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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A138796 Least possible k > 0 with T(k) - T(j) = n, j > 0, where T(i) > 0 are the triangular numbers A000217.

Original entry on oeis.org

2, 3, 4, 3, 6, 4, 8, 4, 10, 6, 5, 7, 5, 6, 16, 9, 6, 10, 6, 8, 7, 12, 9, 7, 8, 7, 28, 15, 8, 16, 32, 8, 10, 8, 13, 19, 11, 9, 10, 21, 9, 22, 9, 10, 13, 24, 17, 10, 12, 11, 10, 27, 10, 13, 11, 12, 16, 30, 11, 31, 17, 11, 64, 11, 18, 34, 12, 14, 13, 36, 12, 37, 20, 12, 13, 12, 21, 40, 18
Offset: 2

Views

Author

Peter Pein (petsie(AT)dordos.net), Mar 30 2008

Keywords

Comments

For T(k) see A138797, for j see A138798 and for T(j) see A138799.
The number of ways n can be written as difference of two triangular numbers is sequence A136107
Note that n = t(k)-t(j) implies 2n = (k-j)(k+j+1), where (k-j) and (k+j+1) are of opposite parity. Let d be the odd element of { k-j, k+j+1 }. Then d is an odd divisor of n and k = ( d + 2n/d - 1 ) / 2. Therefore a(n) = ( min{ d + 2n/d } - 1 ) / 2 where d runs through all odd divisors of n, except perhaps (sqrt(8*n+1) +- 1)/2 which correspond to j=0. See PARI program. The restriction that j > 0 seems artificial. If it is removed we get A212652. - Max Alekseyev, Mar 31 2008

Examples

			a(30)=8, because 30 = T(30) - T(29) = T(11) - T(8) = T(9) - T(5) = T(8) - T(3) and 8 is the least index of the minuends.

Links

Crossrefs

Cf. A000217, A109814, A118235, A136107, A138797, A138798, A138799, A212652.

Programs

  • Mathematica
    T=#(#+1)/2&;Min[k/.{ToRules[Reduce[{T[k]-T[j]\[Equal]#,0
  • PARI
    { a(n) = local(m); m=2*n+1; fordiv(n/2^valuation(n,2),d,if((2*d+1)^2!=8*n+1&&(2*d-1)^2!=8*n+1,m=min(m,d+(2*n)\d))); (m-1)\2 }
vector(100,n,a(n)) \\ Max Alekseyev, Mar 31 2008

A138797 Least possible T(k) with T(k)-T(j)=n, where T(i)>0 are the triangular numbers A000217.

Original entry on oeis.org

3, 6, 10, 6, 21, 10, 36, 10, 55, 21, 15, 28, 15, 21, 136, 45, 21, 55, 21, 36, 28, 78, 45, 28, 36, 28, 406, 120, 36, 136, 528, 36, 55, 36, 91, 190, 66, 45, 55, 231, 45, 253, 45, 55, 91, 300, 153, 55, 78, 66, 55, 378, 55, 91, 66, 78, 136, 465, 66, 496, 153, 66, 2080, 66, 171
Offset: 2

Views

Author

Peter Pein (petsie(AT)dordos.net), Mar 30 2008

Keywords

Comments

For k see A138796, for j see A138798 and for T(j) see A138799.
The number of ways n can be written as difference of two triangular numbers is sequence A136107

Examples

			a(4)=10 because T(A138796(4))=10.

Crossrefs

Cf. A000217, A109814, A118235, A136107, A138796, A138798, A138799.

Programs

  • Mathematica
    T=#(#+1)/2&;T[Min[k/.{ToRules[Reduce[{T[k]-T[j]\[Equal]#,0

A138798 Values of j corresponding to least possible k>0 with T(k)-T(j)=n, where T(i)>0 are the triangular numbers A000217.

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 1, 9, 4, 2, 5, 1, 3, 15, 7, 2, 8, 1, 5, 3, 10, 6, 2, 4, 1, 27, 13, 3, 14, 31, 2, 6, 1, 10, 17, 7, 3, 5, 19, 2, 20, 1, 4, 9, 22, 14, 3, 7, 5, 2, 25, 1, 8, 4, 6, 12, 28, 3, 29, 13, 2, 63, 1, 14, 32, 4, 8, 6, 34, 3, 35, 16, 2, 5, 1, 17, 38, 13, 4, 18, 40, 6, 3, 19, 11, 2, 43, 1
Offset: 2

Views

Author

Peter Pein (petsie(AT)dordos.net), Mar 30 2008

Keywords

Comments

For k see A138796, for T(k) see A138797 and for T(j) see A138799.
The number of ways n can be written as difference of two triangular numbers is sequence A136107

Examples

			a(30)=3 because 30 = T(30)-T(29)=T(11)-T(8)=T(9)-T(5)=T(8)-T(3) and 3 is the least index of the subtrahends.

Crossrefs

Cf. A000217, A109814, A118235, A136107, A138796, A138797, A138799.

Programs

  • Mathematica
    T=#(#+1)/2&;Sort[{k,j}/.{ToRules[Reduce[{T[k]-T[j]\[Equal]#,0

A138799 Values of T(j) corresponding to least possible T(k) with T(k)-T(j)=n, where T(i)>0 are the triangular numbers A000217.

