A368900 -id:A368900 - cp's OEIS frontend

A369032 LCM-transform of permutation A241909.

1, 2, 2, 3, 2, 3, 2, 5, 1, 3, 2, 5, 2, 3, 1, 7, 2, 1, 2, 5, 1, 3, 2, 7, 1, 3, 1, 5, 2, 1, 2, 11, 1, 3, 1, 1, 2, 3, 1, 7, 2, 1, 2, 5, 1, 3, 2, 11, 1, 1, 1, 5, 2, 1, 1, 7, 1, 3, 2, 1, 2, 3, 1, 13, 1, 1, 2, 5, 1, 1, 2, 1, 2, 3, 1, 5, 1, 1, 2, 11, 1, 3, 2, 1, 1, 3, 1, 7, 2, 1, 1, 5, 1, 3, 1, 13, 2, 1, 1, 1, 2, 1, 2, 7, 1

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

See comments in A368900.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A014963, A122111, A368900, A369031, A369033.

up_to = 2^18;
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
v369032 = LCMtransform(vector(up_to,i,A241909(i)));
A369032(n) = v369032[n];

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A369032(n) = A014963(A241909(n));

a(1) = 1, for n > 1, a(n) = lcm {1..A241909(n)} / lcm {1..A241909(n-1)}.

a(n) = A014963(A241909(n)). [A241909 satisfies the property S defined in A368900]

A369033 LCM-transform of permutation A241916.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 1, 5, 11, 3, 13, 7, 1, 2, 17, 1, 19, 5, 1, 11, 23, 3, 1, 13, 1, 7, 29, 1, 31, 2, 1, 17, 1, 1, 37, 19, 1, 5, 41, 1, 43, 11, 1, 23, 47, 3, 1, 1, 1, 13, 53, 1, 1, 7, 1, 29, 59, 1, 61, 31, 1, 2, 1, 1, 67, 17, 1, 1, 71, 1, 73, 37, 1, 19, 1, 1, 79, 5, 1, 41, 83, 1, 1, 43, 1, 11, 89, 1, 1, 23, 1, 47

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

See comments in A368900.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A014963, A122111, A368900, A369031, A369032.

up_to = 2^16;
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
A209229(n) = (n && !bitand(n,n-1));
A241916(n) = if(1==A209229(n), n, my(f = factor(2*n), nbf = #f~, igp = primepi(f[nbf,1]), g = f); for(i=1,nbf,g[i,1] = prime(1+igp-primepi(f[i,1]))); factorback(g)/2); \\ Antti Karttunen, Jan 12 2024
v369033 = LCMtransform(vector(up_to,i,A241916(i)));
A369033(n) = v369033[n];

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A369033(n) = A014963(A241916(n));

a(1) = 1, for n > 1, a(n) = lcm {1..A241916(n)} / lcm {1..A241916(n-1)}.

a(n) = A014963(A241916(n)). [A241916 satisfies the property S defined in A368900]

A368901 The least common multiple of the first n terms of Doudna sequence, A005940.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 60, 180, 360, 2520, 2520, 2520, 2520, 12600, 12600, 37800, 75600, 831600, 831600, 831600, 831600, 831600, 831600, 831600, 831600, 5821200, 5821200, 5821200, 5821200, 29106000, 29106000, 87318000, 174636000, 2270268000, 2270268000, 2270268000, 2270268000, 2270268000, 2270268000, 2270268000, 2270268000

Offset: 0

Antti Karttunen and David James Sycamore, Jan 10 2024

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..16384

Cf. A005940, A368900.

Cf. also A003418.

Array[Set[a[#], #] &, 2]; m = 2; {1, 1, 2}~Join~Reap[Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2] ] ] ] ]; Sow[m = LCM[m, a[n]]], {n, 3, 40}] ][[-1, 1]] (* Michael De Vlieger, Jun 24 2025 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t) };
A368901(n) = lcm(vector(n,i,A005940(i)));

a(0) = 1, and for n > 0, a(n) = lcm(A005940(n), a(n-1)).

a(0) = 1, and for n > 0, a(n) = A368900(n) * a(n-1).

A369045 LCM-transform of binary invert permutation (A054429).

