A086064 -id:A086064 - cp's OEIS frontend

A332703 Lexicographically earliest sequence of distinct positive terms such that a(n) is a substring of a(n)*a(n+1).

1, 10, 11, 100, 21, 58, 51, 69, 39, 87, 33, 101, 1000, 31, 102, 501, 699, 1001, 10000, 41, 59, 27, 103, 1002, 5001, 6999, 8144, 626, 739, 506, 693, 533, 404, 251, 767, 752, 126, 894, 793, 981, 202, 1003, 10001, 100000, 61, 92, 26, 151, 1004, 2501, 7697, 5037, 6453, 20001, 59998, 50001, 69999

Offset: 1

Eric Angelini and Carole Dubois, Feb 20 2020

			1, 10, 11, 100, 21, 58, 51, 69,
a(1)*a(2) = 1*10 = 100 with 1 substring of 100;
a(2)*a(3) = 10*11 = 110 with 10 substring of 110;
a(3)*a(4) = 11*100 = 1100 with 11 substring of 1100;
a(4)*a(5) = 100*21 = 2100 with 100 substring of 2100;
a(5)*a(6) = 21*58 = 1218 with 21 substring of 1218;
a(6)*a(7) = 58*51 = 2958 with 58 substring of 2958;
a(7)*a(8) = 51*69 = 3519 with 51 substring of 3519; etc.

Scott Shannon, Table of n, a(n) for n = 1..3656

Cf. A086064.

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

Original entry on oeis.org

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

A087217 In decimal representation: smallest multiple of n containing it as substring.

Original entry on oeis.org

10, 12, 30, 24, 15, 36, 70, 48, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 120, 210, 220, 230, 240, 125, 260, 270, 280, 290, 300, 310, 320, 330, 340, 350, 360, 370, 380, 390, 240, 410, 420, 430, 440, 450, 460, 470, 480, 490, 150, 510, 520, 530

Offset: 1

Reinhard Zumkeller, Aug 26 2003

a(n) = n*A086064(n).

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

smn[n_]:=Module[{k=2},While[SequenceCount[IntegerDigits[k*n],IntegerDigits[ n]]<1,k++];k*n]; Array[smn,60] (* Harvey P. Dale, Oct 02 2021 *)

def a(n):
    s = str(n)
    return next(mn for mn in range(2*n, 11*n, n) if s in str(mn))
print([a(n) for n in range(1, 55)]) # Michael S. Branicky, Feb 23 2025

A371895 a(n) is the least k>0 such that n*k contains the n-th prime as a substring.

Original entry on oeis.org

2, 15, 5, 18, 22, 22, 25, 24, 26, 29, 21, 31, 32, 31, 98, 96, 27, 34, 88, 355, 13, 36, 21, 79, 39, 39, 189, 383, 376, 371, 41, 41, 416, 41, 426, 42, 425, 43, 43, 433, 419, 431, 237, 44, 266, 433, 45, 465, 464, 458, 83, 46, 423, 417, 468, 47, 472, 468, 47, 469, 103, 473, 488, 486, 202, 481, 348, 496, 63, 499

Offset: 1

Giorgos Kalogeropoulos, Apr 11 2024

From Robert Israel, Jun 06 2024: (Start)
If n is divisible by 2*10^k or 5*10^k, a(n) >= prime(n)*10^(k+1)/n.
Let n = 2^b * 5^c * k with k coprime to 10, and suppose prime(n) has d digits. If b <= c, then f(n) < 10^d * 2^(c-b).  If b > c, then f(n) < 10^d * 5^(b-c).
a(10^k) = prime(10^k).
a(2*10^k) = 5 * prime(2*10^k).
a(5*10^k) = 2 * prime(5*10^k). (End)

			a(8) = 24 because 24 is the least positive integer such that 24*8 = 192 contains the prime(8) = 19.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A086064.

f:= proc(n) local t,k;
  t:= convert(ithprime(n),string);
  for k from 1 do
     if StringTools:-Search(t, convert(n*k,string)) > 0 then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Jun 05 2024

a[n_]:=(k=1;While[!StringContainsQ[ToString[n*k],ToString@Prime@n],k++];k); Array[a,70]

a(n) = my(k=1, s=Str(prime(n))); while(#strsplit(Str(k*n), s) < 2, k++); k; \\ Michel Marcus, Apr 11 2024

from sympy import prime
from itertools import count
def a(n): t=str(prime(n)); return next(k for k in count(1) if t in str(n*k))
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Apr 11 2024

# faster for initial segment of sequence
from sympy import nextprime
from itertools import count, islice
def agen(): # generator of terms
    pn = 2
    for n in count(1):
        t = str(pn)
        yield next(k for k in count(1) if t in str(n*k))
        pn = nextprime(pn)
print(list(islice(agen(), 70))) # Michael S. Branicky, Apr 11 2024

cp's OEIS Frontend

A332703 Lexicographically earliest sequence of distinct positive terms such that a(n) is a substring of a(n)*a(n+1).

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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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PARI

Python

A087217 In decimal representation: smallest multiple of n containing it as substring.

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Mathematica

Python

A371895 a(n) is the least k>0 such that n*k contains the n-th prime as a substring.

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Maple

Mathematica

PARI

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Python