A014134 -id:A014134 - cp's OEIS frontend

A140867 Start of the first run of exactly n integers in A014134.

8, 33, 88, 132, 1265, 768, 2657, 11413, 8755, 50965, 8453, 57258, 124486, 609703, 791190, 955961, 826855, 984638, 2095061, 4196516, 5776666, 6481631, 22355042, 42972320, 11873562, 70956435, 47737822, 57344057, 57651771, 191061462

Offset: 1

David W. Wilson, Jan 05 2009

nonn

Donovan Johnson, Table of n, a(n) for n = 1..56 (terms < 10^12)

is_A014134(n)=for(k=0,sqrtint(n*2),issquare(n-k*(k+1)/2) && return);1
/* length of run starting at n */
rl(n)=local(t=n);while(is_A014134(n++),);n-t
/* print [ k, a[k] ] as defined above */
{s=0;for(n=1,10^6,is_A014134(n)|next;t=n;while(is_A014134(n++),);bittest(s,n-t)&next;s+=1<<(n-t);print1([n-t,t]","))}

More terms and PARI code from M. F. Hasler, Jan 05 2009

a(19)-a(30) from Donovan Johnson, Feb 07 2009

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

Original entry on oeis.org

2, 5, 8, 12, 23, 33

Offset: 1

Jud McCranie, Mar 19 2000

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

			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.

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".

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

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.

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 *)

Complement of A061208.

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

A014133 Sum of a square and a triangular number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 28, 29, 30, 31, 32, 35, 36, 37, 39, 40, 42, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 61, 64, 65, 66, 67, 70, 71, 72, 74, 75, 77, 78, 79, 80, 81, 82, 84, 85, 87, 91

Offset: 0

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A014134 (complement).

isA014133 := proc(n)
    local c,t ;
    for c from 0 do
        t := c*(c+1)/2 ;
        if t > n then
            return false;
        end if;
        if issqr(n-t) then
            return true;
        end if;
    end do:
end proc:
for n from 0 to 100 do
    if isA014133(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 11 2014

With[{nn=20},Select[Union[Flatten[Outer[Plus,Range[0,nn]^2,Accumulate[ Range[ 0,nn]]]]], #<=(nn(nn+1))/2&]] (* Harvey P. Dale, Dec 29 2019 *)

{k: A101428(k) > 0} .- R. J. Mathar, Apr 28 2020

A141945 Primes congruent to 23 mod 25.

Original entry on oeis.org

23, 73, 173, 223, 373, 523, 673, 773, 823, 1123, 1223, 1373, 1423, 1523, 1723, 1823, 1873, 1973, 2273, 2423, 2473, 3023, 3323, 3373, 3623, 3673, 3823, 3923, 4073, 4273, 4373, 4423, 4523, 4673, 4723, 4973, 5023, 5273, 5323, 5573, 5623, 5923, 6073, 6173, 6323

Offset: 1

N. J. A. Sloane, Jul 11 2008

Primes with the last 2 decimal digits in {23, 73}. - Chai Wah Wu, Apr 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A014134.

[p: p in PrimesUpTo(8000) | p mod 25 eq 23 ]; // Vincenzo Librandi, Aug 16 2012

select(isprime, [seq(23+50*j, j=0..1000)]); # Robert Israel, Mar 17 2015

Select[Range[23, 20000, 25], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2011 *)

is(n)=isprime(n) && n%25==23 \\ Charles R Greathouse IV, Jul 01 2016

A264101 Numbers that can't be represented as the sum of two squares, two triangular numbers, or a square and a triangular number.

Original entry on oeis.org

23, 33, 47, 62, 63, 86, 118, 134, 138, 143, 158, 167, 188, 195, 203, 204, 209, 223, 230, 243, 248, 275, 283, 294, 318, 323, 348, 368, 383, 385, 395, 398, 408, 411, 413, 418, 419, 426, 437, 440, 448, 454, 467, 473, 476, 489, 492, 503, 508, 518, 523, 558, 563, 566, 572, 608

Offset: 1

Alex Ratushnyak, Nov 03 2015

nonn

Intersection of A014134, A020757, A022544.

			Since 22 = 16+6, because 16 is a square and 6 is a triangular number, 22 is not a term.
23 is a term because there is no representation as S+T or S1+S2 or T1+T2, where S, S1, S2 are squares, and T, T1, T2 are triangular numbers.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A014134, A020757, A022544, A264118.

