A359256 -id:A359256 - cp's OEIS frontend

A359557 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-2) + a(n-1) are factors of a(n).

1, 2, 3, 5, 4, 6, 10, 8, 12, 20, 14, 34, 18, 26, 22, 24, 46, 70, 58, 16, 74, 30, 52, 82, 134, 36, 170, 206, 94, 60, 154, 214, 92, 102, 194, 148, 114, 262, 188, 90, 278, 138, 78, 42, 120, 48, 84, 66, 150, 54, 204, 258, 462, 180, 642, 822, 366, 132, 498, 210, 354, 282, 318, 240, 186, 426, 306

Offset: 1

Scott R. Shannon, Jan 05 2023

All terms other than 3 and 5 are even. As a(114) = 510 and a(115) = 570 both contain 2 and 5 as prime factors, all subsequent terms are multiples of 10. Likewise after 1994 terms all terms contain 2, 3, 5, 7, 11 as factors, so all subsequent terms are multiples of 2*3*5*7*11 = 2310.
The terms grow rapidly in size, e.g., a(2459) = 28318290. The smallest number not to appear is 7.
a(n) = k*m such that k = A007947(a(n-2)+a(n-1)) and m >= 1 produces the smallest k*m != a(j), j < n. - Michael De Vlieger, Jan 07 2023

			a(5) = 4 as a(3) + a(4) = 3 + 5 = 8 which contains 2 as its only distinct prime factor, and 4 is the smallest unused number to contain 2 as a factor.
a(12) = 34 as a(10) + a(11) = 20 + 14 = 34 which contains 2 and 17 as distinct prime factors, and 34 is also the smallest unused number to contain 2 and 17 as factors.
a(13) = 18 as a(11) + a(12) = 14 + 34 = 48 which contains 2 and 3 as distinct prime factors, and 18 is the smallest unused number to contain 2 and 3 as factors.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing records in red, highlighting composite prime powers in gold, squarefree semiprimes in large green, 5-smooth numbers that are not p-smooth, p > 5, in small green, 7-smooth numbers that are not p-smooth, p > 7, in magenta, 11-smooth numbers that are not p-smooth, p > 11, in cyan, and 13-smooth numbers that are not p-smooth, p > 13, in blue.

Cf. A359256, A027748, A064413, A352867, A007947.

nn = 2^7; c[] = False; q[] = 1; f[n_] := Times @@ FactorInteger[n][[All, 1]]; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; Set[{i, j, k}, {a[1], a[2], f[a[1] + a[2]]}]; Do[m = q[k]; While[c[k m], m++]; m *= k; While[c[k q[k]], q[k]++]; Set[{a[n], c[m], i, j, k}, {m, True, j, m, f[j + m]}], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Jan 07 2023 *)

from math import prod
from sympy import factorint
from itertools import count, islice
def agen():
    i, j, aset = 1, 2, {1, 2}; yield from [i, j]
    while True:
        m = prod(factorint(i+j))
        an = next(k*m for k in count(1) if m*k not in aset)
        i, j = j, an; aset.add(an); yield an
print(list(islice(agen(), 61))) # Michael S. Branicky, Jan 16 2023

A359857 a(1) = 1, a(2) = 2, and let i,j represent a(n-2), a(n-1) respectively. For n > 2: If only one of i,j is prime, a(n) = least novel multiple of i+j. If i,j are both prime, a(n) = least novel multiple of i*j. If both i,j are nonprime, a(n) is least novel k prime to both i and j.

Original entry on oeis.org

1, 2, 3, 6, 9, 5, 14, 19, 33, 52, 7, 59, 413, 472, 11, 483, 494, 17, 511, 528, 13, 541, 7033, 7574, 15, 23, 38, 61, 99, 160, 29, 189, 218, 25, 21, 4, 31, 35, 66, 37, 103, 3811, 3914, 27, 41, 68, 109, 177, 286, 43, 329, 372, 53, 425, 478, 39, 47, 86, 133, 45, 8

Offset: 1

David James Sycamore, Jan 16 2023

nonn

A lexicographically earliest sequence. If the "least novel" restrictions on the first two conditions are removed the result is a sequence initially identical to this one, but eventually repeat terms occur, whereas in this constrained version there are no repeats. The plots appear to distinguish three distinct zones, corresponding to the conditions of the Name. The first (upper) relates to terms following the occurrence of two primes (i*j), the middle zone relates to the occurrence of a prime and a nonprime (i+j), and the third (lowest) refers to terms following the occurrence of consecutive nonprime terms, where the coprime condition introduces the least unused numbers, which are initially (until a(25) = 15) all primes. It is not clear if every positive integer appears, but this does seem likely (note the late appearance of 4 at a(36)).

