A351495 -id:A351495 - cp's OEIS frontend

A361330 Smallest prime that does not divide A351495(n).

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

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 18 2023

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A351495.

A361332 Where n appears in A351495, or -1 if it never occurs.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 23, 7, 8, 9, 161, 10, 1771, 12, 11, 16, 23023, 13, 391391, 14, 15, 19, 7436429, 17, 18, 21, 20, 24, 171037867, 22, 4960098143, 26, 25, 29, 28, 27

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 19 2023

Rémy Sigrist, C++ program

Cf. A351495, A361333.

a(29)-a(36) from Rémy Sigrist, May 04 2023

A361333 Index of prime(n) in A351495.

Original entry on oeis.org

2, 3, 6, 23, 161, 1771, 23023, 391391, 7436429, 171037867, 4960098143

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 19 2023

Rémy Sigrist, C++ program

Cf. A351495, A361332.

a(9)-a(11) from Rémy Sigrist, Mar 20 2023

A361334 Index of 2^n in A351495.

Original entry on oeis.org

1, 2, 4, 7, 16, 26, 52, 100, 200, 394, 788, 1572, 3144, 6282, 12564, 25124, 50248, 100490, 200980, 401956, 803912, 1607818, 3215636, 6431268, 12862536, 25725066, 51450132, 102900260, 205800520, 411601034, 823202068, 1646404132, 3292808264, 6585616522, 13171233044

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Mar 19 2023

nonn

Rémy Sigrist, C++ program

Cf. A351495, A361332, A361333.

More terms from Rémy Sigrist, Mar 20 2023

A361693 Index of where prime(n) first appears as a divisor of any term in A351495.

Original entry on oeis.org

2, 3, 6, 12, 19, 21, 29, 31, 39, 47, 49, 58, 65, 67, 75, 85, 93, 95, 104, 111, 113, 123, 131, 139, 150, 157, 159, 167, 169, 177, 196, 203, 213, 215, 231, 233, 242, 251, 259, 269, 277, 279, 295, 297, 305, 307, 325, 343, 351, 353, 361, 369, 371, 387, 397, 407, 415, 417, 426, 433, 435, 453, 472, 479

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 20 2023

nonn

Cf. A351495, A361333, A361330, A361332.

A359804 a(1) = 1, a(2) = 2; thereafter let p be the smallest prime that does not divide a(n-2)*a(n-1), then a(n) is the smallest multiple of p that is not yet in the sequence.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 10, 7, 9, 8, 15, 14, 11, 12, 20, 21, 22, 25, 18, 28, 30, 33, 35, 16, 24, 40, 42, 44, 45, 49, 26, 27, 50, 56, 36, 55, 63, 32, 60, 70, 66, 13, 65, 34, 39, 75, 38, 77, 48, 80, 84, 88, 85, 51, 46, 90, 91, 99, 52, 95, 54, 98, 100, 57, 105, 58, 110, 69, 112, 115, 72, 119, 120, 121

Offset: 1

David James Sycamore, Mar 08 2023

nonn

Let i = a(n-2), j = a(n-1). For k > 1, m >= 1, a(n) = m*prime(k) iff rad(i*j) = primorial(k-1), and this is the m-th such occurrence. This suggests the late appearance of most primes (namely those >= 7), apparent in the lowest part of scatterplot, where for example a(717126), a(63056215) = 31, 37 respectively.
As Scott R. Shannon has just observed, the following proof is incomplete, since it requires a proof that every even number appears. Even the induction step seems a little dubious. - N. J. A. Sloane, Mar 18 2023
All multiples of all primes appear in the sequence, for if not there is a least prime p such that m*p is not a term for any [some?] m >= 1. Choose any prime q < p; then every multiple of q must appear, so then p*q must be a term; contradiction since this is a multiple of p. [But what if p = 2?]
Corollary: This sequence is a permutation of the positive integers. [This question appears to be still open. - N. J. A. Sloane, Mar 18 2023]
Conjecture: The primes appear in their natural order.

			a(3) must be 3 because a(1,2) = 1,2 and 3 is the least prime which does not divide 2.
a(4) = 5 since this is the least multiple of the smallest prime which does not divide 2*3 = 6.
a(8) = 7 because a(6,7) = 6,10 and 7 is the smallest prime which does not divide 60, rad(60) = 2*3*5 = 30.
a(19,20) = 18,28, and 5 is the smallest prime not dividing rad(18*28) = 42. Since multiples of 5 have appeared 5 times already, a(20) = 6*5 = 30.

