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.

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A173977 Integers k > 1 for which A020639(2*k-1) < A020639(2*k-3).

Original entry on oeis.org

5, 8, 11, 13, 14, 17, 20, 23, 25, 26, 28, 29, 32, 35, 38, 41, 43, 44, 46, 47, 50, 53, 56, 58, 59, 62, 65, 67, 68, 71, 73, 74, 77, 80, 83, 85, 86, 88, 89, 92, 95, 98, 101, 103, 104, 107, 110, 113, 116, 118, 119, 122, 125, 127, 128, 130, 131, 133, 134, 137, 140, 143, 146, 148
Offset: 1

Views

Author

Vladimir Shevelev, Mar 04 2010

Keywords

Comments

Every number m == 2 (mod 3), m > 2, is in the sequence (see A016789).

Crossrefs

Cf. A016789, A020639, A104275.

Programs

  • Maple
    A020639 := proc(n) if n = 1 then 1; else min(op(numtheory[factorset](n)) ) ; end if; end proc:
isA173977 := proc(n) A020639(2*n-1) < A020639(2*n-3) ; end proc:
for n from 2 to 400 do if isA173977(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Mar 25 2010
  • Mathematica
    lpf[n_] := lpf[n] = FactorInteger[n][[1, 1]]; Select[Range[2, 150], lpf[2*#-1] < lpf[2*#-3] &] (* Amiram Eldar, Oct 24 2024 *)

Extensions

More terms from R. J. Mathar, Mar 25 2010

A186041 Numbers of the form 3*k + 2, 5*k + 3, or 7*k + 4.

Original entry on oeis.org

2, 3, 4, 5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 62, 63, 65, 67, 68, 71, 73, 74, 77, 78, 80, 81, 83, 86, 88, 89, 92, 93, 95, 98, 101, 102, 103, 104, 107, 108, 109, 110, 113, 116, 118, 119, 122
Offset: 1

Views

Author

Klaus Brockhaus, Feb 11 2011, Mar 09 2011

Keywords

Comments

n is in the sequence iff n is in A016789 or in A016885 or in A017029.
First differences are periodic with period length 57. Least common multiple of 3, 5, 7 is 105; number of terms <= 105 is 57.
Sequence is not essentially the same as A053726: a(n) = A053726(n-3) for 3 < n < 33, a(34)=62, A053726(34-3)=61.
Sequence is not essentially the same as A104275: a(n) = A104275(n-2) for 3 < n < 33, a(34)=62, A104275(34-3)=61.

Links

Crossrefs

Cf. A016789, A016885, A017029, A053726, A104275.

Programs

  • Magma
    IsA186041:=func< n | exists{ k: k in [0..n div 3] | n in [3*k+2, 5*k+3, 7*k+4] } >; [ n: n in [1..200] | IsA186041(n) ];
  • Mathematica
    Take[With[{no=50},Union[Join[3Range[0,no]+2,5Range[0,no]+3,7Range[0,no]+4]]],70]  (* Harvey P. Dale, Feb 16 2011 *)

Formula

a(n) = a(n-57) + 105.
a(n) = a(n-1) + a(n-57) - a(n-58).
G.f.: x*(x^57 + x^56 + x^55 + x^54 + 3*x^53 + 3*x^52 + 2*x^51 + x^50 + 3*x^49 + x^48 + 2*x^47 + 3*x^46 + 2*x^45 + x^44 + 2*x^43 + x^42 + 3*x^41 + x^40 + 2*x^39 + 3*x^38 + x^37 + 2*x^36 + 2*x^35 + x^34 + 2*x^33 + x^32 + x^31 + 2*x^30 + 3*x^29 + 3*x^28 + 2*x^27 + x^26 + x^25 + 2*x^24 + x^23 + 2*x^22 + 2*x^21 + x^20 + 3*x^19 + 2*x^18 + x^17 + 3*x^16 + x^15 + 2*x^14 + x^13 + 2*x^12 + 3*x^11 + 2*x^10 + x^9 + 3*x^8 + x^7 + 2*x^6 + 3*x^5 + 3*x^4 + x^3 + x^2 + x + 2) / ((x - 1)^2*(x^2 + x + 1)*(x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^36 - x^35 + x^33 - x^32 + x^30 - x^29 + x^27 - x^26 + x^24 - x^23 + x^21 - x^20 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1)).

A224710 The number of unordered partitions {a,b} of 2n-1 such that a and b are composite.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 7, 8, 8, 8, 9, 9, 10, 11, 11, 12, 13, 13, 13, 14, 15, 15, 16, 16, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 28, 29, 30, 31, 32, 33, 33, 34, 34, 35, 36, 36
Offset: 1

Views

Author

J. Stauduhar, Apr 16 2013

Keywords

Comments

Except for the initial terms, the same sequence as A210469.

Examples

			n=7: 13 has a unique representation as the sum of two composite numbers, namely 13 = 4+9, so a(7)=1.

Links

Crossrefs

Subsequence of A224708. Cf. A210469.

Programs

  • Mathematica
    Table[Length@ Select[IntegerPartitions[2 n - 1, {2}] /. n_Integer /; ! CompositeQ@ n -> Nothing, Length@ # == 2 &], {n, 71}] (* Version 10.2, or *)
Table[If[n == 1, 0, n - 2 - PrimePi[2 n - 4]], {n, 71}] (* Michael De Vlieger, May 03 2016 *)

Formula

a(n) = n - 2 - primepi(2n-4) for n>1. - Anthony Browne, May 03 2016
a(A104275(n+2) + 1) = n. - Anthony Browne, May 25 2016

A283616 a(n) = Product_{k=2..floor(sqrt(2n-1)/2)+1} (2n-1) mod (2k-1).

Original entry on oeis.org

1, 1, 2, 1, 0, 2, 1, 0, 4, 4, 0, 6, 0, 0, 8, 1, 0, 0, 4, 0, 12, 3, 0, 20, 0, 0, 24, 0, 0, 24, 5, 0, 0, 32, 0, 16, 9, 0, 0, 56, 0, 72, 0, 0, 320, 0, 0, 0, 84, 0, 24, 240, 0, 512, 160, 0, 90, 0, 0, 0, 0, 0, 0, 12, 0, 500, 0, 0, 160, 672, 0, 0, 0, 0, 2880, 1792, 0, 0, 72, 0, 0, 378
Offset: 1

Views

Author

Zhandos Mambetaliyev, Mar 11 2017

Keywords

Comments

For n>1, if a(n) > 0 then 2n-1 is prime.
From Robert G. Wilson v, Mar 15 2017: (Start)
Except for n=1, a(n)=0 iff 2n-1 is not prime (A104275).
a(n) is prime for n: 3, 6, 22 & 31. (End)

Links

Crossrefs

Cf. A180491.

Programs

  • Mathematica
    Table[Product[Mod[(2 n - 1), (2 k - 1)], {k, 2, Floor[Sqrt[2 n - 1]/2] + 1}], {n, 80}] (* Michael De Vlieger, Mar 15 2017 *)
  • PARI
    a(n)=my(t=2*n-1); prod(k=2,sqrtint(t\4)+1, t%(2*k-1)) \\ Charles R Greathouse IV, Mar 22 2017
Previous Showing 11-14 of 14 results.

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