A183534 -id:A183534 - cp's OEIS frontend

A183527 An Ulam-type sequence: a(n) = n if n<=4; for n>4, a(n) = least number > a(n-1) which is a unique sum of 4 distinct earlier terms.

1, 2, 3, 4, 10, 16, 17, 18, 19, 22, 64, 65, 66, 68, 69, 128, 132, 188, 190, 191, 194, 252, 253, 255, 313, 314, 318, 374, 376, 377, 436, 441, 496, 497, 499, 500, 502, 560, 561, 563, 621, 622, 626, 682, 684, 685, 687, 745, 746, 805, 811

Offset: 1

Jonathan Vos Post and Alois P. Heinz, Jan 05 2011

nonn

An Ulam-type sequence - see A002858 for further information.

			a(5) = 10 = 1 + 2 + 3 + 4 = 4*5/2, because it is the least number >4 with a unique sum of 4 distinct earlier terms.
a(6) = 16 = 1 + 2 + 3 + 10 = 4^2, because it is the least number >10 with a unique sum of 4 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for Ulam numbers

Column k=4 of A183534.

Cf. A002858, A007086, A183528-A183533, A007300, A135737.

Maple
```
# see A183534 for programs.
```

Conjectured G.f.: (-61*x^124-56*x^118+53*x^117+3*x^116 -x^115+2*x^114-65*x^113-58*x^112+57*x^111 -56*x^110-x^109-57*x^108+54*x^106 +3*x^105+58*x^104-52*x^102+50*x^101-55*x^100 +56*x^95-53*x^94-3*x^93+x^92-2*x^91 +2*x^90+x^87-x^86 +41*x^84-51*x^83-66*x^82-58*x^81 -52*x^79+4*x^78-58*x^77 -x^74-x^73-x^72-54*x^71 -6*x^70-59*x^69-x^68-58*x^67-2*x^66 -x^65-2*x^64-56*x^63-4*x^62-x^61 -58*x^60-2*x^59-59*x^58-x^57-x^56 -2*x^55-x^54-x^53-54*x^52-6*x^51 -59*x^50-x^49-58*x^48-2*x^47-x^46 -2*x^45-56*x^44-4*x^43-x^42-58*x^41 -2*x^40-x^39-58*x^38-2*x^37-x^36 -2*x^35-x^34-55*x^33-5*x^32-59*x^31 -x^30-2*x^29-56*x^28-4*x^27-x^26 -58*x^25-2*x^24-x^23-58*x^22-3*x^21 -x^20-2*x^19-56*x^18-4*x^17-59*x^16 -x^15-2*x^14-x^13-x^12-42*x^11 -3*x^10-x^9-x^8-x^7-6*x^6 -6*x^5-x^4-x^3-x^2-x) / (-x^74+x^73+x-1). (This has been verified for n up to 1000.)

A183533 An Ulam-type sequence: a(n) = n if n<=10; for n>10, a(n) = least number > a(n-1) which is a unique sum of 10 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 55, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 145, 163, 190, 217, 235, 271, 280, 1740, 1741, 1744, 1745, 1799, 1804, 1805, 1824, 1825, 1831, 1859, 1869, 1913, 1914, 3554, 10521, 10522, 10526, 10527, 10537, 10563, 10564

Offset: 1

Jonathan Vos Post and Alois P. Heinz, Jan 05 2011

nonn

An Ulam-type sequence - see A002858 for further information.

			a(11) = 55 = 1 + ... + 10 = 10*11/2, because it is the least number >10 with a unique sum of 10 distinct earlier terms.
a(12) = 100 = 1 + ... + 9 + 55 = 10^2, because it is the least number >55 with a unique sum of 10 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..120
Index entries for Ulam numbers

Column k=10 of A183534.

Cf. A002858, A007086, A183527-A183532, A135737.

Maple
```
# see A183534 for programs.
```

A183532 An Ulam-type sequence: a(n) = n if n<=9; for n>9, a(n) = least number > a(n-1) which is a unique sum of 9 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 45, 81, 82, 83, 84, 85, 86, 87, 88, 89, 117, 133, 153, 177, 189, 221, 225, 1325, 1326, 1328, 1329, 1373, 1378, 1379, 1391, 1392, 1398, 1423, 1427, 2717, 2718, 4031, 4032, 4035, 4037, 4039, 5316, 5319, 5346, 5352, 5353, 5354, 5361

Offset: 1

Jonathan Vos Post and Alois P. Heinz, Jan 05 2011

nonn

An Ulam-type sequence - see A002858 for further information.

			a(10) = 45 = 1 + ... + 9 = 9*10/2, because it is the least number >9 with a unique sum of 9 distinct earlier terms.
a(11) = 81 = 1 + ... + 8 + 45 = 9^2, because it is the least number >45 with a unique sum of 9 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..200
Index entries for Ulam numbers