Winston de Greef, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a million terms showing primes in red.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither prime power nor squarefree in blue.

Cf. A002110, A007947, A053669.
See also A361502, A361503, A361504, A361505.
A351495 has a very similar definition.

R:= 1,2: S:= {1,2}:
for i from 3 to 100 do
  s:= R[i-2]*R[i-1]:
  p:= 2;
  while s mod p = 0 do p:= nextprime(p) od:
  for r from p by p while member(r,S) do od:
  R:= R,r; S:= S union {r}
od:
R; # Robert Israel, Mar 08 2023

nn = 2^10; c[] = False; q[] = 1;
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
Set[{i, j}, {a[1], a[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}, {k, True, j, k}];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Mar 08 2023 *)

findp(n) = forprime(p=2, , if (n%p, return(p)));
lista(nn) = my(va = vector(nn, k, if (k<=2, k))); for (n=3, nn, my(vsa = vecsort(va), p=findp(va[n-1]*va[n-2]), k=p); while (vecsearch(vsa, k), k+=p); va[n] = k;); va; \\ Michel Marcus, Mar 09 2023

from itertools import count, islice
from sympy import prime, primefactors, primepi
def A359804_gen(): # generator of terms
    aset, bset, cset = set(), {1}, {1,2}
    yield from (1,2)
    while True:
        for i in count(1):
            if not (i in aset or i in bset):
                p = prime(i)
                for j in count(1):
                    if (m:=j*p) not in cset:
                        yield m
                        cset.add(m)
                        break
                break
        aset, bset = bset, set(map(primepi,primefactors(m)))
A359804_list = list(islice(A359804_gen(),30)) # Chai Wah Wu, Mar 18 2023

A368133 a(1,2,3) = 1,2,3; let j = a(n-1), M(n) = Product_{i = 1..n-2} { p a distinct prime: p | a(i), gcd(p, j) = 1 }. For n > 3, a(n) is the least novel multiple of M(n) if M(n) > 1; otherwise a(n) is the least novel multiple of A053669(j), the smallest prime which does not divide j.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 12, 10, 9, 20, 15, 8, 30, 7, 60, 14, 45, 28, 75, 42, 25, 84, 35, 18, 70, 21, 40, 63, 50, 105, 16, 210, 11, 420, 22, 315, 44, 525, 66, 140, 33, 280, 99, 350, 132, 175, 198, 245, 264, 385, 24, 770, 27, 1540, 36, 1155, 26, 2310, 13, 4620, 39, 3080

Offset: 1

David James Sycamore, Dec 13 2023

nonn

M(n) is a squarefree number whose prime factors are the distinct primes which divide a(m), m <= n-2, but do not divide j. M(n) > 1 implies there exists at least one term prior to j having a prime divisor which does not divide j, and M(n) is the product of all such primes. If, for any term a(m), m <= n-2, every prime factor of a(m) also divides j, then M(n) = 1, the empty product.
Primorial a(n-1) implies prime a(n); see Formula.
Conjectured to be a permutation of the positive integers.
Compare with A368108 which has a slightly different definition but works in a similar way.
From Michael De Vlieger, Jan 05 2024: (Start)
This sequence is the same as A362855 for 91306 terms.
A362855(91306) = a(91306) = A002110(17),
A362855(91307) = 53 = prime(16), a(91307) = 61 = prime(18),
A362855(91308) = A002110(17)/prime(16), a(91308) = 2*A002110(17).
Thereafter the sequences diverge. It seems unlikely that the 2 sequences will become coincident again as n increases beyond 91308. (End)

