A034706 -id:A034706 - cp's OEIS frontend

A050941 Numbers that are not the sum of consecutive triangular numbers.

2, 5, 7, 8, 11, 12, 13, 14, 17, 18, 22, 23, 24, 26, 27, 29, 30, 32, 33, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 51, 53, 54, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 79, 82, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 97

Offset: 1

N. J. A. Sloane, Jan 02 2000

nonn

Numbers that are not the difference of two tetrahedral numbers. - Franklin T. Adams-Watters, Dec 16 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A034706.

import Data.List.Ordered (minus)
a050941 n = a050941_list !! (n-1)
a050941_list = minus [0..] a034706_list
-- Reinhard Zumkeller, May 12 2015

lim = 20; Take[#, Floor[Length[#]/lim]] &@ Complement[Range@ Max@ #, #] &@ Union[Subtract @@@ Map[Sort[#, Greater] &, Permutations[Table[Binomial[n + 2, 3], {n, 0, lim}], {2}]]] (* Michael De Vlieger, Dec 17 2015, in part after Zerinvary Lajos at A000292 *)

A269414 Prime numbers that are the sum of one or more consecutive triangular numbers.

Original entry on oeis.org

3, 19, 31, 83, 109, 199, 251, 409, 571, 631, 683, 829, 1091, 1489, 1999, 2341, 2531, 2971, 3529, 4621, 4789, 5051, 7039, 7211, 7669, 8779, 9721, 10459, 10711, 11171, 13681, 14851, 15131, 16069, 16381, 16883, 17659, 18731, 20011, 20359, 21683, 23251, 24851

Offset: 1

Vaclav Kotesovec, Feb 25 2016

nonn

Vaclav Kotesovec and Chai Wah Wu, Table of n, a(n) for n = 1..5000 (terms 1..100 from Vaclav Kotesovec)

Cf. A257974, A000217, A125602, A034706.

A309783 Numbers that are sums of one or more consecutive positive triangular numbers in more than one way.

Original entry on oeis.org

10, 36, 55, 64, 100, 120, 136, 164, 210, 276, 361, 435, 460, 514, 560, 596, 676, 760, 1176, 1225, 1320, 1326, 1460, 1484, 1485, 1505, 1540, 1684, 1736, 1770, 1891, 1936, 2014, 2080, 2145, 2180, 2314, 2485, 2596, 2890, 3156, 3244, 3275, 3364, 3486, 3570, 3710, 3916

Offset: 1

Ilya Gutkovskiy, Aug 17 2019

nonn

The first number that is the sum in three ways is 2180. The first that is the sum in four ways is 10053736. - Robert Israel, Aug 20 2019

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (first 2759 terms from Robert Israel)

Cf. A000217, A034706, A130052, A307666.

N:= 10000: # for terms <= N
V:= Vector(N):
for i from 1 while i*(i+1)/2 <= N do
  s:= i*(i+1)*(i+2)/6;
  for j from i-1 to 0 by -1 do
    t:= j*(j+1)*(j+2)/6;
    if s-t > N then break fi;
    V[s-t]:= V[s-t]+1
  od;
od:
select(t -> V[t]>1, [$1..N]); # Robert Israel, Aug 20 2019

A307666(a(n)) > 1.

A385656 Numbers k such that the sum of the decimal digits of k^2 divides k^2.

Original entry on oeis.org

1, 2, 3, 6, 9, 10, 12, 15, 18, 20, 21, 24, 30, 36, 39, 42, 45, 48, 49, 51, 52, 54, 60, 63, 65, 66, 68, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 111, 112, 117, 120, 126, 132, 138, 140, 144, 148, 150, 156, 162, 168, 174, 180, 182, 190, 198, 200, 201, 204, 207

Offset: 1

Vighnesh Patil, Jul 06 2025

			15 is a term 15^2 = 225; digit sum of 225 = 2 + 2 + 5 = 9; 225 mod 9 = 0, so 15 is included.
18 is a term 18^2 = 324; digit sum of 324 = 3 + 2 + 4 = 9; 324 mod 9 = 0, so 16 is included.

Vighnesh Patil, Table of n, a(n) for n = 1..1000

Cf. A005349, A007953, A034706, A118547, A385640.

digitSum := n -> add(i,i=convert(n, base, 10)):
isok := n -> modp(n^2, digitSum(n^2)) = 0:
select(isok, [$1..400])[];

DigitSum[n_] := Total[IntegerDigits[n]];
Select[Range[400], Mod[#^2, DigitSum[#^2]] == 0 &]

isok(k) = (k^2 % sumdigits(k^2)) == 0; \\ Michel Marcus, Jul 06 2025

def digit_sum(n): return sum(int(d) for d in str(n))
def ok(n): return (n**2) % digit_sum(n**2) == 0
print([n for n in range(1, 1000) if ok(n)])

a(n) = sqrt(A118547(n)). - Michel Marcus, Jul 07 2025

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A050941 Numbers that are not the sum of consecutive triangular numbers.

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A269414 Prime numbers that are the sum of one or more consecutive triangular numbers.

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A309783 Numbers that are sums of one or more consecutive positive triangular numbers in more than one way.

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A385656 Numbers k such that the sum of the decimal digits of k^2 divides k^2.

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