A076031 -id:A076031 - cp's OEIS frontend

A076027 Initial members of groups in A076031.

1, 2, 3, 5, 9, 14, 19, 25, 32, 40, 49, 59, 70, 82, 95, 109, 124, 140, 157, 175, 194, 215, 236, 258, 281, 305, 330, 356, 383, 411, 440, 470, 501, 533, 566, 600, 635, 671, 708, 746, 785, 825, 866, 908, 951, 995, 1040, 1086, 1133, 1181, 1230, 1280

Offset: 1

Amarnath Murthy, Oct 01 2002

nonn

Cf. A076028, A076029, A076030, A076031.

Corrected and extended by Hans Havermann, Nov 09 2002

A076028 Final members of groups in A076031.

Original entry on oeis.org

1, 8, 12, 210, 1430, 1785, 168245, 409045, 119629510, 293493662, 7454211765, 8777087598279, 41162302753, 184604382197975454, 3700467339975406, 20717472096185651930, 150297492924069082735, 588852089711321510

Offset: 1

Amarnath Murthy, Oct 01 2002

nonn

Cf. A076027, A076029, A076030, A076031.

Corrected and extended by Hans Havermann, Nov 09 2002

Name corrected by Rémy Sigrist, Dec 04 2022

A076029 Products of members of groups in A076031.

Original entry on oeis.org

1, 16, 144, 44100, 18404100, 1835265600, 16304474894400, 5421097109610000, 296757450928652313600, 178616855049415417958400, 1411446839588456089779360000, 638977904981699070278114213990400

Offset: 1

Amarnath Murthy, Oct 01 2002

nonn

Cf. A076027, A076028, A076030, A076031.

Corrected and extended by Hans Havermann, Nov 09 2002

A076030 Square roots of products of members of groups in A076031.

Original entry on oeis.org

1, 4, 12, 210, 4290, 42840, 4037880, 73628100, 17226649440, 422630873280, 37569227295600, 25278012283043520, 37342441057521600, 1860812172555592576320, 6714127941651376646400, 13782919836150390515990400, 1090799084645723774857536000

Offset: 1

Amarnath Murthy, Oct 01 2002

nonn

a(n) = A076029(n)^(1/2).

Cf. A076027, A076028, A076029, A076031.

Corrected and extended by Hans Havermann, Nov 09 2002

A076096 Final terms of rows in A076099.

Original entry on oeis.org

1, 8, 18, 9261000, 329422500, 13456677968449745006250, 11022732501667945875061568782593750, 99325923481814861327531258459131254450000

Offset: 1

Amarnath Murthy, Oct 07 2002

nonn

Cf. A076027, A076028, A076029, A076030, A076031, A076095, A076097, A076098, A076099.

More terms from David Wasserman, Apr 01 2005
Edited by Ray Chandler, May 09 2007
Offset corrected by Sean A. Irvine, Mar 19 2025

A076097 Products of rows in A076099.

Original entry on oeis.org

1, 16, 216, 1944810000, 3913539300000, 9992390792252042651841000000, 1068203306732039166959846576540631630000000, 1316372224808054302435728765171857144838608100000000

Offset: 1

Amarnath Murthy, Oct 08 2002

nonn

Cf. A076027, A076028, A076029, A076030, A076031, A076095, A076096, A076098, A076099.

More terms from David Wasserman, Apr 01 2005
Edited by Ray Chandler, May 09 2007
Offset corrected by Sean A. Irvine, Mar 19 2025

A076098 Geometric mean of n-th row in A076099.

Original entry on oeis.org

1, 4, 6, 210, 330, 46410, 1009470, 2454270, 358888530, 8804809860, 14908423530, 87770875982790, 8644083578130, 6461153376929140890, 55507010099631090, 435066914019898690530, 12624989405621802949740, 1768322825403098494530

Offset: 0

Amarnath Murthy, Oct 08 2002

nonn

Cf. A076027, A076028, A076029, A076030, A076031, A076095, A076096, A076097, A076099.

a(n)= A076097(n)^(1/n).

More terms from David Wasserman, Apr 01 2005
Edited by Ray Chandler, May 09 2007
Offset corrected by Sean A. Irvine, Mar 19 2025

A076099 Triangle read by rows in which the n-th row contains n numbers not occurring in the previous rows whose product is an n-th power. The first (n-1) numbers of the n-th row are the smallest number not occurring earlier and the n-th term is chosen to satisfy the requirement.

