A212616 -id:A212616 - cp's OEIS frontend

A212617 Least pentagonal number that is the product of n pentagonal numbers greater than 1.

5, 10045, 20475, 836640, 12397000, 1331330000, 143820000, 213051960000, 94724270640000, 3908675145375000, 104284286367187500, 43867845932728125000000, 12399293137277921875000

Offset: 1

T. D. Noe, Jun 12 2012

10^21 < a(12) <= pen(171012369792) = 43867845932728125000000 = pen(2)^9 * pen(32) * pen(132) * pen(19439). - Donovan Johnson, Jun 14 2012

			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

Lars Blomberg, Table of n, a(n) with solutions for n=1..13

Cf. A000326 (pentagonal numbers).

Cf. A212616, A225066-A225070 (5- to 10-gonal cases).

a(11) from Donovan Johnson, Jun 14 2012

a(12)-a(13) from Lars Blomberg, Sep 21 2013

A225066 Least hexagonal number that is the product of n hexagonal numbers greater than 1.

Original entry on oeis.org

6, 2850, 79800, 2162160, 134734320, 15518903400, 174626020800, 19794628854000, 659394533191680, 659394533191680, 659394533191680, 38022361747469489295360

Offset: 1

T. D. Noe, May 01 2013

			Let hex(n) = n*(2n-1). Then
a(1) = 6 = hex(2).
a(2) = 2850 = hex(38) = hex(3) * hex(10).
a(3) = 79800 = hex(200) = hex(3) * hex(4) * hex(10).
a(4) = 2162160 = hex(1040) = hex(2)^2 * hex(4) * hex(33).
a(5) = 134734320 = hex(8208) = hex(2)^2 * hex(3) * hex(4) * hex(67).
a(6) = 15518903400 = hex(88088) = hex(2) * hex(3) * hex(6)^2 * hex(7) * hex(15).

Lars Blomberg, Table of n, a(n) with solutions for n=1..12

Cf. A000384 (hexagonal numbers).

Cf. A212616, A212617, A225066-A225070 (3-, 5- to 10-gonal cases).

Corrected a(6) and added a(7)-a(12) by Lars Blomberg, Sep 21 2013

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

T. D. Noe, May 01 2013

			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).

Lars Blomberg, Table of n, a(n) with solutions for n=1..11

Cf. A001107 (decagonal numbers).

Cf. A212616, A212617, A225066-A225069 (3-, 5- to 9-gonal cases).

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

T. D. Noe, May 01 2013

			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).

Lars Blomberg, Table of n, a(n) with solutions for n=1..11

Cf. A001106 (9-gonal or nonagonal numbers).

Cf. A212616, A212617, A225066-A225070 (3-, 5- to 10-gonal cases).

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

T. D. Noe, May 01 2013

			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).

Lars Blomberg, Table of n, a(n) with solutions for n=1..12

Cf. A000566 (heptagonal numbers).

Cf. A212616, A212617, A225066-A225070 (3-, 5- to 10-gonal cases).

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

T. D. Noe, May 01 2013

			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).

Lars Blomberg, Table of n, a(n) with solutions for n=1..10

Cf. A000567 (octagonal numbers).

Cf. A212616, A212617, A225066-A225070 (3-, 5- to 10-gonal cases).

Corrected a(4) and added a(7)-a(10) by Lars Blomberg, Sep 21 2013

A296098 a(n) is the smallest triangular number (A000217) that can be represented as a product of k triangular numbers greater than 1 for all k = 1,2,...,n. a(n)=-1 if no such triangular number exists.

Original entry on oeis.org

3, 36, 630, 25200, 25200, 2821500, 55030954895280, 55030954895280, 55030954895280, 55030954895280, 55030954895280, 55030954895280

Offset: 1

Alex Ratushnyak, Dec 04 2017

From Jon E. Schoenfield, Apr 21 2018: (Start)
Of the 482 triangular numbers < 55030954895280 that can be represented as a product of seven triangular numbers greater than 1, the only one that can be represented as a product of two triangular numbers greater than 1 is 218434391280, which cannot be represented as a product of 3 triangular numbers greater than 1. Thus, a(n) >= 55030954895280 for all n >= 7.
However, 55030954895280 can be represented (see Example section) as a product of k triangular numbers greater than 1 for all k in 1,2,...,12 (but not for k=13), so a(7) = a(8) = ... = a(12) = 55030954895280 (and, for each n > 12, a(n) > 55030954895280, or a(n) = -1 if no such number exists).
If, for some integer N > 12, it could be proved that a(N) = -1, then it would also be established that a(n) = -1 for every n > N. (End)

			25200 is the smallest triangular number representable as a product of 2, 3 and 4 triangular numbers, 25200 = 210 * 120 = 120 * 21 * 10 = 28 * 15 * 10 * 6.
Therefore a(4)=25200.
Also, 25200 = 28 * 10 * 10 * 3 * 3, and therefore a(5)=25200.
From _Jon E. Schoenfield_, Apr 21 2018: (Start)
Let f(k_1, k_2, ..., k_m) = Product_{j=1..m} A000217(k_j) = Product_{j=1..m} (k_j*(k_j + 1)/2). Then, since no smaller number can be represented as a product of k triangular numbers greater than 1 for all k in 1,2,...,7,
a(7) = 55030954895280
     = f(10491039)
     = f(2261, 6560)
     = f(6, 493, 6560)
     = f(28, 39, 81, 323)
     = f(17, 18, 27, 40, 116)
     = f(4, 8, 17, 28, 38, 81)
     = f(2, 3, 17, 18, 26, 28, 40)
     = f(2, 2, 2, 2, 2, 17, 144, 532)
     = f(2, 2, 2, 2, 12, 17, 18, 28, 40)
     = f(2, 2, 2, 2, 2, 2, 3, 3, 40, 2261)
     = f(2, 2, 2, 2, 2, 2, 2, 2, 16, 29, 532)
     = f(2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 40, 493)
and a(7) = a(8) = a(9) = a(10) = a(11) = a(12).
(End)

Cf. A000217, A188630, A212616, A295769, A296097.

a(n) >= A212616(n) (unless a(n) = -1). - Jon E. Schoenfield, Apr 21 2018

a(7)-a(12) from Jon E. Schoenfield, Apr 21 2018

cp's OEIS Frontend

A212617 Least pentagonal number that is the product of n pentagonal numbers greater than 1.

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A225066 Least hexagonal number that is the product of n hexagonal numbers greater than 1.

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A225070 Least decagonal (10-gonal) number that is the product of n decagonal numbers greater than 1.

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A225069 Least nonagonal (9-gonal) number that is the product of n nonagonal numbers greater than 1.

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A225067 Least heptagonal (7-gonal) number that is the product of n heptagonal numbers greater than 1.

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A225068 Least octagonal (8-gonal) number that is the product of n octagonal numbers greater than 1.

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A296098 a(n) is the smallest triangular number (A000217) that can be represented as a product of k triangular numbers greater than 1 for all k = 1,2,...,n. a(n)=-1 if no such triangular number exists.

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