			a(3) = 3 since only one of a(1), a(2) is prime and i+j = 3 has not occurred previously.
a(4) = 6 since a(2) = 2 and a(3) = 3 are both prime and i*j = 6 has not occurred previously.
a(6) = 5 since a(4) = 6 and a(5) = 9 are both composite, and 5 is the least novel number prime to both.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A359256, A359557.

from math import gcd
from sympy import isprime
from itertools import count, islice
def agen():
    i, j, pi, pj, mink, aset = 1, 2, 0, 1, 3, {1, 2}
    yield from [i, j]
    while True:
        if pi^pj:
            k, m = max(mink//(i+j), 1), i+j
            while m*k in aset: k += 1
        elif pi&pj:
            k, m = max(mink//(i*j), 1), i*j
            while m*k in aset: k += 1
        else:
            k, m = mink, 1
            while k in aset or gcd(k, i) != 1 or gcd(k, j) != 1: k += 1
        an = m*k
        i, j, pi, pj = j, an, pj, int(isprime(an)); yield an; aset.add(an)
        while mink in aset: mink += 1
print(list(islice(agen(), 36))) # Michael S. Branicky, Jan 16 2023

a(31)=29 inserted and a(38) and beyond from Michael S. Branicky, Jan 16 2023

A361593 a(1) = 1, a(2) = 2, a(3) = 3; for n > 3, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-3) + a(n-2) + a(n-1) are factors of a(n).

Original entry on oeis.org

1, 2, 3, 6, 11, 10, 9, 30, 7, 46, 83, 34, 163, 70, 267, 20, 357, 322, 699, 1378, 2399, 2238, 6015, 5326, 13579, 6230, 25135, 106, 31471, 14178, 45755, 15234, 75167, 68078, 8341, 151586, 228005, 193966, 573557, 248882, 1016405, 306474, 1571761, 361830, 2240065, 1043414, 3645309, 3464394

Offset: 1

Scott R. Shannon and Michael De Vlieger, Mar 16 2023

nonn

This is a variation of A359557 where the previous three terms are added instead of two. Unlike A359557 the terms here do no rapidly reach a regime where all terms share one or more prime factors, and it is unknown if this ever occurs.

			a(6) = 10 as a(3) + a(4) + a(5) = 3 + 6 + 11 = 20 = 2*2*5, and the smallest unused number containing 2 and 5 as factors is 10.

Michael De Vlieger, Table of n, a(n) for n = 1..750
Michael De Vlieger, Scatterplot of Log_10(a(n)), n = 1..750, showing records in red.
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..600, showing primes in red, prime powers in gold, and squarefree composites in green.

Cf. A359557, A359256, A027748, A064413, A352867, A007947.

nn = 120; c[] = False; q[] = 1;
f[n_] := Times @@ FactorInteger[n][[All, 1]]; t = 3;
Array[Set[{a[#], c[#]}, {#, True}] &, t]; Set[{i, j, k, x}, {a[t - 2],
   a[t - 1], a[t], f[a[t - 2] + a[t - 1] + a[t]]}];
Do[m = q[x];
  While[c[x m], m++];
  m *= x; While[c[x q[x]], q[x]++];
  Set[{a[n], c[m], i, j, k, x}, {m, True, j, k, m, f[j + k + m]}], {n,
t + 1, nn}]; Array[a, nn] (* Michael De Vlieger, Mar 20 2023 *)

cp's OEIS Frontend

A359557 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-2) + a(n-1) are factors of a(n).

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A359857 a(1) = 1, a(2) = 2, and let i,j represent a(n-2), a(n-1) respectively. For n > 2: If only one of i,j is prime, a(n) = least novel multiple of i+j. If i,j are both prime, a(n) = least novel multiple of i*j. If both i,j are nonprime, a(n) is least novel k prime to both i and j.

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A361593 a(1) = 1, a(2) = 2, a(3) = 3; for n > 3, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-3) + a(n-2) + a(n-1) are factors of a(n).

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Mathematica