A373333 -id:A373333 - cp's OEIS frontend

A373330 a(n) is the difference between T = A000217(n^2) and the greatest square not exceeding T.

0, 0, 1, 9, 15, 1, 41, 0, 55, 72, 9, 156, 36, 204, 262, 144, 135, 289, 209, 316, 111, 117, 406, 309, 527, 261, 342, 860, 804, 36, 954, 1200, 624, 605, 1257, 969, 1400, 741, 849, 1856, 1639, 0, 1721, 2076, 855, 701, 1770, 1101, 1719, 397, 426, 1980, 1416, 2449, 1142

Offset: 0

Hugo Pfoertner, Jun 02 2024

Michael De Vlieger, Table of n, a(n) for n = 0..9998
Hugo Pfoertner, Logarithmic plot of a(n) vs n, n <= 10^5, lower envelope of terms > 0 shown in red, zoom for details.

Cf. A000217, A000290, A002315, A037270, A061288, A373329, A373333.

A373331 and A373332 are the coordinates of the observed lower envelope of this sequence.

Array[PolygonalNumber[#^2] - Floor[Sqrt[(#^4 + #^2)/2]]^2 &, 55, 0] (* Michael De Vlieger, Jun 02 2024 *)

a(n) = my(T=(n^4+n^2)/2); T-sqrtint(T)^2

from sympy import integer_nthroot
def A373330(n): return (T:=(n**4 + n**2) // 2)-(integer_nthroot(T,2)[0])**2
# Karl-Heinz Hofmann, Jul 01 2024

a(n) = A000217(n^2) - A373329(n)^2.

a(A002315(n)) = 0.

A373331 X-coordinates of the lower envelope of A373330.

Original entry on oeis.org

5, 10, 136, 153, 306, 428, 623, 763, 4072, 8580, 9964, 22373, 25571, 109408, 190619, 276532, 314223, 463097, 2256531, 2791875, 4325972, 14492546, 20201957, 28465976, 64084574, 72478694, 109821726, 543521049, 65927493216, 105027578781, 128041940756

Offset: 1

Hugo Pfoertner, Jun 02 2024

a(32) >= 24882280056497.

Note the remark in A374175 regarding the lack of a proof of the increasing minimum of the terms of A373330. The absence of smaller minima was checked up to 4.5*10^12.

Hugo Pfoertner, Observed lower envelope of A373330.

A373332 gives the corresponding y-coordinates.

Cf. A373330, A373333, A374175.

A373332 Y-coordinates of the lower envelope of A373330.

Original entry on oeis.org

1, 9, 15, 536, 545, 559, 784, 1029, 1296, 9864, 19972, 25281, 61372, 71919, 254305, 264196, 501809, 577720, 4825277, 5615100, 8943039, 23666905, 23765244, 24555772, 126237882, 157076310, 264463290, 341853680, 1219695647, 95714023805, 220113045816

Offset: 1

Hugo Pfoertner, Jun 02 2024

The terms A373330(A002315(k)) = 0 for all k are excluded when determining the lower envelope.
a(32) <= 415683424401.
Note the remark in A374175 regarding the lack of a proof of the increasing minimum of the terms of A373330.

A373331 gives the corresponding x-coordinates.

Cf. A373330, A373333, A374175.

A374175 a(n) is the conjectured number of occurrences of n in A373330.

Original entry on oeis.org

2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Hugo Pfoertner, Jun 30 2024

nonn

The sequence must be considered as conjectural, since so far no proof is known for the non-occurrence of arbitrarily small terms > 0 for very large n in A373330, despite the growing distance reserve observed in A373331 and A373332.

			Some observed positions of n in A373330:
      n   positions
     41   6         (A000217(6^2)=666, next smaller square = 625, 41 = 666 - 625)
      1   2     5
      9   3    10
     15   4   136
    ...
  25281 197  1590  22373
 264196 725 65684 276532.
No other terms = 3 or greater are known.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A373330, A373333 (positions of terms > 0).

a373330(n) = {my(T=(n^4+n^2)/2); T-sqrtint(T)^2};
a374175(nmax,slimit) = {my(hits=vectorsmall(nmax)); for (k=0, slimit, my (j = a373330(k)); if(j>0 && j<=nmax, hits[j]++)); hits};

a(0) = oo.

cp's OEIS Frontend

A373330 a(n) is the difference between T = A000217(n^2) and the greatest square not exceeding T.

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A373331 X-coordinates of the lower envelope of A373330.

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A373332 Y-coordinates of the lower envelope of A373330.

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A374175 a(n) is the conjectured number of occurrences of n in A373330.

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