A077348 -id:A077348 - cp's OEIS frontend

A077346 Triangle in which n-th row contains n smallest squares beginning with n.

1, 25, 225, 36, 324, 361, 4, 49, 400, 441, 529, 576, 5041, 5184, 5329, 64, 625, 676, 6084, 6241, 6400, 729, 784, 7056, 7225, 7396, 7569, 7744, 81, 841, 8100, 8281, 8464, 8649, 8836, 80089, 9, 900, 961, 9025, 9216, 9409, 9604, 9801, 90000

Offset: 1

Amarnath Murthy, Nov 05 2002

A018796 contains initial terms of rows. Subsidiary sequence: There can be a rearrangement of squares in groups so that the n-th group contains n squares beginning with n and not occurring earlier. E.g. the initial term of row 25 would be 256 and not 25.

			Triangle begins:
    1;
   25, 225;
   36, 324,  361;
    4,  49,  400,  441;
  529, 576, 5041, 5184, 5329;
  ...

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Cf. A018796, A077347, A077348, A077349.

g:= proc(n,d)
    local i,r;
    r:= ilog10(n)+1;
    seq(i^2, i = ceil(sqrt(n*10^(d-r))) .. ceil(sqrt((n+1)*10^(d-r)))-1)
end proc:
f:= proc(n)
   local R,count,v,d;
   count:= 0; R:= NULL;
   for d from 1 + ilog10(n) do
     v:= g(n,d);
     R:= R,v;
     count:= count + nops([v]);
     if count >= n then return R[1..n] fi
   od
end proc:
f(1):= 1:
for i from 1 to 10 do f(i) od; # Robert Israel, Jan 28 2025

row(n) = my(list=List(), k=1); while (#list != n, if (strsplit(Str(k^2), Str(n))[1] == "", listput(list, k^2)); k++); Vec(list); \\ Michel Marcus, Feb 08 2023

More terms from Michel Marcus, Feb 08 2023

A077347 Final terms of rows of A077346.

Original entry on oeis.org

1, 225, 361, 441, 5329, 6400, 7744, 80089, 90000, 101124, 112896, 122500, 135424, 145924, 157609, 165649, 179776, 1806336, 1901641, 2013561, 2114116, 2220100, 2325625, 2427364, 2528100, 2637376, 2742336, 2845969, 2951524, 3059001

Offset: 1

Amarnath Murthy, Nov 05 2002

a(n) = n-th square beginning with n. A018796 contains initial terms of rows. Subsidiary sequence: There can be a rearrangement of squares in groups so that the n-th group contains n squares beginning with n and not occurring earlier. E.g., the initial term of row 25 would be 256 and not 25 and accordingly the final terms would differ.

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

Cf. A018796, A077346, A077348, A077349.

startsWith := proc(n,dig) local nshft ; nshft := n ; while nshft > dig do nshft := floor(nshft/10) ; od ; if dig = nshft then RETURN(true) ; else RETURN(false) ; fi ; end: A077347 := proc(n) local candid,c; candid := 1 ; for c from 1 to n do while not startsWith(candid^2,n) do candid := candid+1 ; od ; if c = n then RETURN(candid^2) ; fi ; candid := candid+1 ; od ; end: for n from 1 to 50 do printf("%a,",A077347(n)) ; od ; # R. J. Mathar, Nov 12 2006

More terms from R. J. Mathar, Nov 12 2006

A077349 a(n) = A077347(n)^(1/2).

Original entry on oeis.org

1, 15, 19, 21, 73, 80, 88, 283, 300, 318, 336, 350, 368, 382, 397, 407, 424, 1344, 1379, 1419, 1454, 1490, 1525, 1558, 1590, 1624, 1656, 1687, 1718, 1749, 1779, 1806, 1836, 1865, 1895, 1918, 1949, 6165, 6247, 6326, 6409, 6487, 6565, 6641, 6720, 6795, 6868, 6943, 7013, 7089

Offset: 1

Amarnath Murthy, Nov 05 2002

Andrew Howroyd, Table of n, a(n) for n = 1..500

Main diagonal of A077348.

Cf. A077346, A018796, A077347.

a(n)=row(n)[n] \\ row(n) defined in A077348. - Andrew Howroyd, Dec 11 2024

a(8) onwards from Andrew Howroyd, Dec 11 2024

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A077346 Triangle in which n-th row contains n smallest squares beginning with n.

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A077347 Final terms of rows of A077346.

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A077349 a(n) = A077347(n)^(1/2).

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