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 1-8 of 8 results.

A141164 Numbers having exactly 1 divisor of the form 8*k + 7.

Original entry on oeis.org

7, 14, 15, 21, 23, 28, 30, 31, 35, 39, 42, 45, 46, 47, 49, 55, 56, 60, 62, 69, 70, 71, 75, 77, 78, 79, 84, 87, 90, 91, 92, 93, 94, 95, 98, 103, 110, 111, 112, 115, 117, 120, 124, 127, 133, 138, 140, 141, 142, 143, 147, 150, 151, 154, 155, 156, 158, 159, 167, 168, 174, 180, 182, 183, 184, 186, 188, 190, 191, 196, 199
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 26 2011

Keywords

Examples

			a(1) = A188226(1) = 7.

Links

Crossrefs

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), this sequence (m=1, i=7).
Indices of 1 in A188172.
A007522 is a subsequence.
Cf. A004771.

Programs

  • Haskell
    import Data.List (elemIndices)
a141164 n = a141164_list !! (n-1)
a141164_list = map succ $ elemIndices 1 $ map a188172 [1..]
  • Mathematica
    okQ[n_] := Length[Select[Divisors[n] - 7, Mod[#, 8] == 0 &]] == 1; Select[Range[200], okQ]
  • PARI
    res(n, a, b) = sumdiv(n, d, (d%a) == b)
isA141164(n) = (res(n, 8, 7) == 1) \\ Jianing Song, Apr 06 2021

Formula

A188172(a(n)) = 1.

A343108 Numbers having no divisor of the form 8*k + 3.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 20, 23, 25, 26, 28, 29, 31, 32, 34, 37, 40, 41, 46, 47, 49, 50, 52, 53, 56, 58, 61, 62, 64, 65, 68, 71, 73, 74, 79, 80, 82, 85, 89, 92, 94, 97, 98, 100, 101, 103, 104, 106, 109, 112, 113, 116, 119, 122, 124, 125, 127, 128
Offset: 1

Views

Author

Jianing Song, Apr 05 2021

Keywords

Comments

Numbers not divisible by at least one of 3, 11, 19, ...

Examples

			7 is a term since it has no divisor congruent to 3 modulo 8.

Links

Crossrefs

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), this sequence (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).
Indices of 0 in A188170.

Programs

  • PARI
    res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343108(n) = (res(n,8,3) == 0)

A343109 Numbers having no divisor of the form 8*k + 5.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 17, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 34, 36, 38, 41, 43, 44, 46, 47, 48, 49, 51, 54, 56, 57, 59, 62, 64, 66, 67, 68, 71, 72, 73, 76, 79, 81, 82, 83, 86, 88, 89, 92, 94, 96, 97, 98, 99, 102, 103, 107, 108, 112, 113
Offset: 1

Views

Author

Jianing Song, Apr 05 2021

Keywords

Comments

Numbers not divisible by at least one of 5, 13, 21, ...

Examples

			9 is a term since it has no divisor congruent to 5 modulo 8.

Links

Crossrefs

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), this sequence (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).
Indices of 0 in A188171.

Programs

  • PARI
    res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343109(n) = (res(n,8,5) == 0)

A343110 Numbers having no divisor of the form 8*k + 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 22, 24, 25, 26, 27, 29, 32, 33, 34, 36, 37, 38, 40, 41, 43, 44, 48, 50, 51, 52, 53, 54, 57, 58, 59, 61, 64, 65, 66, 67, 68, 72, 73, 74, 76, 80, 81, 82, 83, 85, 86, 88, 89, 96, 97, 99, 100, 101, 102
Offset: 1

Views

Author

Jianing Song, Apr 05 2021

Keywords

Comments

Numbers not divisible by at least one of 7, 15, 23, ...

Examples

			9 is a term since it has no divisor congruent to 7 modulo 8.

Links

Crossrefs

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), this sequence (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).
Indices of 0 in A188172.

