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.

Previous Showing 11-16 of 16 results.

A079088 Number of divisors of (prime(n)+1)*(prime(n+1)+1)/4.

Original entry on oeis.org

2, 4, 6, 8, 8, 6, 12, 16, 18, 20, 10, 8, 16, 20, 20, 20, 16, 8, 24, 18, 16, 40, 32, 18, 12, 24, 32, 32, 16, 28, 32, 24, 32, 32, 36, 12, 8, 32, 36, 32, 48, 48, 24, 12, 54, 24, 24, 48, 32, 24, 64, 48, 36, 32, 36, 60, 64, 16, 8, 16, 24, 32, 64, 24, 8, 16, 12, 24, 48, 24, 48, 72, 32, 32
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 22 2002

Keywords

Links

Crossrefs

Cf. A000005, A079079, A079085, A079086.

Programs

  • Haskell
    a079088 = a000005 . a079079  -- Reinhard Zumkeller, Oct 08 2012
  • Mathematica
    a[n_] := DivisorSigma[0, (Prime[n]+1) * (Prime[n+1]+1) / 4]; Array[a, 100] (* Amiram Eldar, Apr 06 2025 *)
  • PARI
    a(n) = numdiv((prime(n)+1)*(prime(n+1)+1)/4); \\ Amiram Eldar, Apr 06 2025

Formula

a(n) = A000005(A079079(n)).

A079095 Squarefree numbers of the form (prime(k)+1)*(prime(k+1)+1)/4.

Original entry on oeis.org

3, 6, 42, 399, 462, 930, 1054, 3135, 4830, 6478, 13110, 19599, 20022, 24963, 26394, 35530, 38805, 39999, 41205, 44310, 52899, 71002, 74254, 81510, 94863, 95790, 103362, 109230, 111547, 114243, 135790, 144399, 146685, 157206, 166866, 183183
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 22 2002

Keywords

Examples

			(A000040(12)+1)*(A000040(12+1)+1)/4 = (37+1)*(41+1)/4 = 399 = 3*7*19, therefore 399 is a term.

Links

Crossrefs

Intersection of A005117 and A079079.
Cf. A000040, A008966, A079093.

Programs

  • Haskell
    a079095 n = a079095_list !! (n-1)
a079095_list = filter ((== 1) . a008966) a079079_list
-- Reinhard Zumkeller, Oct 09 2012
  • Mathematica
    With[{p = Prime[Range[150]]}, Select[(Most[p]+1) * (Rest[p]+1) / 4, SquareFreeQ]] (* Amiram Eldar, Apr 06 2025 *)

Formula

A008966(a(n)) = 1.

A079083 Smallest odd prime factor of (prime(n)+1)*(prime(n+1)+1)/4.

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 19, 3, 3, 3, 3, 3, 3, 17, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 19, 41, 3, 3, 3, 3, 3, 3, 3, 3, 5, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 17, 3, 3, 3, 3, 3, 3, 3, 3, 13, 3, 3, 3, 3, 3, 11, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 131, 137
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 22 2002

Keywords

Links

Crossrefs

Cf. A078701, A079079, A079084.

Programs

  • Haskell
    a079083 = a078701 . a079079  -- Reinhard Zumkeller, Oct 08 2012
  • Mathematica
    odd[n_] := n / 2^IntegerExponent[n, 2]; a[n_] := FactorInteger[odd[(Prime[n]+1)*(Prime[n+1]+1)]][[1, 1]]; Array[a, 100] (* Amiram Eldar, Apr 06 2025 *)
  • PARI
    odd(n) = n >> valuation(n, 2);
a(n) = factor(odd((prime(n)+1)*(prime(n+1)+1)))[1, 1]; \\ Amiram Eldar, Apr 06 2025

Formula

a(n) = A078701(A079079(n)).

A079087 Maximum exponent in prime factorization of (prime(n)+1)*(prime(n+1)+1)/4.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 2, 3, 2, 4, 4, 1, 1, 4, 4, 4, 1, 1, 3, 2, 3, 4, 3, 2, 2, 2, 3, 3, 1, 6, 7, 2, 1, 3, 2, 2, 1, 3, 2, 3, 2, 5, 5, 2, 2, 3, 5, 5, 1, 2, 3, 3, 2, 3, 2, 4, 3, 3, 1, 1, 2, 3, 3, 2, 1, 1, 2, 2, 2, 2, 3, 5, 3, 1, 7, 6, 1, 1, 1, 2, 1, 3, 3, 2, 3, 3, 2, 1, 3, 4, 5, 6, 3, 3, 3, 3, 3, 2, 1, 1, 2, 3, 2, 1, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 22 2002

Keywords

Links

Crossrefs

Cf. A051903, A079079, A079085.

Programs

  • Haskell
    a079087 = a051903 . a079079  -- Reinhard Zumkeller, Oct 08 2012
  • Mathematica
    With[{pms=Times@@#/4&/@Partition[Prime[Range[110]]+1,2,1]}, Max[ Transpose[ FactorInteger[#]][[2]]]&/@pms] (* Harvey P. Dale, Nov 10 2011 *)
  • PARI
    a(n) = vecmax(factor((prime(n)+1)*(prime(n+1)+1)/4)[,2]); \\ Amiram Eldar, Sep 08 2024

Formula

a(n) = A051903(A079079(n)).

A079092 Euler's totient of (prime(n)+1)*(prime(n+1)+1)/4.

Original entry on oeis.org

2, 2, 4, 8, 12, 36, 24, 32, 48, 64, 144, 216, 120, 160, 216, 216, 240, 480, 384, 432, 576, 384, 432, 1008, 1344, 768, 864, 720, 1440, 1152, 1280, 1320, 1056, 1200, 1440, 2808, 3120, 1920, 2016, 2016, 1728, 2304, 3072, 5760, 2400, 4160, 4992, 3456, 3168, 6336
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 22 2002

Keywords

Links

Crossrefs

Cf. A000010, A079079.

Programs

  • Haskell
    a079092 = a000010 . a079079  -- Reinhard Zumkeller, Oct 09 2012
  • Mathematica
    a[n_] := EulerPhi[(Prime[n]+1) * (Prime[n+1]+1) / 4]; Array[a, 100] (* Amiram Eldar, Apr 06 2025 *)
  • PARI
    a(n) = eulerphi((prime(n)+1)*(prime(n+1)+1)/4); \\ Amiram Eldar, Apr 06 2025

Formula

a(n) = A000010(A079079(n)).

A079094 Arithmetic derivative of (prime(n)+1)*(prime(n+1)+1)/4.

Original entry on oeis.org

1, 5, 16, 44, 41, 51, 123, 244, 336, 608, 624, 211, 493, 1280, 1836, 1647, 991, 623, 2724, 2256, 2556, 4496, 3483, 2541, 1694, 3896, 7236, 5319, 2122, 12352, 16576, 5925, 5891, 8275, 10180, 6396, 3479, 13780, 13476, 13581, 12993, 26672, 26480
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 22 2002

Keywords

Links

Crossrefs

Cf. A003415, A079079, A079093.

Programs

  • Haskell
    a079094 = a003415 . a079079  -- Reinhard Zumkeller, Oct 09 2012
  • Mathematica
    ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := ad[(Prime[n]+1) * (Prime[n+1]+1) / 4]; Array[a, 100] (* Amiram Eldar, Apr 06 2025 *)

Formula

a(n) = A003415(A079079(n)).
Previous Showing 11-16 of 16 results.

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