			a(1, 2, 3) = 1, 2, 3. M(4) = 2 because 2 | a(2) but does not divide a(3); 2 is the only a(m), m < 3, with this property, so a(4) = 4, the least novel multiple of 2.
Now we have a(1,2,3,4) = 1,2,3,4. M(5) = 3 because 3 | a(3) but does not divide a(4); 3 is the only a(m), m < 4, with this property, so a(5) = 2*3 = 6, the least novel multiple of 3.
We now have a(1..5) = 1, 2, 3, 4, 6. M(6) = 1, the empty product, because there is no prime which divides some a(m), m < 5, which does not also divide a(n-1) = 6. This situation invokes the second condition of the definition, so a(6) = 1*5, the least novel multiple of A053669(6) = 5, the smallest prime which does not divide 6. Consequently a(7) = 2*6 = 12 because no prime dividing a(1..5) also divides 5.
The same situation arises again at a(13) = 30 = 2*3*5; every prime divisor of a(m), m < 13, is 2, 3, or 5, which again invokes the second condition, M(14) = 1, the empty product, so a(14) = 1*7, since A053669(30) = 7. Consequently a(15) = 2*7 = 14.
a(91307) = 61 (whereas A362855(91307) = 53; point of divergence from A362855).

Michael De Vlieger, Log log scatterplot of a(n) and b(n), n = 1..2^20, where a(n) is shown in red, b(n) = A362855(n) is shown in blue. Black indicates where a(n) = b(n).

Cf. A002110, A053669, A351495, A362855, A368108.

nn = 10^5; c[] := False; m[] := 1;
Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 3]; j = 3;
s = {2}; r = Max[s]; c[3] = False;
q[x_] := Block[{qq = 2}, While[Divisible[x, qq], qq = NextPrime[qq]]; qq];
Do[(If[# == 1,
        Set[k, NextPrime[r]],
        While[Or[c[#], # == j] &[# m[#]], m[#]++];
          Set[k, # m[#]]] &[Times @@ Complement[s, #]];
          s = Union[s, #];
     If[Last[#] > r, r = Last[#]]) &@ FactorInteger[j][[All, 1]];
  Set[{a[n], c[j], j}, {k, True, k}], {n, 4, nn}];
Array[a, nn] (* Michael De Vlieger, Jan 05 2024 *)

When for some m, a(m) = A002110(n), a primorial number, a(m+1) = prime(n+1), a(m+2) = 2*A002110(n), and a(m+3) = 2*prime(n+1); see Example.

a(n) = A362855(n), for 1 <= n <= 91306 (see link and Example).

More terms from Michael De Vlieger, Jan 05 2024

A362754 a(1) = 1, a(2) = 6; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and also contains as a factor the smallest prime that is not a factor of a(n-1).

Original entry on oeis.org

1, 6, 10, 12, 15, 18, 20, 24, 30, 14, 21, 28, 36, 40, 42, 35, 50, 45, 48, 60, 56, 54, 70, 63, 66, 55, 22, 33, 44, 72, 75, 78, 65, 26, 39, 52, 84, 80, 90, 98, 96, 100, 102, 85, 34, 51, 68, 108, 105, 110, 99, 88, 114, 95, 38, 57, 76, 120, 112, 126, 130, 117, 104, 132, 135, 138, 115, 46, 69, 92, 144

Offset: 1

Scott R. Shannon, May 02 2023

nonn

No term can be a prime power as each term must contain at least two distinct prime factors. This make the sequence similar to A360519 and A361606. A close examination of the lines of concentrated terms, see the attached images, shows they have a slight downward curvature. In the first 250000 terms the only fixed points are 1, 69, 87, 116825, although it is possible more exist for very large values of n.

