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 1-4 of 4 results.

A385410 Multiples k of b that are not perfect powers and whose trailing digits form a power of b, where 1 < b < k.

Original entry on oeis.org

12, 14, 15, 18, 21, 22, 24, 28, 33, 34, 35, 38, 39, 42, 44, 45, 48, 51, 52, 54, 55, 58, 62, 63, 65, 66, 68, 69, 72, 74, 75, 77, 78, 82, 84, 85, 88, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105, 108, 110, 111, 112, 114, 115, 116, 118, 120, 122, 123, 124, 126, 129
Offset: 1

Views

Author

Stefano Spezia and Michael De Vlieger, Jun 28 2025

Keywords

Examples

			In general, no prime p is a term since they are a power of base p.
Numbers having a single digit are not terms:
  1 is not a term since 1 is a power of all bases b;
  Composites 4, 6, and 9 are not in the sequence since 4 = 2^2, 6 = 2*3, and 9 = 3^2.
10 is not a term since it ends in a single zero, and zero is not a power of another number.
a(1) = 12 since it is not a perfect power, 2 | 12, and 12 mod 10 is a power of 2.
a(2) = 14 since it is not a perfect power, 2 | 14, and 14 mod 10 is a power of 2.
20 is not a term since it ends with a zero, and zero is not a power of another number.
26 is not a term since 6 does not divide 26.
1100 is a term since it is not a perfect power, 100 = 100^1, and 100 | 1100.

Links

Crossrefs

Subset of A002808, A007916, and A106543.
Cf. A384714, A385411, A385412.

Programs

  • Mathematica
    nn = 130; t = Union@ Flatten@ Table[m = 10^IntegerLength[b] + b; If[m > nn, Nothing, s = b^Range[0, Floor@ Log[b, nn]]; Flatten@ Reap[Map[(w = IntegerDigits[#]; i = 0; While[Set[k, FromDigits@ Join[IntegerDigits[i], w]] <= nn, If[And[Divisible[k, b], FreeQ[s, k]], Sow[k]]; i++]) &, s] ][[-1]]], {b, 2, nn}]; Select[t, GCD @@ FactorInteger[#][[;; , -1]] === 1 &]

A385411 Numbers k that are not perfect powers, not divisible by some b, and whose trailing digits form a power of b, where 1 < b < k.

Original entry on oeis.org

11, 13, 14, 17, 18, 19, 21, 23, 26, 28, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 54, 56, 57, 58, 59, 61, 67, 68, 69, 71, 73, 74, 76, 78, 79, 83, 86, 87, 89, 91, 94, 97, 98, 101, 103, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124
Offset: 1

Views

Author

Stefano Spezia and Michael De Vlieger, Jun 29 2025

Keywords

Examples

			Numbers having a single digit are not terms:
  k=1..2 are not terms since b < k could not be a valid base;
  Numbers k=3..9 are not in the sequence since there is not b < k with the same digit of k.
10 is not a term since it ends in a single zero, and zero is not a power of another number.
a(1) = 11 since it is a prime and 11 mod 10 = 1 = b^0 for all bases b in [10] \ 1.
12 is not a term since for all 1 < b < 12 either b | 12 or 12 mod 10 = 2 <> b^e, with e > 0.
a(2) = 13 since it is a prime and 13 mod 10 is a power of 3.
All primes p greater than 7 are terms since they are not perfect powers and are not divisible by 1 < b < p.
20 is not a term because it ends with a zero, and zero is not a power of another number.
26 is a term since it is not a perfect power, 6 does not divide 26, and 26 mod 10 = 6^1.
116 is a term since it is not a perfect power (116 = 2^2*29), 16 does not divide 116, and 116 mod 100 = 16^1.

Links

Crossrefs

Subset of A007916.
Cf. A384714, A385410, A385412.

Programs

  • Mathematica
    nn = 125; t = Union@ Flatten@ Table[m = 10^IntegerLength[b] + b; If[m > nn, Nothing, s = b^Range[0, Floor@ Log[b, nn]]; Flatten@ Reap[Map[(w = IntegerDigits[#]; i = 0; While[Set[k, FromDigits@ Join[IntegerDigits[i], w]] <= nn, If[And[!Divisible[k, b], FreeQ[s, k]], Sow[k]]; i++]) &, s] ][[-1]]], {b, 2, nn}]; Select[t, GCD @@ FactorInteger[#][[;; , -1]] === 1 &]

A385412 Numbers k that are not perfect powers and whose trailing digits form a power of b, where 1 < b < k.

Original entry on oeis.org

11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92
Offset: 1

Views

Author

Stefano Spezia and Michael De Vlieger, Jun 30 2025

Keywords

Examples

			Numbers having a single digit are not terms:
  1 is not a term since 1 is a power of all bases b;
  Numbers k=2..9 are not in the sequence since k^e mod 10 <> b, with e > 1.
10 is not a term since it ends in a single zero, and zero is not a power of another number.
a(1) = 11 since it is a prime and 11 mod 10 is a power of all bases b.
a(2) = 12 since it is not a perfect power, and 12 mod 10 is a power of 2.
a(3) = 13 since it is a prime and 13 mod 10 is a power of 3.
All primes p greater than 7 are terms since they are not perfect powers.
20 is not a term because it ends with a zero, and zero is not a power of another number.
26 is a term since it is not a perfect power, and 26 mod 10 = 6^1.

Links

Crossrefs

Union of A385410 and A385411.
Subset of A007916.
Cf. A384714.

Programs

  • Mathematica
    nn = 130; t = Union@ Flatten@ Table[m = 10^IntegerLength[b] + b; If[m > nn, Nothing, s = b^Range[0, Floor@ Log[b, nn]]; Flatten@ Reap[Map[(w = IntegerDigits[#]; i = 0; While[Set[k, FromDigits@ Join[IntegerDigits[i], w]] <= nn, If[FreeQ[s, k], Sow[k]]; i++]) &, s] ][[-1]]], {b, 2, nn}]; Select[t, GCD @@ FactorInteger[#][[;; , -1]] === 1 &]

A385289 Numbers whose trailing digits form a power of 2.

Original entry on oeis.org

1, 2, 4, 8, 11, 12, 14, 16, 18, 21, 22, 24, 28, 31, 32, 34, 38, 41, 42, 44, 48, 51, 52, 54, 58, 61, 62, 64, 68, 71, 72, 74, 78, 81, 82, 84, 88, 91, 92, 94, 98, 101, 102, 104, 108, 111, 112, 114, 116, 118, 121, 122, 124, 128, 131, 132, 134, 138, 141, 142, 144, 148
Offset: 1

Views

Author

Stefano Spezia, Jun 24 2025

Keywords

Links

Crossrefs

Union of A000079 and A384714.
Cf. A209229.

Programs

  • Mathematica
    Select[Range[150],Sum[Boole[IntegerQ[Log2[FromDigits[Drop[IntegerDigits[#],i]]]]],{i,0,IntegerLength[#]}]>0 &]
  • PARI
    isp2(k) = k==1<A209229
isok(k) = if (isp2(k), return(1)); for (i=1, oo, my(z=k % 10^i); if (z==k, return(0), if (z && isp2(z), return(1)))); return(0); \\ Michel Marcus, Jun 24 2025
Showing 1-4 of 4 results.

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