Original entry on oeis.org

1, 3, 6, 1, 15, 3, 28, 1, 45, 10, 3, 15, 1, 6, 120, 28, 3, 36, 1, 15, 6, 55, 21, 3, 10, 1, 378, 91, 6, 105, 496, 3, 21, 1, 55, 153, 28, 6, 15, 190, 3, 210, 1, 10, 45, 253, 105, 6, 28, 15, 3, 325, 1, 36, 10, 21, 78, 406, 6, 435, 91, 3, 2016, 1, 105, 528, 10, 36, 21, 595, 6, 630
Offset: 2

Views

Author

Peter Pein (petsie(AT)dordos.net), Mar 30 2008

Keywords

Comments

For k see A138796, for T(k) see A138797 and for j see A138798.
The number of ways n can be written as difference of two triangular numbers is sequence A136107

Examples

			a(30)=6 because 30 = T(30)-T(29)=T(11)-T(8)=T(9)-T(5)=T(8)-T(3) and T(3)=6 is the least minuend.

Crossrefs

Cf. A000217, A109814, A118235, A136107, A138796, A138797, A138798.

Programs

  • Mathematica
    T=#(#+1)/2&;T[Sort[{k,j}/.{ToRules[Reduce[{T[k]-T[j]\[Equal]#,0

A111787 a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 0, 5, 3, 4, 0, 3, 5, 4, 3, 7, 0, 3, 0, 0, 3, 4, 5, 3, 0, 4, 3, 5, 0, 3, 0, 8, 3, 4, 0, 3, 7, 5, 3, 8, 0, 3, 5, 7, 3, 4, 0, 3, 0, 4, 3, 0, 5, 3, 0, 8, 3, 5, 0, 3, 0, 4, 3, 8, 7, 3, 0, 5, 3, 4, 0, 3, 5, 4, 3, 11, 0, 3, 7, 8, 3, 4, 5, 3, 0, 7, 3, 5, 0, 3, 0, 13
Offset: 1

Views

Author

Jaap Spies, Aug 16 2005

Keywords

Comments

a(n)=0 if n is an odd prime or a power of 2. For numbers of the third kind we proceed as follows: suppose n is to be written as sum of k consecutive integers starting with m, then 2n = k(2m + k - 1). Let p be the smallest odd prime divisor of n then a(n) = min(p,2n/p).

Examples

			a(15)=3 because 15=4+5+6 (k=3) and 15=2+3+4+5 (k=4)

References

  • Nieuw Archief voor Wiskunde 5/6 nr. 2 Problems/UWC Problem C part 3, Jun 2005, pp. 181-182

Links

Crossrefs

Cf. A111774, A111775, A109814.

Programs

  • Maple
    ispoweroftwo := proc(n) local a, t; t := 1; while (n > t) do t := 2*t end do; if (n = t) then a := true else a:= false end if; return a;end proc; A111787:= proc(n) local d, k; k:=0; if isprime(n) or ispoweroftwo(n) then return(0); fi; for d from 3 by 2 to n do if n mod d = 0 then k:=min(d,2*n/d); break; fi; od; return(k); end proc; seq(A111787(i),i=1..150);

A218621 a(n) = unique divisor d of n such that d + (n/d - 1)/2 is minimal and integral.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 4, 1, 2, 3, 16, 1, 2, 1, 4, 3, 2, 1, 8, 5, 2, 3, 4, 1, 6, 1, 32, 3, 2, 5, 4, 1, 2, 3, 8, 1, 6, 1, 4, 5, 2, 1, 16, 7, 10, 3, 4, 1, 6, 5, 8, 3, 2, 1, 4, 1, 2, 7, 64, 5, 6, 1, 4, 3, 10, 1, 8, 1, 2, 5, 4, 7, 6, 1, 16, 9, 2, 1, 4, 5
Offset: 1

Views

Author

L. Edson Jeffery, Feb 18 2013

Keywords

Comments

Differs from A079891 starting at a(18).
For integers M, k, with 0<=k<=M, consider a representation of n as n = T(M) - T(M-k) = M + (M-1) + ... + (M-k+1), in which k is maximal, where T(r) = r*(r+1)/2 is the r-th triangular number. Then k = A109814(n), and M = A212652(n) = a(n) + (n/a(n) - 1)/2 is minimal.
Conjecture. For n, p, v, j natural numbers, the conditions on a(n) seem to be the following:
1. If n is an odd prime, then a(n) = 1.
2. If n is odd and composite, then
a(n) = max(p : p | n, p <= sqrt(n), p is a prime).
3. If n is equal to a power of 2, then a(n) = n.
4. If n = 2^j*v, with v odd, v>1 and j>1, then a(n) = 2^j.
5. If n = 2*v, with v odd and composite, then
a(n) = 2*p, where p is the least prime such that p | v.
6. If n = 2*p, for p an odd prime, then a(n) = 2.

Links

Crossrefs

Cf. A000217, A109814, A118235, A141419, A209260, A212652.

Programs

  • Mathematica
    Table[d = Divisors[n]; mn = Infinity; best = 0; Do[q = i + (n/i - 1)/2; If[IntegerQ[q] && q < mn, mn = q; best = i], {i, d}]; best, {n, 100}] (* T. D. Noe, Feb 21 2013 *)
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