			a(3) = 10 as a(2) = 6 = 2*3, and 10 is the smallest unused number that shares a factor with 6 while also containing 5 as a prime factor, the smallest prime not a factor of 6.
a(4) = 12 as a(3) = 10 = 2*5, and 12 is the smallest unused number that shares a factor with 10 while also containing 3 as a prime factor, the smallest prime not a factor of 10.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing squarefree composites in green, numbers neither squarefree nor prime powers in blue, and numbers in A286708 in large light blue.
Scott R. Shannon, Image of the first 250000 terms. The green line is a(n) = n.
Scott R. Shannon, Image of the first 250000 terms in color. Terms with a lowest prime factor 2, 3, 5, 7, 11, >=13 are colored white, red, yellow, green, blue, violet and light gray respectively.

Cf. A337687, A351495, A064413, A360519, A361606.

nn = 120; c[] := False; m[] := 2; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, {1, 6}]; j = a[2]; Do[q = 2; While[Divisible[j, q], q = NextPrime[q]]; k = m[q]; While[Or[c[#], PrimePowerQ[#], CoprimeQ[j, k]] &[q k], k++]; k *= q; While[c[m[q] q], m[q]++]; Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, May 02 2023 *)

A364154 Lexicographically earliest sequence of distinct positive integers such that a(n) is least novel multiple m of the product of all primes less than the greatest prime factor of a(n-1) which do not divide a(n-1); a(1) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 30, 8, 9, 10, 12, 11, 210, 13, 2310, 14, 15, 16, 17, 30030, 18, 19, 510510, 20, 21, 40, 24, 22, 105, 26, 1155, 28, 45, 32, 23, 9699690, 25, 36, 27, 34, 15015, 38, 255255, 42, 35, 48, 29, 223092870, 31, 6469693230, 33, 70, 39, 770, 51, 10010

Offset: 1

David James Sycamore, Jul 11 2023

nonn

It follows from the definition that the sequence is infinite, and that the records (outside of the first 7 terms) are all primorial numbers, meaning that it grows very quickly.
When there are no primes less than the greatest prime factor of a(n-1) which do not divide a(n-1) then m is the least novel multiple of 1, the empty product, and therefore a(n) = u, the least unused number in the sequence so far. The only way a prime can enter the sequence is as u. When a(n-1) = prime(k), a(n) is A002110(k-1), and any primorial term is followed by u. Thus: prime —> primorial —> u.
Sequence is a permutation of the positive integers since by the definition no number appears more than once and m = 1 eventually introduces any number not already placed by the first part of the definition (m > 1).

			a(1) = 1 and there are no primes < 1 which divide 1 therefore m = 1 so a(2) = 2, the least unused number. Likewise a(3) = 3.
a(4) = 2*2 = 4 since 2 is the only prime < 3 which does not divide 3 and 2 has already occurred.
Since a(7) = 7, a(8) = 2*3*5 = 30.

Michael De Vlieger, Table of n, a(n) for n = 1..4992
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2^16.
Michael De Vlieger, Plot p^e | a(n) at (x,y) = (n, pi(p)) for n = 1..2^11, with a color function representing e = 1 as black, e = 2 as red, ..., the maximum value of e in the dataset as magenta. Under the plot, we indicate the empty product in black, primes in red, composite prime powers in gold, squarefree composites in green, and all other numbers in blue.

Cf. A002110, A083720, A351495, A359804, A363195.

nn = 120; c[] := False; m[] := 1; a[1] = j = 1; c[1] = True;
Do[k = Times @@ Complement[Prime@ Range[PrimePi@ Last[#] - 1], #] &[
   FactorInteger[j][[All, 1]] ];
 While[c[k m[k]], m[k]++]; k *= m[k];
 Set[{a[n], c[k], j}, {k, True, k}], {n, 2, nn}];
Array[a, nn]

lista(nn) = my(c, m, v=List([1, 2])); for(k=3, nn, c=m=1; forprime(p=2, vecmax(factor(v[k-1])[, 1]), if(v[k-1]%p, m*=p)); while(setsearch(Set(v), c*m), c++); listput(v, c*m)); Vec(v) \\ Jinyuan Wang, Jul 11 2023

