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A342326 a(n) is the smallest nonnegative integer that can be written as a sum of two distinct nonzero triangular numbers in exactly n ways or -1 if no such integer exists. 1
0, 4, 16, 81, 471, 2031, 1381, 11781, 6906, 17956, 34531, 123256, 40056, 305256, 863281, 448906, 200281, 1957231, 520731, 10563906, 1001406, 11222656, 7631406, 3454506, 1482081, 75865156, 7172606106, 8852431, 25035156, 334020781, 13018281, 38531031, 7410406, 7014160156 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Conjecture: This sequence has a positive a(n) for every positive integer n, and each sequence in the infinite indexed family, of which this sequence offers the initial terms, is infinite, as well.

From David A. Corneth, Mar 08 2021: (Start)

a(40) = 37052031, a(45) = 221310781, a(48) = 60765331, a(39) <= 2782318906, a(42) <= 325457031, a(47) <= 927577056, a(50) <= 2200089531, a(54) <= 327539956, a(56) <= 926300781, a(60) <= 481676406, a(63) <= 4598740656, a(64) <= 303826656, a(71) <= 4579579956, a(72) <= 789949306, a(80) <= 1519133281, a(96) <= 3220562556. Terms for n <= 96 not listed here and terms for which only upper bounds are known are >= 3*10^8.

Is a(n) == 6 (mod 25) for n >= 5? It holds for all terms known to date.

The triangular numbers mod 25 are periodic with period 25. Constructing all 25*25 = 625 sums of two distinct triangular numbers mod 25 gives 65 cases for 6 (mod 25). The second largest occurs 40 times. (End)

a(47) = 550240551, a(59) = 7629645156, a(67) = 6418012656, a(81) = 9498658731, a(90) = 8188498906. All upper bounds listed in the above comments for n other than 47 are the exact values of a(n). For all n for which no value is listed here or above, a(n) > 10^10 (or a(n) = -1). - Jon E. Schoenfield, Mar 09 2021

From Martin Ehrenstein, Mar 09 2021: (Start)

a(44) = 15646972656. For n<=51, all terms not mentioned here or above, a(n) >= 6.5*10^10 (or a(n) = -1).

a(47) == 1 (mod 25) and a(95) = 47652012541 == 16 (mod 25). Thus the answer to Corneth's question is 'No'. (End)

LINKS

Martin Ehrenstein, Table of n, a(n) for n = 0..37

FORMULA

a(n) = min { m >= 0 : A307597(m) = n }. - Alois P. Heinz, Mar 08 2021

EXAMPLE

a(1) =  4 = 1 +  3;

a(2) = 16 = 1 + 15 =  6 + 10;

a(3) = 81 = 3 + 78 = 15 + 66 = 36 + 45.

MATHEMATICA

r = 125000; (* generates the first 12 terms of the sequence *)

lst = Table[0, {r}];

lim = Floor[Sqrt[2r]];

Do[ num = (i^2 + i)/2 + (j^2 + j)/2;

If[num <= r, lst[[num]]++], {i, lim}, {j,  i - 1}];

First /@ (Flatten@Position[lst, #] & /@ Range[Max[lst]])

PROG

(PARI) upto(n) = {my(v = vector(n)); res = vector(10); for(i = 1, (sqrtint(8*n + 1)-1)\2, bi = binomial(i + 1, 2); for(j = i+1, (sqrtint(8*(n - bi))-1)\2, v[bi + binomial(j+1, 2)]++ ) ); for(i = 1, #v, if(v[i] > 0, if(v[i] > #res, res = concat(res, vector(v[i] - #res)); ); if(res[v[i]] == 0, res[v[i]] = i ) ) ); concat(0, res) } \\ David A. Corneth, Mar 08 2021

CROSSREFS

Cf. A000217, A260647, A051533, A262749, A265140, A265134, A265136, A265138, A307597.

Sequence in context: A337839 A001257 A133053 * A110378 A076997 A092304

Adjacent sequences:  A342323 A342324 A342325 * A342327 A342328 A342329

KEYWORD

nonn

AUTHOR

Robert G. Root, Mar 08 2021

EXTENSIONS

a(13)-a(18) from Alois P. Heinz, Mar 08 2021

a(19)-a(25) from David A. Corneth, Mar 08 2021

a(26)-a(33) from Jon E. Schoenfield, Mar 09 2021 (some terms first found by David A. Corneth)

STATUS

approved

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Last modified August 4 12:33 EDT 2021. Contains 346447 sequences. (Running on oeis4.)