A212616
Least triangular number that is the product of n triangular numbers greater than 1.
Original entry on oeis.org
3, 36, 300, 1485, 3240, 265356, 265356, 21520080, 21520080, 193720086, 1743362676, 141214502520, 141214502520, 11438393835996, 11438393835996, 926510072902560, 926510072902560, 75047317454789316, 75047317454789316, 6078832727785072200, 6078832727785072200
Offset: 1
Contribution from _Donovan Johnson_, Jun 11 2012: (Start)
Let tri(n) = n*(n+1)/2. Then
a(1) = 3 = tri(2).
a(2) = 36 = tri(3)^2.
a(3) = 300 = tri(2) * tri(4)^2.
a(4) = 1485 = tri(2)^3 * tri(10).
a(5) = 3240 = tri(2)^2 * tri(3)^2 * tri(4).
a(6) = 265356 = tri(2)^4 * tri(8) * tri(13).
265356 = tri(2)^3 * tri(3) * tri(6) * tri(12).
a(7) = 265356 = tri(2)^4 * tri(3)^2 * tri(13).
a(8) = 21520080 = tri(2)^6 * tri(8) * tri(40).
a(9) = 21520080 = tri(2)^6 * tri(3)^2 * tri(40).
a(10) = 193720086 = tri(2)^7 * tri(3) * tri(6) * tri(37).
a(11) = 1743362676 = tri(2)^8 * tri(3)^2 * tri(121).
a(12) = 141214502520 = tri(2)^10 * tri(8) * tri(364).
141214502520 = tri(2)^8 * tri(3) * tri(5) * tri(13) * tri(72).
a(13) = 141214502520 = tri(2)^10 * tri(3)^2 * tri(364).
141214502520 = tri(2)^10 * tri(4) * tri(13) * tri(72).
a(14) = 11438393835996 = tri(2)^12 * tri(8) * tri(1093).
a(15) = 11438393835996 = tri(2)^12 * tri(3)^2 * tri(1093).
a(16) = 926510072902560 = tri(2)^14 * tri(8) * tri(3280).
a(17) = 926510072902560 = tri(2)^14 * tri(3)^2 * tri(3280).
a(18) = 75047317454789316 = tri(2)^16 * tri(8) * tri(9841).
a(19) = 75047317454789316 = tri(2)^16 * tri(3)^2 * tri(9841).
a(20) = 6078832727785072200 = tri(2)^18 * tri(8) * tri(29524).
a(21) = 6078832727785072200 = tri(2)^18 * tri(3)^2 * tri(29524). (End)
A212617
Least pentagonal number that is the product of n pentagonal numbers greater than 1.
Original entry on oeis.org
5, 10045, 20475, 836640, 12397000, 1331330000, 143820000, 213051960000, 94724270640000, 3908675145375000, 104284286367187500, 43867845932728125000000, 12399293137277921875000
Offset: 1
Let pen(n) = n*(3*n-1)/2. Then
a(1) = pen(2) = 5.
a(2) = pen(82) = 10045 = 35 * 287 = pen(5) * pen(14).
a(3) = pen(117) = 20475 = 5 * 35 * 117 = pen(2) * pen(5) * pen(9).
a(4) = pen(747) = 836640 = 5 * 12 * 12 * 1162
= pen(2) * pen(3)^2 * pen(28).
a(5) = pen(2875) = 12397000 = pen(2) * pen(4) * pen(5)^2 * pen(8).
a(6) = pen(29792) = 1331330000 = pen(2)^2 * pen(5)^2 * pen(11) * pen(13).
a(7) = pen(9792) = 143820000 = pen(2)^4 * pen(3) * pen(6) * pen(16).
a(8) = pen(376875) = 213051960000
= pen(2)^4 * pen(3)^2 * pen(4) * pen(268).
a(9) = pen(7946667) = 94724270640000
= pen(2)^3 * pen(3)^3 * pen(6) * pen(10) * pen(199).
a(10)= pen(51046875) = 3908675145375000
= pen(2)^5 * pen(4) * pen(6) * pen(8) * pen(26) * pen(90).
a(11)= pen(263671875) = 104284286367187500
= pen(2)^7 * pen(7)^2 * pen(30) * pen(369). - _Donovan Johnson_, Jun 14 2012
A225070
Least decagonal (10-gonal) number that is the product of n decagonal numbers greater than 1.