Original entry on oeis.org

1, 3, 2, 7, 1, 5, 2, 1, 1, 13, 1, 11, 1, 3, 2, 31, 1, 29, 1, 3, 1, 5, 1, 23, 1, 1, 1, 19, 1, 17, 2, 1, 1, 61, 1, 59, 1, 1, 1, 1, 1, 53, 1, 1, 1, 7, 1, 47, 1, 1, 1, 43, 1, 41, 1, 1, 1, 37, 1, 1, 1, 1, 2, 127, 1, 5, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 113, 1, 1, 1, 109, 1, 107, 1, 1, 1, 103, 1, 101, 1, 1, 1, 97, 1, 1, 1

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

Binary invert permutation, A054429, is a self-inverse permutation related to the binary expansion of n that keeps all the numbers of range [2^k, 2^(1+k)[ in the same range, i.e., for all n >= 1, A000523(A054429(n)) = A000523(n), from which it immediately follows that A054429 has the property S mentioned in the comments of A368900, and therefore this sequence is equal to A014963(A054429(n)), for n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A014963, A054429, A368900, A369041, A369042, A369043, A369044.

up_to = 65537;
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
A054429(n) = ((3<<#binary(n\2))-n-1);
v369045 = LCMtransform(vector(up_to,i,A054429(i)));
A369045(n) = v369045[n];

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A054429(n) = ((3<<#binary(n\2))-n-1);
A369045(n) = A014963(A054429(n));

a(n) = lcm {1..A054429(n)} / lcm {1..A054429(n-1)}.
a(n) = A014963(A054429(n)). [See comments.]
For n >= 1, Product_{d|n} a(A054429(d)) = n. [Implied by above.]

A369010 Exponential of Mangoldt function M(n) applied to primorial base exp-function: a(n) = A014963(A276086(n)).

Original entry on oeis.org

1, 2, 3, 1, 3, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Jan 14 2024

nonn

Also LCM-transform of A276086, because A276086 has the S-property explained in the comments of A368900.

Antti Karttunen, Table of n, a(n) for n = 0..30030
Index entries for sequences related to primorial base

Cf. A014963, A060735 (positions of terms > 1), A276086, A368900.

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A369010(n) = A014963(A276086(n));

up_to = 510511; \\ = 1+A002110(7);
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
v369010 = LCMtransform(vector(up_to,n,A276086(n-1)));
A369010(n) = v369010[1+n];

a(n) = A014963(A276086(n)).

For n > 0, a(n) = lcm {1..A276086(n)} / lcm {1..A276086(n-1)}.

A369060 LCM-transform of Sycamore's D(3) variant of Doudna sequence (A356867).

Original entry on oeis.org

1, 2, 3, 5, 2, 1, 1, 2, 3, 7, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 2, 3, 11, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 2, 3, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jan 13 2024

nonn

See discussion at A368900.

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A014963, A356867, A368900.

up_to = 3^9; \\ Checked up to (3^12)
A356867list(up_to) = { my(v=vector(up_to),met=Map(),h=0,ak); for(i=1,#v,if(1==sumdigits(i,3), v[i] = i; h = i, ak = v[i-h]; forprime(p=2,,if(3!=p && !mapisdefined(met,p*ak), v[i] = p*ak; break))); mapput(met,v[i],i)); (v); };
v356867 = A356867list(up_to);
A356867(n) = v356867[n];
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
v369060 = LCMtransform(v356867);
A369060(n) = v369060[n];
A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };

a(n) = lcm {1..A356867(n)} / lcm {1..A356867(n-1)}.

a(n) = A014963(A356867(n)). [This holds because A356867 satisfies the property S explained in A368900]

A369686 LCM-transform of A359804 (see Comment and links).

Original entry on oeis.org

1, 2, 3, 5, 2, 1, 1, 7, 3, 2, 1, 1, 11, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 7, 13, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 17, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 11, 1, 5, 1, 1, 1, 1, 1, 1, 3, 31, 1

Offset: 1

David James Sycamore, Jan 28 2024

nonn

Let b(k) be the Least Common Multiple (LCM) of the first k terms of A359804, then a(n) = b(n)/b(n-1), where sequence b(n) is A369685.

