 2006

For n >= 1, a(n+1) = gcd(n+1,c(n)), where c(n) is the n-th composite.
First occurrence of k: 1, 2, 3, 4, 35, 48, 56, 40, 18, 20, 22, 120, 130, 140, 375, ..., . - Robert G. Wilson v
Question: What is the reason for the conspicuous arc-like structures in the scatter plot? - Antti Karttunen, Mar 02 2023

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A064814.

NonPrime[n_Integer] := FixedPoint[n + PrimePi@# &, n + PrimePi@n]; f[n_] := GCD[n, NonPrime@n]; Array[f, 103] (* Robert G. Wilson v *)

A002808(maxn)= { local(a); a=[4]; for(n=5,maxn, if( !isprime(n), a=concat(a,n); ); ); return(a); } A118310(maxn)= { local(nonppo,a,newa,nonppol); a=[;]; nonppo=concat(1,A002808(maxn)); nonppol=matsize(nonppo); for(n=1,nonppol[2], newa= gcd(n, nonppo[n]); a=concat(a,newa); ); return(a); } print(A118310(180)); \\ R. J. Mathar

A118310(n) = if(1==n, n, my(x=n-1); for(k=2, oo, if(!isprime(k), x--; if(!x, return(gcd(n,k)))))); \\ Antti Karttunen, Mar 02 2023

More terms from Robert G. Wilson v and R. J. Mathar, May 16 2006

cp's OEIS Frontend

A073258 Numbers n such that gcd(c(n),n) = gcd(A002808(n),n) = A064814(n)=1 where c(n) is the n-th composite number.

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A073257 Smallest k such that gcd(c(k),k) = gcd(A002808(k),k) = A064814(k) = n.

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A073259 Number of iterations of f(n,k) = n+pi(k)+1 starting from f(n,n) until a fixed point is reached.

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A337490 a(0)=1; for n > 0, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

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A333980 Partial sums of A337490.

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A071224 LCM of n and n-th composite number.

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A333826 a(1)=1; for n>1, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

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A334880 Numbers k such that gcd(k, k-th composite number) > 1.

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A073260 Length of FixedPointList leading to value of [10^n]-th composite number.

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