N:= 1000: # for terms <= N
S:= [seq(i^2,i=0..floor(sqrt(N)))]: nS:= nops(S):
T:= [seq(i*(i+1)/2, i=0..floor(sqrt(2*N)))]: nT:= nops(T):
sort(convert({$1..N} minus {seq(seq(S[i]+S[j], j=1..i),i=1..nS),
seq(seq(S[i]+T[j],i=1..nS),j=1..nT),
seq(seq(T[i]+T[j],j=1..i),i=1..nT)}, list)); # Robert Israel, May 19 2020

mx = 610; Complement[ Range@ mx, Union@ Flatten@ Table[{i^2 + j^2, i(i + 1)/2 + j^2, i(i + 1)/2 + j(j + 1)/2}, {i, 0, Sqrt[2 mx]}, {j, 0, Sqrt[2 mx]}]] (* Robert G. Wilson v, Nov 29 2015 *)

A231353 Least k>=0 such that n - k^2 is a triangular number, or -1 if no such k exists.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, -1, 3, 0, 1, 3, -1, 2, 0, 1, 4, -1, 2, -1, 0, 1, -1, 3, 2, 4, -1, 0, 1, 3, 4, 2, -1, -1, 5, 0, 1, -1, 6, 2, -1, 6, -1, 4, 0, 1, -1, -1, 2, 7, 6, 4, 5, 3, 0, 1, 6, -1, 2, -1, 4, -1, -1, 3, 8, 0, 1, -1, -1, 2, 4, 6, -1, 8, 3, -1, 7, 0, 1, 5, 6, 2

Offset: 0

Alex Ratushnyak, Nov 08 2013

sign

Indices of -1's: A014134.
Indices of 0's: A000217.
Indices of 1's: A000124.

Cf. A000217, A000290, A014134, A000124, A231354.

A231354 Least k such that n - triangular(k) is a square, or -1 if no such k exists.

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 3, 2, -1, 0, 1, 4, 2, -1, 4, 3, 0, 1, -1, 2, -1, 6, 3, -1, 5, 0, 1, -1, 2, 7, 6, 3, 7, -1, -1, 4, 0, 1, -1, 2, 5, -1, 3, -1, 7, 8, 4, -1, -1, 0, 1, 5, 2, 7, 9, 3, 10, 6, -1, 4, -1, 8, -1, -1, 0, 1, 11, 2, -1, -1, 3, 10, 8, -1, 4, 11, -1, 7, 12, 5, 10, 0

Offset: 0

Alex Ratushnyak, Nov 08 2013

sign

Indices of -1's: A014134.
Indices of 0's: A000290.
Indices of 1's: A002522.

Cf. A000217, A000290, A002522, A014134, A231353.

import math
for n in range(333):
  for k in range(1000000):
    t = n - k*(k+1)/2
    if t<0:
      print('-1', end=', ')
      break
    r = int(math.sqrt(t))
    if r*r==t:
      print(str(k), end=', ')
      break

A076769 Integers not expressible as the sum of a positive triangular number and a square.

Original entry on oeis.org

8, 9, 13, 18, 20, 23, 27, 33, 34, 38, 41, 43, 47, 48, 58, 60, 62, 63, 68, 69, 73, 76, 83, 86, 88, 89, 90, 93, 97, 98, 99, 108, 111, 112, 113, 118, 123, 125, 132, 133, 134, 135, 138, 139, 143, 144, 146, 148, 151, 158, 160, 163, 164, 167, 168, 173, 174, 177, 181, 182

Offset: 1

Jason Earls, Nov 14 2002

nonn

Cf. A000217, A000290, A014134.

isok(n)={for(k=1, sqrtint(2*n), if(issquare(n-binomial(k+1,2)), return(0))); 1} \\ Andrew Howroyd, Sep 18 2024

Name clarified and offset changed by Andrew Howroyd, Sep 18 2024

cp's OEIS Frontend

A140867 Start of the first run of exactly n integers in A014134.

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

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A014133 Sum of a square and a triangular number.

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Maple

Mathematica

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A141945 Primes congruent to 23 mod 25.

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Magma

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A264101 Numbers that can't be represented as the sum of two squares, two triangular numbers, or a square and a triangular number.

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Maple

Mathematica

A231353 Least k>=0 such that n - k^2 is a triangular number, or -1 if no such k exists.

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A231354 Least k such that n - triangular(k) is a square, or -1 if no such k exists.

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Python

A076769 Integers not expressible as the sum of a positive triangular number and a square.

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Author

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Programs

PARI

Extensions