The property S (as defined in A368900) refers to what is observed in the positive integers (A000027), and also in the Doudna sequence (A005940), whereby each prime power appears prior to any of its multiples. The present sequence does not have this property since, for example, 26 = a(31) precedes 13 = a(42). Thus A369804 represents a significant disturbance of A000027 in that whereas it is conjectured to be a permutation of the positive integers, it does not preserve one of the basic properties of that sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Andrzej Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
Andrzej Nowicki, Strong divisibility and LCM-sequences, Am. Math. Monthly (2015) Vol. 122, 958-966.

Cf. A014963, A359804, A368900, A369685.

nn = 120; c[] = False; q[] = 1;
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
Set[{i, j}, {1, 2}]; m = 2; u = 3;
Do[
  (k = q[#]; While[c[k #], k++]; k *= #; While[c[# q[#]], q[#]++]) &[
  (p = 2; While[Divisible[i j, p], p = NextPrime[p]]; p)];
  Set[{a[n], c[k], i, j, m}, {#/m, True, j, k, #}] &[LCM[m, k]];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Jan 29 2024 *)

a(n) = A369685(n)/A369685(n-1).

More terms from Michael De Vlieger, Jan 29 2024

A371985 For n a power of 2, a(n) = n. Otherwise a(n) is the smallest novel multiple of a(n - 2^m), where 2^m is the greatest power of 2 not exceeding n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 12, 20, 15, 18, 27, 16, 11, 14, 21, 24, 25, 30, 36, 40, 28, 50, 48, 60, 45, 54, 81, 32, 13, 22, 33, 44, 35, 42, 63, 56, 49, 70, 72, 80, 75, 90, 108, 96, 55, 84, 105, 120, 100, 150, 144, 160, 112, 200, 192, 180, 135, 162, 243, 64, 17

Offset: 1

David James Sycamore, Apr 15 2024

nonn

Reminiscent of the Doudna sequence A005940; also of A052330 and A269848.
All powers of 2 (a(n) = n) are assigned first in order to avoid the second part of the definition giving a(n) = 2^k for some n which is not a power of 2 (see Example for a(12) = 20).
It follows from the definition that all powers of 2, all primes and all multiples of all primes are terms so this sequence is a permutation of the positive integers (A000027), with primes in order.
Each prime power appears before any of its multiples, meaning that this sequence has "property S" as defined in A368900.

			a(3) = 3, because 2 is the greatest power of 2 not exceeding 3 and 3-2 = 1, so a(3) = 3, the least novel multiple of a(1) = 1.
a(12) is the smallest novel multiple of a(12-8) = a(4) = 4, and at this point in the sequence 4,8,12 are all prior terms and a(16) = 16 is already taken, so a(12) = 20.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
David A. Corneth, PARI program
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Fan style binary tree showing a(n), n = 1..8192, with a color function showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple. Purple represents powerful numbers that are not prime powers.
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A005940, A052330, A269848, A368900.

nn = 10; c[] := False; m[] := 1; a[1] = 1; c[1] = True;
Do[If[i == 0,
   k = 2^j + i,
   (While[Set[k, m[#] #]; Or[c[k], IntegerQ@ Log2[k]], m[#]++]) &@ a[i]];
  Set[{a[2^j + i], c[k]}, {k, True}], {j, nn}, {i, 0, 2^j - 1}];
Array[a, 2^(nn + 1) - 1] (* Michael De Vlieger, Apr 15 2024 *)

PARI
```
\\ See PARI link
```

a(2^k + 1) = prime(k+1).

A383258 LCM-transform of A064664 (the inverse of the EKG-sequence).

Original entry on oeis.org

1, 2, 5, 3, 1, 2, 7, 2, 1, 3, 1, 1, 1, 13, 11, 17, 1, 1, 37, 1, 1, 19, 43, 2, 1, 3, 1, 1, 1, 23, 61, 31, 1, 2, 5, 1, 67, 1, 29, 1, 1, 1, 3, 41, 1, 1, 89, 1, 1, 1, 1, 47, 1, 1, 53, 7, 1, 1, 107, 1, 1, 1, 1, 2, 1, 59, 2, 1, 1, 1, 1, 1, 1, 1, 1, 71, 1, 1, 151, 1, 1, 73, 1, 1, 1, 1, 1, 79, 167, 83, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 197

Offset: 1

Antti Karttunen, Apr 21 2025

nonn

As the sequence A064664 has no S-property defined in the comments of A368900, therefore this is not equal to A014963(A064664(n)).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to EKG sequence.