Column k=9 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

Maple
```
# see A183534 for programs.
```

A183528 An Ulam-type sequence: a(n) = n if n<=5; for n>5, a(n) = least number > a(n-1) which is a unique sum of 5 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 15, 25, 26, 27, 28, 29, 35, 43, 45, 165, 171, 172, 174, 180, 181, 328, 333, 338, 339, 340, 341, 493, 499, 500, 647, 652, 657, 658, 659, 660, 661, 662, 663, 815, 818, 819, 971, 1127, 1137, 1138, 1139, 1140, 1141, 1142

Offset: 1

Jonathan Vos Post and Alois P. Heinz, Jan 05 2011

nonn

An Ulam-type sequence - see A002858 for further information.

			a(6) = 15 = 1 + 2 + 3 + 4 + 5 = 5*6/2, because it is the least number >5 with a unique sum of 5 distinct earlier terms.
a(7) = 25 = 1 + 2 + 3 + 4 + 15 = 5^2, because it is the least number >15 with a unique sum of 5 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for Ulam numbers

Column k=5 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

Maple
```
# see A183534 for programs.
```

A184533 a(n) = floor(1/{(2+n^3)^(1/3)}), where {}=fractional part.

Original entry on oeis.org

2, 6, 13, 24, 37, 54, 73, 96, 121, 150, 181, 216, 253, 294, 337, 384, 433, 486, 541, 600, 661, 726, 793, 864, 937, 1014, 1093, 1176, 1261, 1350, 1441, 1536, 1633, 1734, 1837, 1944, 2053, 2166, 2281, 2400, 2521, 2646, 2773, 2904, 3037, 3174, 3313, 3456, 3601

Offset: 1

Clark Kimberling, Jan 16 2011

Column 2 of the array at A184532.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A183532, A183534. Essenitally the same as A032528.

p[n_]:=FractionalPart[(n^3+2)^(1/3)]; q[n_]:=Floor[1/p[n]]; Table[q[n],{n,1,120}]
Join[{2},Table[(6*n^2 - (1-(-1)^n))/4,{n,2,50}]] (* or *) Join[{2}, LinearRecurrence[{2,0,-2,1},{6, 13, 24, 37},50]] (* G. C. Greubel, Feb 20 2017 *)

a(n)=my(x=sqrtn(n^3+2,3));x-=n;1/x\1 \\ Charles R Greathouse IV, Aug 23 2011

concat([2], for(n=2,25, print1((6*n^2 - (1-(-1)^n))/4, ", "))) \\ G. C. Greubel, Feb 20 2017

a(n) = floor(1/{(2+n^3)^(1/3)}), where {}=fractional part.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) = (6*n^2 - (1-(-1)^n))/4 for n>1.
From Alexander R. Povolotsky, Aug 22 2011: (Start)
a(n+1) +a(n) = 3*n^2 + 3*n + 1.
G.f.: x*(-2 - 2*x - x^2 - 2*x^3 + x^4)/((-1 + x)^3*(1 + x)). (End)
a(n) = floor(1/((n^3+2)^(1/3)-n)). - Charles R Greathouse IV, Aug 23 2011
E.g.f.: (3*x*(x + 1)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x) + 2*x)/2. - Stefano Spezia, Apr 19 2025

A183529 An Ulam-type sequence: a(n) = n if n<=6; for n>6, a(n) = least number > a(n-1) which is a unique sum of 6 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 21, 36, 37, 38, 39, 40, 41, 51, 61, 66, 284, 285, 289, 290, 297, 298, 299, 310, 312, 559, 561, 562, 570, 571, 574, 575, 834, 836, 837, 838, 839, 840, 841, 849, 850, 1109, 1124, 1125, 1126, 1127, 1386, 1401, 1402, 1661, 1676, 1677, 1936, 1951

Offset: 1

Jonathan Vos Post and Alois P. Heinz, Jan 05 2011

nonn

An Ulam-type sequence - see A002858 for further information.

			a(7) = 21 = 1 + ... + 6 = 6*7/2, because it is the least number >6 with a unique sum of 6 distinct earlier terms.
a(8) = 36 = 1 + ... + 5 + 21 = 6^2, because it is the least number >21 with a unique sum of 6 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..700
Index entries for Ulam numbers

Column k=6 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

Maple
```
# see A183534 for programs.
```

A183530 An Ulam-type sequence: a(n) = n if n<=7; for n>7, a(n) = least number > a(n-1) which is a unique sum of 7 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 28, 49, 50, 51, 52, 53, 54, 55, 70, 82, 91, 109, 112, 555, 556, 563, 564, 572, 573, 576, 583, 584, 591, 593, 1103, 1104, 1105, 1111, 1112, 1124, 1632, 1637, 1642, 1643, 1648, 1653, 1654, 1655, 1656, 1657, 1660, 1661, 1662, 1671, 1672, 2184

