A333811 -id:A333811 - cp's OEIS frontend

A333410 a(n) is the smallest positive integer not yet appearing in the sequence such that n*a(n) contains n as a substring.

1, 6, 10, 11, 3, 16, 21, 23, 22, 31, 100, 26, 87, 51, 41, 73, 69, 66, 63, 36, 58, 101, 97, 52, 5, 102, 103, 46, 79, 61, 107, 76, 192, 151, 81, 38, 201, 89, 164, 35, 59, 34, 173, 126, 99, 184, 74, 135, 153, 7, 167, 176, 29, 251, 121, 28, 168, 148, 27, 56, 92, 123, 137, 57, 141, 207, 25, 113

Offset: 1

Scott R. Shannon, Apr 11 2020

			a(2) = 6 as 6 has not appeared previously and 2 * 6 = 12 which contains '2' as a substring.
a(6) = 16 as 16 has not appeared previously and 6 * 16 = 96 which contains '6' as a substring.
a(7) = 21 as 21 has not appeared previously and 7 * 21 = 147 which contains '7' as a substring.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A333410

Cf. A086064, A333774, A333775, A333811, A332795, A332703.

PARI
```
See Links section.
```

from itertools import count, islice
def agen(): # generator of terms
    s, mink, aset, concat = 1, 2, {1}, "1"
    yield from [1]
    for n in count(2):
        an, sn = mink, str(n)
        while an in aset or not sn in str(n*an): an += 1
        aset.add(an); s += an; concat += str(an); yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 68))) # Michael S. Branicky, Feb 08 2024

A333774 a(0) = 0; for n > 0, a(n) = the smallest positive integer not yet appearing in the sequence such that a(n-1) + a(n) contains as a substring either a(n-1) or a(n).

Original entry on oeis.org

0, 1, 9, 10, 2, 18, 100, 3, 20, 4, 30, 5, 40, 6, 50, 7, 60, 8, 70, 200, 11, 99, 300, 12, 108, 972, 107, 963, 106, 954, 105, 945, 104, 936, 103, 927, 102, 918, 101, 909, 1000, 13, 117, 1053, 116, 1044, 115, 1035, 114, 1026, 113, 1017, 112, 1008, 111, 999, 110, 990, 109, 981, 2000, 14

Offset: 0

Scott R. Shannon, Apr 05 2020

			a(1) = 1 as a(0) = a(1) = 0 + 1 = 1 which contains '1' as a substring.
a(2) = 9 as a(1) + a(2) = 1 + 9 = 10 which contains '1' as a substring.
a(4) = 2 as a(3) + a(4) = 10 + 2 = 12 which contains '2' as a substring
a(49) = 1026 as a(48) + a(49) = 114 + 1026 = 1140 which contains '114' as a substring.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A333774

Cf. A333775, A333811, A332795, A332703.

PARI
```
See Links section.
```

A333775 a(0) = 0; for n > 0, a(n) is the smallest positive integer not yet appearing in the sequence such that a(n-1) * a(n) contains as a substring either a(n-1) or a(n).

Original entry on oeis.org

0, 1, 2, 6, 4, 10, 3, 5, 7, 11, 8, 16, 20, 21, 9, 22, 51, 12, 26, 24, 52, 76, 28, 46, 40, 31, 30, 41, 15, 50, 13, 25, 17, 69, 34, 42, 94, 100, 14, 82, 71, 60, 36, 38, 89, 55, 61, 35, 80, 56, 87, 33, 75, 29, 53, 101, 18, 66, 151, 32, 91, 43, 102, 19, 63, 137, 83, 96, 126, 44, 59, 27, 103

Offset: 0

Scott R. Shannon, Apr 05 2020

			a(1) = 1 as a(0) * a(1) = 0 * 1 = 0 which contains '0' as a substring.
a(4) = 4 as a(3) * a(4) = 6 * 4 = 24 which contains '4' as a substring.
a(18) = 26 as a(17) * a(18) = 12 * 26 = 312 which contains '12' as a substring.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A333775

Cf. A333774, A333811, A332795, A332703.

PARI
```
See Links section.
```

A333923 a(n) is the smallest positive integer such that n^a(n) is divisible by n+a(n).

Original entry on oeis.org

2, 6, 4, 20, 3, 42, 8, 18, 6, 110, 4, 156, 14, 10, 16, 272, 6, 342, 5, 6, 10, 506, 3, 100, 6, 54, 4, 812, 6, 930, 32, 48, 30, 14, 12, 1332, 26, 42, 10, 1640, 6, 1806, 20, 30, 18, 2162, 6, 294, 14, 30, 12, 2756, 10, 66, 8, 24, 6, 3422, 4, 3660, 62, 18, 64, 60, 6, 4422

Offset: 2

Scott R. Shannon, Apr 10 2020

nonn

As in A063427, if n is a prime then a(n^k) = (n-1)*n^k for k>=1. This sequence also matches A063427 for numerous other nonprime terms for small values of n.

For n below 10000 the values where n = a(n), other than n being a power of 2, are n = 14, 62, 122, 254, 508, 1018, 2038, 2042, 8182, 8186.

			a(2) = 2 as 2 ^ 2 = 4 is divisible by 2 + 2 = 4.
a(3) = 6 as 3 ^ 6 = 729 is divisible by 3 + 6 = 9.
a(4) = 4 as 4 ^ 4 = 256 is divisible by 4 + 4 = 8.
a(5) = 20 as 5 ^ 20 = 95367431640625 is divisible by 5 + 20 = 25.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A063427, A333811, A332795, A332703.

spi[n_]:=Module[{k=1},While[PowerMod[n,k,n+k]!=0,k++];k]; Array[spi,70,2] (* Harvey P. Dale, Jan 16 2022 *)

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A333410 a(n) is the smallest positive integer not yet appearing in the sequence such that n*a(n) contains n as a substring.

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A333774 a(0) = 0; for n > 0, a(n) = the smallest positive integer not yet appearing in the sequence such that a(n-1) + a(n) contains as a substring either a(n-1) or a(n).

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A333775 a(0) = 0; for n > 0, a(n) is the smallest positive integer not yet appearing in the sequence such that a(n-1) * a(n) contains as a substring either a(n-1) or a(n).

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A333923 a(n) is the smallest positive integer such that n^a(n) is divisible by n+a(n).

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