Cf. A064413, A064664, A064954.

Cf. also A265576, A368900.

a(n) = lcm {1..A064664(n)} / lcm {1..A064664(n-1)}.

A372699 a(1) = 1, j = a(n-1). For n > 1, a(n) is the least novel m such that rad(m*j) is the smallest primorial number divisible by A007947(j).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 12, 10, 9, 8, 18, 15, 16, 24, 20, 27, 32, 36, 25, 30, 7, 60, 14, 45, 40, 48, 50, 54, 75, 64, 72, 80, 81, 96, 90, 21, 70, 42, 35, 84, 100, 108, 120, 28, 105, 56, 135, 128, 144, 125, 150, 49, 180, 63, 140, 126, 160, 162, 200, 192, 225, 216, 240

Offset: 1

David James Sycamore and Michael De Vlieger, May 14 2024

nonn

a(n) depends upon if rad(j) = A002110(k) for some k (equivalently A083720(j) = 1), or not. If so a(n) is least novel m such that rad(m*j) = A002110(k+1). Otherwise a(n) = least novel m such that rad(m*j) = A002110(A000720(q)), where q = gpf(j).
Put otherwise, if p = nextprime(q), and A = A083720, then for n > 1 if A(j) = 1, a(n) is the least novel p-smooth number divisible by p, and if A(j) = w > 1, a(n) is the least novel q-smooth number divisible by w.
If j is a term in A002110, a(n) = smallest prime which has not yet appeared in the sequence (e.g., 1-->2, 2-->3, 6-->5, 30-->7, 210-->11, and so on).
Primes are in order and if p is prime and p|a(n) there is an i <= n such that a(i) = p (no multiple of p appears prior to p). Sequence is conjectured to have "Property S" of A368900. Also, for integers x, y with x < y and rad(x) = rad(y), x appears in the sequence before y. Conjecture: Sequence is a permutation of the positive integers which preserves the above mentioned properties of A000027.

			a(1) = 1 = A002110(0) so a(2) = 2 since rad(2*1) = 2 = A002110(1).
a(2) = 2 = A002110(1) so a(3) = 3 since then rad(3*2) = 6 = A002110(2).
a(6) = 5 is not primorial so a(7) = 12, since rad(12*5) = 30 = A002110(3) and no smaller novel number has this property.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10800.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10800, showing primes in red, perfect powers of primes in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, with purple indicating powerful numbers that are not prime powers. Primorials P(k) = A002110(k) and primes are labeled.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^18, with a color function indicating greatest prime factor p = gpf(a(n)), with red indicating p = 2, orange p = 3, ..., magenta p = 29, and pink p = 31.

Cf. A000027, A002110, A007947, A083720, A368900.

nn = 120;
c[] := False; m[] := 1;
Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = a[2];
f[x_] := f[x] = FactorInteger[x][[All, 1]];
Do[(If[Divisible[j, Set[q, Times @@ Prime@ Range[#2]]],
       q = Times @@ Prime@ Range[#2 + 1]];
       k = q/#3; mm = m[k];
       While[Or[c[k*mm], ! Divisible[q, Times @@ f[mm]]], mm++];
       k *= mm; While[c[k*m[k]], m[k]++]) & @@
     {#, PrimePi@ Last[#], Times @@ #} &[f[j]];
  Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}];
Array[a, nn]

a(n) = A002110(k) --> a(n+1) = prime(k+1); a(n+2) = 2*a(n), a(n+3) = 2*prime(k+1) (n >= 1).

More terms from Michael De Vlieger, May 18 2024

cp's OEIS Frontend

A369032 LCM-transform of permutation A241909.

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A369033 LCM-transform of permutation A241916.

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A368901 The least common multiple of the first n terms of Doudna sequence, A005940.

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A369045 LCM-transform of binary invert permutation (A054429).

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A369010 Exponential of Mangoldt function M(n) applied to primorial base exp-function: a(n) = A014963(A276086(n)).

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A369060 LCM-transform of Sycamore's D(3) variant of Doudna sequence (A356867).

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A369686 LCM-transform of A359804 (see Comment and links).

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A371985 For n a power of 2, a(n) = n. Otherwise a(n) is the smallest novel multiple of a(n - 2^m), where 2^m is the greatest power of 2 not exceeding n.

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