cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Scott R. Shannon, Apr 11 2020

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    See Links section.
  • Python
    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

Views

Author

Scott R. Shannon, Apr 05 2020

Keywords

Examples

			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.

Links

Crossrefs

Cf. A333775, A333811, A332795, A332703.

Programs

  • PARI
    See Links section.

A333811 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, 5, 3, 7, 4, 6, 8, 9, 11, 15, 12, 14, 13, 10, 16, 17, 21, 20, 25, 18, 23, 19, 22, 24, 26, 29, 30, 28, 31, 27, 32, 33, 35, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 46, 48, 50, 49, 51, 52, 53, 55, 54, 56, 57, 59, 58, 61, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70
Offset: 0

Views

Author

Scott R. Shannon, Apr 05 2020

Keywords

Comments

a(3) = 5 as a(2) ^ a(3) = 2 ^ 5 = 32 which contains '2' as a substring.
a(4) = 3 as a(3) ^ a(4) = 5 ^ 3 = 125 which contains '5' as a substring.
a(5) = 7 as a(4) ^ a(5) = 3 ^ 7 = 2187 which contains '7' as a substring.

Links

Crossrefs

Cf. A333774, A333775, A332795, A332703.

Programs

  • 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

Views

Author

Scott R. Shannon, Apr 05 2020

Keywords

Examples

			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.

Links

Crossrefs

Cf. A333774, A333811, A332795, A332703.

Programs

  • 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

Views

Author

Scott R. Shannon, Apr 10 2020

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A063427, A333811, A332795, A332703.

Programs

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

A342127 Numbers m such that the product of m and the string m in reverse contains m as a substring.

Original entry on oeis.org

0, 1, 5, 6, 10, 47, 50, 60, 75, 78, 100, 125, 152, 457, 500, 600, 750, 1000, 1025, 1052, 1250, 1520, 5000, 5625, 6000, 7500, 10000, 10025, 10052, 10250, 10520, 12266, 12500, 15200, 23258, 43567, 50000, 56250, 60000, 62656, 75000, 82291, 90625, 98254, 100000, 100025, 100052, 100250, 100520
Offset: 1

Views

Author

Scott R. Shannon, Mar 01 2021

Keywords

Comments

Numerous patterns exist in the terms, e.g., all numbers of the form 1*10^k, 5*10^k, 6*10^k, 75*10^k, 10^(k+2)+25, where k>=0, are in the sequence.

Examples

			6 is a term as 6*reverse(6) = 6*6 = 36 contains '6' as a substring.
47 is a term as 47*reverse(47) = 47*74 = 3478 contains '47' as a substring.
1052 is a term as 1052*reverse(1052) = 1052*2501 = 2631052 contains '1052' as a substring.

Links

Crossrefs

Cf. A061205, A342130 (base 2), A001477, A004086, A181721, A203565, A332795.

Programs

  • Maple
    filter:= proc(n) local L,d,Lp,r,i;
  L:= convert(n,base,10);
  d:= nops(L);
  r:= add(L[-i]*10^(i-1),i=1..d);
  Lp:= convert(n*r,base,10);
  ormap(t -> Lp[t..t+d-1] = L, [$1..nops(Lp)+1-d])
end proc:
select(filter, [$0..120000]); # Robert Israel, Mar 24 2024
  • Mathematica
    Select[Range[0,110000],SequenceCount[IntegerDigits[# IntegerReverse[#]],IntegerDigits[#]]>0&] (* Harvey P. Dale, Apr 20 2024 *)
  • PARI
    isok(m) = #strsplit(Str(m*fromdigits(Vecrev(digits(m)))), Str(m)) > 1; \\ Michel Marcus, Mar 01 2021
  • Python
    def ok(n): return (s:=str(n)) in str(n*int(s[::-1]))
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Mar 25 2024

A342130 Decimal numbers m such that the product of the binary string of m and the binary string of m in reverse contains the binary string of m as a substring.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 27, 32, 54, 64, 108, 128, 139, 165, 256, 512, 815, 1024, 1630, 2048, 2821, 3167, 3693, 3941, 4096, 4747, 5642, 6334, 7737, 7881, 8192, 9494, 10837, 11284, 12479, 13363, 16384, 18988, 22568, 24669, 24958, 27945, 31205, 32768, 38869, 40861, 45136, 48367, 49338, 49535, 55121
Offset: 1

Views

Author

Scott R. Shannon, Mar 01 2021

Keywords

Comments

All numbers of the form 2^k, k>=0, are in the sequence.

Examples

			8 is a term as bin(8)*reverse(bin(8)) = 100_2*1_2 = 100_2 contains '100' as a substring.
27 is a term as bin(27)*reverse(bin(27)) = 11011_2*11011_2 = 1011011001_2 contains '11011' as a substring.
108 is a term as bin(108)*reverse(bin(108)) = 1101100_2*11011_2 = 101101100100_2 contains '1101100' as a substring.
139 is a term as bin(139)*reverse(bin(139)) = 10001011_2*11010001_2 = 111000101111011_2 contains '10001011' as a substring.

Crossrefs

Cf. A342127 (base 10), A305989, A007088, A000079, A203565, A332795.

Programs

  • PARI
    strbin(x) = Str(fromdigits(binary(x), 10));
isok(m) = {my(p = m*fromdigits(Vecrev(binary(m)), 2)); #strsplit(strbin(p), strbin(m)) > 1;} \\ Michel Marcus, Mar 01 2021
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.