Original entry on oeis.org

1, 2, 8, 3, 4, 18, 5, 6, 7, 9261000, 9, 10, 11, 12, 329422500, 13, 14, 15, 16, 17, 13456677968449745006250, 19, 20, 21, 22, 23, 24, 11022732501667945875061568782593750, 25, 26, 27, 28, 29, 30, 31

Offset: 1

Amarnath Murthy, Oct 08 2002

Another rearrangement of the natural numbers in which the product of next n numbers is an n-th power.

The first n-1 elements of the n-th group are the smallest n-1 numbers that haven't already appeared, say u1, u2, ..., u_(n-1) and let u_n be the unknown final element of the n-th group. Let u1*u2*u3*...*u_(n-1) = (p1^e1)(p2^e2)...(pr^er). Then u_n = product(i=1 to r) p_i^(ei + n*floor(ei/n) - n) ...unless this has already appeared in the sequence (probably this never happens). More simply, I conjecture that u_n = product(i=1 to r) p_i^(ei - n). - Sam Alexander, Dec 31 2003

			Triangle begins:
1
2 8
3 4 18
5 6 7 9261000
9 10 11 12 329422500
...

Cf. A076027, A076028, A076029, A076030, A076031, A076095, A076096, A076097, A076098.

Corrected and extended by Sam Alexander, Dec 31 2003
Edited by Ray Chandler, May 09 2007
Offset corrected by Sean A. Irvine, Mar 19 2025

A076095 Initial terms of rows in A076099.

Original entry on oeis.org

1, 2, 3, 5, 9, 13, 19, 25, 32, 40, 49, 59, 70, 82, 95, 109, 124, 140, 157, 175, 194, 214, 235, 257, 280, 304, 329, 355, 382, 410, 439, 469, 500, 532, 565, 599, 634, 670, 707, 745, 784, 824, 865, 907, 950, 994, 1039, 1085, 1132, 1180, 1229, 1279, 1330, 1382

Offset: 1

Amarnath Murthy, Oct 07 2002

nonn

This sequence is not the same as A076027.

Cf. A076027, A076028, A076029, A076030, A076031, A076096, A076097, A076098, A076099.

More terms from David Wasserman, Apr 01 2005
Edited by Ray Chandler, May 09 2007
Offset corrected by Sean A. Irvine, Mar 19 2025

A076596 Triangle read by rows: n-th row contains n numbers whose product is a square; all numbers in the triangle are distinct and the square is as small as possible.

Original entry on oeis.org

1, 2, 8, 3, 4, 12, 5, 7, 10, 14, 6, 9, 15, 18, 20, 11, 13, 16, 22, 25, 26, 17, 19, 21, 27, 28, 34, 38, 23, 24, 30, 33, 35, 42, 44, 46

Offset: 1

Donald S. McDonald, Nov 10 2002

Differs from A076031 and A076696.

			Triangle begins:
  1 = 1^2,
  2 * 8 = 4^2,
  3 * 4 * 12 = 12^2,
  5 * 7 * 10 * 14 = 70^2,
  6 * 9 * 15 * 18 * 20 = 540^2,
  11 * 13 * 16 * 22 * 25 * 26 = 5720^2,
  17 * 19 * 21 * 27 * 28 * 34 * 38 = 81396^2,
  23 * 24 * 30 * 33 * 35 * 42 * 44 * 46 = 1275120^2, ...

Sean A. Irvine, Java program (github)

Cf. A076031, A076696.

Name clarified and a(16)-a(36) from Sean A. Irvine, Apr 09 2025

cp's OEIS Frontend

A076027 Initial members of groups in A076031.

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A076028 Final members of groups in A076031.

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A076029 Products of members of groups in A076031.

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A076030 Square roots of products of members of groups in A076031.

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A076096 Final terms of rows in A076099.

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A076097 Products of rows in A076099.

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A076098 Geometric mean of n-th row in A076099.

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A076099 Triangle read by rows in which the n-th row contains n numbers not occurring in the previous rows whose product is an n-th power. The first (n-1) numbers of the n-th row are the smallest number not occurring earlier and the n-th term is chosen to satisfy the requirement.

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A076095 Initial terms of rows in A076099.

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A076596 Triangle read by rows: n-th row contains n numbers whose product is a square; all numbers in the triangle are distinct and the square is as small as possible.

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