Programs

  • PARI
    res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343110(n) = (res(n,8,7) == 0)

A343111 Numbers having exactly 2 divisors of the form 8*k + 1, that is, numbers with exactly 1 divisor of the form 8*k + 1 other than 1.

Original entry on oeis.org

9, 17, 18, 25, 27, 33, 34, 36, 41, 45, 49, 50, 51, 54, 57, 63, 65, 66, 68, 72, 73, 75, 82, 85, 89, 90, 97, 98, 100, 102, 105, 108, 113, 114, 117, 119, 121, 123, 125, 126, 129, 130, 132, 135, 136, 137, 144, 145, 146, 147, 150, 161, 164, 165, 169, 170, 175
Offset: 1

Views

Author

Jianing Song, Apr 05 2021

Keywords

Examples

			63 is a term since among the divisors of 63 (namely 1, 3, 7, 9, 21 and 63), the only divisors congruent to 1 modulo 8 are 1 and 9.

Links

Crossrefs

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), this sequence (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).
Indices of 2 in A188169.
A007519 is a subsequence.

Programs

  • PARI
    res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343111(n) = (res(n,8,1) == 2)

A343112 Numbers having exactly 1 divisor of the form 8*k + 3.

Original entry on oeis.org

3, 6, 9, 11, 12, 15, 18, 19, 21, 22, 24, 30, 35, 36, 38, 39, 42, 43, 44, 45, 48, 55, 59, 60, 63, 67, 69, 70, 72, 76, 77, 78, 83, 84, 86, 87, 88, 90, 91, 93, 95, 96, 107, 110, 111, 115, 117, 118, 120, 121, 126, 131, 133, 134, 138, 139, 140, 141, 143, 144
Offset: 1

Views

Author

Jianing Song, Apr 05 2021

Keywords

Examples

			63 is a term since among the divisors of 63 (namely 1, 3, 7, 9, 21 and 63), the only divisor congruent to 3 modulo 8 is 3.

Links

Crossrefs

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), this sequence (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).
Indices of 1 in A188170.
A007520 is a subsequence.

Programs

  • PARI
    res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343112(n) = (res(n,8,3) == 1)

A343113 Numbers having exactly 1 divisor of the form 8*k + 5.

Original entry on oeis.org

5, 10, 13, 15, 20, 21, 25, 26, 29, 30, 35, 37, 39, 40, 42, 50, 52, 53, 55, 58, 60, 61, 63, 69, 70, 74, 75, 77, 78, 80, 84, 87, 91, 93, 95, 100, 101, 104, 106, 109, 110, 111, 115, 116, 120, 122, 126, 133, 138, 140, 141, 143, 147, 148, 149, 150, 154, 155
Offset: 1

Views

Author

Jianing Song, Apr 05 2021

Keywords

Examples

			52 is a term since among the divisors of 52 (namely 1, 2, 4, 13, 26 and 52), the only divisor congruent to 5 modulo 8 is 13.

Links

Crossrefs

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), this sequence (m=1, i=5), A141164 (m=1, i=7).
Indices of 1 in A188171.
A007521 is a subsequence.

Programs

  • PARI
    res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343113(n) = (res(n,8,5) == 1)

A364542 Numbers k for which A005940(k) >= k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95
Offset: 1

Views

Author

Antti Karttunen, Jul 28 2023

Keywords

Comments

Sequence A005941(A364560(.)) sorted into ascending order.
A029747 is included as a subsequence, because it gives the known fixed points of map n -> A005940(n).
Differs from A343107 for the first time at a(22) = 25, which term is not present in A343107. On the other hand, 35 is the first term of A343107 that is not present in this sequence.

Crossrefs

Positions of nonnegative terms in A364499.
Complement of A364540.
Cf. A005940, A005941, A029747 (subsequence), A343107 (not a subsequence), A364560.

Programs

  • Mathematica
    nn = 95; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Select[Range[nn], a[#] >= # &] (* Michael De Vlieger, Jul 28 2023 *)
  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
isA364542(n) = (A005940(n)>=n);
Showing 1-8 of 8 results.

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

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