More terms from Jinyuan Wang, Jul 11 2023

A368108 a(1,2,3) = 1,2,3. For n > 3, a(n) is the smallest of the least novel multiples of all primes which divide an earlier term but do not divide a(n-1). If the prime divisors of all prior terms also divide a(n-1), a(n) is the least novel multiple of the smallest prime which does not divide a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 9, 10, 12, 15, 14, 18, 7, 16, 20, 21, 22, 24, 11, 25, 26, 27, 13, 28, 30, 33, 32, 35, 34, 36, 17, 38, 39, 19, 40, 42, 44, 45, 46, 48, 23, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 29, 62, 63, 31, 64, 65, 66, 68, 69, 70, 72, 75, 74, 76, 37, 77

Offset: 1

David James Sycamore, Dec 12 2023

nonn

Conjectured to be a permutation of the positive integers with primes in order.

Same as A351495 for the first 13 terms; diverges thereafter.

			a(4) = 4, least novel multiple of 2, the smallest prime which does not divide 3.
a(5) = 6, least novel multiple of 3, the smallest prime which does not divide 4.
There is only one occasion where the second condition of the definition applies, namely a(5) = 6, where 2 and 3 have already occurred; therefore a(6) = 5, the smallest prime which does not divide 6.
a(7) = 8 since 2 and 3 do not divide 5, and their least novel multiples are 8, and 9 respectively.
Since a(7) = 8, a(8) is the least novel multiple of 3 (9) or 5 (10), so a(8) = 9.
a(13) = 18 and 5, 7 are the primes which divide prior terms but don't divide 18. The least novel multiple of 5 is 20, and the least novel multiple of 7 is 7, therefore a(14) = 7.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red.

Cf. A351495.

nn = 120; c[] := False; m[] := 1;
Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 3];
j = 3; s = {2}; r = Max[s]; c[3] = False;
Do[(If[Length[#] == 0, Set[k, NextPrime[r]],
        Set[k, Min[#]]] &@
      DeleteCases[Map[(While[c[# m[#]], m[#]++]; # m[#]) &, s], j];
     s = Union[s, #];
     If[Last[#] > r, r = Last[#]]) &@ FactorInteger[j][[All, 1]];
  Set[{a[n], c[j], j}, {k, True, k}], {n, 4, nn}];
Array[a, nn] (* Michael De Vlieger, Dec 12 2023 *)

If a(m) = 2*p where p is a prime > 5 which is not already a term, then a(m+2) = p.

More terms from Michael De Vlieger, Dec 12 2023

cp's OEIS Frontend

A361330 Smallest prime that does not divide A351495(n).

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A361332 Where n appears in A351495, or -1 if it never occurs.

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A361333 Index of prime(n) in A351495.

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A361334 Index of 2^n in A351495.

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A361693 Index of where prime(n) first appears as a divisor of any term in A351495.

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A359804 a(1) = 1, a(2) = 2; thereafter let p be the smallest prime that does not divide a(n-2)*a(n-1), then a(n) is the smallest multiple of p that is not yet in the sequence.

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A368133 a(1,2,3) = 1,2,3; let j = a(n-1), M(n) = Product_{i = 1..n-2} { p a distinct prime: p | a(i), gcd(p, j) = 1 }. For n > 3, a(n) is the least novel multiple of M(n) if M(n) > 1; otherwise a(n) is the least novel multiple of A053669(j), the smallest prime which does not divide j.

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A362754 a(1) = 1, a(2) = 6; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and also contains as a factor the smallest prime that is not a factor of a(n-1).

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A364154 Lexicographically earliest sequence of distinct positive integers such that a(n) is least novel multiple m of the product of all primes less than the greatest prime factor of a(n-1) which do not divide a(n-1); a(1) = 1.

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