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 1-3 of 3 results.

A053614 Numbers that are not the sum of distinct triangular numbers.

Original entry on oeis.org

2, 5, 8, 12, 23, 33
Offset: 1

Views

Author

Jud McCranie, Mar 19 2000

Keywords

Comments

The Mathematica program first computes A024940, the number of partitions of n into distinct triangular numbers. Then it finds those n having zero such partitions. It appears that A024940 grows exponentially, which would preclude additional terms in this sequence. - T. D. Noe, Jul 24 2006, Jan 05 2009

Examples

			a(2) = 5: the 7 partitions of 5 are 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1. Among those the distinct ones are 5, 4+1, 3+2. None contains all distinct triangular numbers.
12 is a term as it is not a sum of 1, 3, 6 or 10 taken at most once.

References

  • Joe Roberts, Lure of the Integers, The Mathematical Association of America, 1992, page 184, entry 33.
  • David Wells in "The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, page 94, states that "33 is the largest number that is not the sum of distinct triangular numbers".

Links

  • Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125.

Crossrefs

Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
Cf. A001422, A121405 (corresponding sequences for square and pentagonal numbers)
Cf. A000217, A002243, A002244, A014134, A014156, A014158, A020757, A050941, A050942, A051611, A007294, A051533, A060773.

Programs

  • Mathematica
    nn=100; t=Rest[CoefficientList[Series[Product[(1+x^(k*(k+1)/2)), {k,nn}], {x,0,nn(nn+1)/2}], x]]; Flatten[Position[t,0]] (* T. D. Noe, Jul 24 2006 *)

Formula

Complement of A061208.

Extensions

Entry revised by N. J. A. Sloane, Jul 23 2006

A034706 Numbers which are sums of consecutive triangular numbers.

Original entry on oeis.org

0, 1, 3, 4, 6, 9, 10, 15, 16, 19, 20, 21, 25, 28, 31, 34, 35, 36, 45, 46, 49, 52, 55, 56, 64, 66, 74, 78, 80, 81, 83, 84, 85, 91, 100, 105, 109, 110, 116, 119, 120, 121, 130, 136, 144, 145, 153, 155, 161, 164, 165, 166, 169, 171, 185, 190, 196, 199, 200, 202, 210
Offset: 1

Views

Author

Erich Friedman

Keywords

Links

Crossrefs

Complement gives A050941.
Cf. A000217 (1 consec), A001110 (2 consec), A129803 (3 consec), A131557 (5 consec), A257711 (7 consec), A034705, A269414 (subsequence of primes).

Programs

  • Haskell
    -- import Data.Set (deleteFindMin, union, fromList); import Data.List (inits)
a034706 n = a034706_list !! (n-1)
a034706_list = f 0 (tail $ inits $ a000217_list) (fromList [0]) where
   f x vss'@(vs:vss) s
     | y < x = y : f x vss' s'
     | otherwise = f w vss (union s $ fromList $ scanl1 (+) ws)
     where ws@(w:_) = reverse vs
           (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 12 2015
  • Maple
    isA034706 := proc(n)
    local a,b;
    for a from 0 do
        if a*(a+1)/2 > n then
            return false;
        end if;
        for b from a do
            tab := (1+b-a)*(a^2+b*a+a+b^2+2*b)/6 ;
            if tab = n then
                return true;
            elif tab > n then
                break;
            end if;
        end do:
    end do:
end proc:
for n from 0 to 100 do
    if isA034706(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 14 2015
  • Mathematica
    M = 1000; (* to get all terms <= M *)
nmax = (Sqrt[8 M + 1] - 1)/2 // Ceiling;
Table[Sum[n(n+1)/2, {n, j, k}], {j, 0, nmax}, {k, j, nmax}] // Flatten // Union // Select[#, # <= M&]& (* Jean-François Alcover, Mar 10 2019 *)

A257974 Prime numbers that are not the sum of one or more consecutive triangular numbers.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 107, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 211, 223, 227, 229, 233, 239, 241, 257, 263, 269, 271, 277, 281, 283
Offset: 1

Views

Author

Vicente Izquierdo Gomez, Nov 05 2015

Keywords

Comments

Subsequence of primes of A050941. - Michel Marcus, Dec 14 2015
Prime numbers that are not the difference of two tetrahedral numbers (A000292). - Franklin T. Adams-Watters, Dec 16 2015

Examples

			From _Michael De Vlieger_, Nov 06 2015: (Start)
3 is a triangular number thus is not a term.
The triangular numbers <= 7 are {1, 3, 6}. None of these are 7. 7 is not found among the sums of adjacent pairs of terms, i.e., {{1, 3}, {3, 6}} = {4, 9}. The sum of all numbers {1, 3, 6} = 10. Thus 7 is a term.
The triangular numbers <= 19 are {1, 3, 6, 10, 15}. 19 is not a triangular number. 19 is not found among sums of pairs of adjacent terms {4, 9, 16, 25} nor among those of quartets of adjacent terms {20, 34}, but is found among sums of triples of adjacent terms {10, 19, 31}. Thus 19 is not a term. (End)

Links

Crossrefs

Cf. A050941, A000217, A000292, A125602, A269414.

Programs

  • Maple
    isA257974 := proc(n)
    if isprime(n) then
        return not isA034706(n) ;
    else
        false ;
    end if;
end proc:
for n from 0 to 400 do
    if isA257974(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 14 2015
  • Mathematica
    t = Array[Binomial[# + 1, 2] &, {10^4}]; fQ[n_] := Block[{s}, s = TakeWhile[t, # <= n &]; AnyTrue[Flatten[Total /@ Partition[s, #, 1] & /@ Range[Length@ s - 1]], # == n &]]; Select[Prime@ Range@ 120, ! fQ@ # &] (* Michael De Vlieger, Nov 06 2015, Version 10 *)

Extensions

More terms from Michael De Vlieger, Nov 06 2015
Showing 1-3 of 3 results.

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