Original entry on oeis.org
10, 69300, 9729720, 4257000, 967412160, 4104100000, 1408951239696000, 59860503846000000, 1542547619019487080000, 39054496014386012160000, 450510331438947780000000
Offset: 1
Let dec(n) = n*(4n-3). Then
a(1) = 10 = dec(2).
a(2) = 69300 = dec(132) = dec(2) * dec(42).
a(3) = 9729720 = dec(1560) = dec(3) * dec(4) * dec(42).
a(4) = 4257000 = dec(1032) = dec(2)^3 * dec(33).
a(5) = 967412160 = dec(15552) = dec(2) * dec(3) * dec(4) * dec(8) * dec(9).
a(6) = 4104100000 = dec(32032) = dec(2)^3 * dec(4) * dec(7) * dec(11).
Corrected a(6) and added a(7)-a(11) by
Lars Blomberg, Sep 20 2013
A225069
Least nonagonal (9-gonal) number that is the product of n nonagonal numbers greater than 1.
Original entry on oeis.org
9, 265926, 9909504, 28123200, 34171875, 9833523682950, 189619679700, 1489258878162739200, 32051313254079000000000, 231538926078057635957250, 5980078350588060426240000
Offset: 1
Let non(n) = n*(7n-5)/2. Then
a(1) = 9 = non(2).
a(2) = 265926 = non(276) = non(4) * non(41).
a(3) = 9909504 = non(1683) = non(3) * non(4) * non(51).
a(4) = 28123200 = non(2835) = non(3)^2 * non(5) * non(14).
a(5) = 34171875 = non(3125) = non(2)^2 * non(5)^3.
a(6) = 9833523682950 = non(1676180) = non(2)^3 * non(6) * non(55) * non(58).
Cf.
A001106 (9-gonal or nonagonal numbers).
Corrected a(4)-a(6) and added a(7)-a(11) by
Lars Blomberg, Sep 21 2013
A225067
Least heptagonal (7-gonal) number that is the product of n heptagonal numbers greater than 1.
Original entry on oeis.org
7, 6426, 35224, 2077992, 3610893055, 14209771072, 118896888880, 6400213601782, 22535310978496008, 22535310978496008, 2418562185097611420000, 2462278542548750181849600
Offset: 1
Let hep(n) = n*(5n-3)/2. Then
a(1) = 7 = hep(2).
a(2) = 6426 = hep(51) = hep(4) * hep(9).
a(3) = 35224 = hep(119) = hep(2) * hep(4) * hep(8).
a(4) = 2077992 = hep(912) = hep(2)^2 * hep(3) * hep(31).
a(5) = 3610893055 = hep(38005) = hep(2)^3 * hep(5) * hep(277).
a(6) = 14209771072 = hep(75392) = hep(2)^4 * hep(31) * hep(32).
Corrected a(6) and added a(7)-a(12) by
Lars Blomberg, Sep 21 2013
A225068
Least octagonal (8-gonal) number that is the product of n octagonal numbers greater than 1.
Original entry on oeis.org
8, 1408, 2165800, 37333296, 19384601600, 69370076160, 69370076160, 56288711711232000, 7917914554368000000, 199449790781142859776
Offset: 1
Let oct(n) = n*(3n-2). Then
a(1) = 8 = oct(2).
a(2) = 1408 = oct(22) = oct(2) * oct(8).
a(3) = 2165800 = oct(850) = oct(4) * oct(5) * oct(17).
a(4) = 37333296 = oct(3528) = oct(3)^2 * oct(8) * oct(13).
a(5) = 19384601600 = oct(80384) = oct(2)^2 * oct(5) * oct(14) * oct(53).
a(6) = 69370076160 = oct(152064) = oct(3)^3 * oct(4) * oct(7) * oct(22).
Corrected a(4) and added a(7)-a(10) by
Lars Blomberg, Sep 21 2013
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