Offset: 1

Jonathan Vos Post and Alois P. Heinz, Jan 05 2011

nonn

An Ulam-type sequence - see A002858 for further information.

			a(8) = 28 = 1 + ... + 7 = 7*8/2, because it is the least number >7 with a unique sum of 7 distinct earlier terms.
a(9) = 49 = 1 + ... + 6 + 28 = 7^2, because it is the least number >28 with a unique sum of 7 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..500
Index entries for Ulam numbers

Column k=7 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

Maple
```
# see A183534 for programs.
```

A183531 An Ulam-type sequence: a(n) = n if n<=8; for n>8, a(n) = least number > a(n-1) which is a unique sum of 8 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 36, 64, 65, 66, 67, 68, 69, 70, 71, 92, 106, 120, 141, 148, 792, 793, 802, 803, 816, 817, 820, 828, 830, 838, 844, 1591, 1592, 1606, 1607, 2356, 2357, 2358, 2359, 2360, 2384, 2395, 2402, 3149, 3150, 3169, 3919, 3920, 3921, 3922, 3923

Offset: 1

Jonathan Vos Post and Alois P. Heinz, Jan 05 2011

nonn

An Ulam-type sequence - see A002858 for further information.

			a(9) = 36 = 1 + ... + 8 = 8*9/2, because it is the least number >8 with a unique sum of 8 distinct earlier terms.
a(10) = 64 = 1 + ... + 7 + 36 = 8^2, because it is the least number >36 with a unique sum of 8 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..500
Index entries for Ulam numbers

Column k=8 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

Maple
```
# see A183534 for programs.
```

A184534 a(n) = floor(1/{(4+n^3)^(1/3)}), where {}=fractional part.

Original entry on oeis.org

1, 3, 7, 12, 18, 27, 36, 48, 60, 75, 90, 108, 126, 147, 168, 192, 216, 243, 270, 300, 330, 363, 396, 432, 468, 507, 546, 588, 630, 675, 720, 768, 816, 867, 918, 972, 1026, 1083, 1140, 1200, 1260, 1323, 1386, 1452, 1518, 1587, 1656, 1728, 1800, 1875, 1950, 2028, 2106, 2187, 2268, 2352, 2436, 2523, 2610, 2700, 2790, 2883, 2976, 3072, 3168, 3267, 3366, 3468, 3570, 3675, 3780, 3888, 3996, 4107, 4218, 4332, 4446, 4563, 4680, 4800, 4920, 5043, 5166, 5292, 5418, 5547, 5676, 5808

Offset: 1

Clark Kimberling, Jan 16 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A183532, A183534.

Table[Floor[1/FractionalPart[(n^3 + 4)^(1/3)]], {n, 1, 120}]

for(n=1, 50, print1(floor(1/frac((4 + n^3)^(1/3))), ", ")) \\ G. C. Greubel, May 14 2017

a(n) = floor[1/{(4+n^3)^(1/3)}], where {}=fractional part.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
From Colin Barker, Oct 07 2012: (Start)
Empirical: a(n) = 3*(1 - (-1)^n + 4*n + 2*n^2)/8 for n>2.
Empirical G.f.: x*(x^6-2*x^5+x^4-x^2-x-1)/((x-1)^3*(x+1)).(End)

cp's OEIS Frontend

A183527 An Ulam-type sequence: a(n) = n if n<=4; for n>4, a(n) = least number > a(n-1) which is a unique sum of 4 distinct earlier terms.

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A183533 An Ulam-type sequence: a(n) = n if n<=10; for n>10, a(n) = least number > a(n-1) which is a unique sum of 10 distinct earlier terms.

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A183532 An Ulam-type sequence: a(n) = n if n<=9; for n>9, a(n) = least number > a(n-1) which is a unique sum of 9 distinct earlier terms.

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A183528 An Ulam-type sequence: a(n) = n if n<=5; for n>5, a(n) = least number > a(n-1) which is a unique sum of 5 distinct earlier terms.

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Maple

A184533 a(n) = floor(1/{(2+n^3)^(1/3)}), where {}=fractional part.

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Mathematica

PARI

PARI

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A183529 An Ulam-type sequence: a(n) = n if n<=6; for n>6, a(n) = least number > a(n-1) which is a unique sum of 6 distinct earlier terms.

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A183530 An Ulam-type sequence: a(n) = n if n<=7; for n>7, a(n) = least number > a(n-1) which is a unique sum of 7 distinct earlier terms.

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Maple

A183531 An Ulam-type sequence: a(n) = n if n<=8; for n>8, a(n) = least number > a(n-1) which is a unique sum of 8 distinct earlier terms.

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