A020342 -id:A020342 - cp's OEIS frontend

A124100 Sum_(x^iy^jz^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (8, 15, 17).

1, 40, 1089, 25160, 531521, 10625640, 204744769, 3844391560, 70827391041, 1286290883240, 23101397290049, 411249127989960, 7269184506192961, 127745926316548840, 2234231991096868929, 38920247688751940360

Offset: 0

Giorgio Balzarotti and Paolo P. Lava, Nov 26 2006

nonn

			a(2) = 1089 because x^2 + y^2 + z^2 + x*y + x*z + y*z = 8^2 + 15^2 + 17^2 + 8*15 + 8*17 + 15*17 = 1089 and x^2 + y^2 = z^2.

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 196.

Harvey P. Dale, Table of n, a(n) for n = 0..811
Index entries for linear recurrences with constant coefficients, signature (40, -511, 2040).

Cf. A019682, A020000, A020340-A020342, A020344-A020346, A021664, A021684, A021844, A025942, A077515.

seq(sum(8^(m-n)*sum(15^p*17^(n-p),p=0..n),n=0..m),m=0..N);

LinearRecurrence[{40,-511,2040},{1,40,1089},30] (* Harvey P. Dale, May 25 2025 *)

a(m) = (x^(m+2)*(z-y) + y^(m+2)*(x-z) + z^(m+2)*(y-x))/((x-y)*(y-z)*(z-x)).
From Chai Wah Wu, Sep 24 2016: (Start)
a(n) = 40*a(n-1) - 511*a(n-2) + 2040*a(n-3) for n > 2.
G.f.: 1/((1 - 8*x)*(1 - 15*x)*(1 - 17*x)). (End)
a(n) = 2^(3*n+6)/63 - 15^(n+2)/14 + 17^(n+2)/18. - Vaclav Kotesovec, Sep 25 2016

A331401 Table of distinct triples (A,B,C) such that A = B * C with B < C and A's digits being distinct and split between B and C.

Original entry on oeis.org

126, 6, 21, 153, 3, 51, 1206, 6, 201, 1260, 6, 210, 1260, 21, 60, 1395, 15, 93, 1435, 35, 41, 1503, 3, 501, 1530, 3, 510, 1530, 30, 51, 1827, 21, 87, 2187, 27, 81, 3159, 9, 351, 3784, 8, 473, 10426, 26, 401, 12384, 3, 4128, 12546, 51, 246, 12843, 3, 4281, 12964, 14, 926, 13950, 15, 930

Offset: 1

Gilles Esposito-Farèse and Eric Angelini, Jan 16 2020

The sequence is finite; it has 23425 triples (A,B,C) and thus 70275 terms. The last triple is (8410593762,9654,871203).

			The first triple is (126,6,21) and we see that 126 = 6 * 21, the digits of 126 being distinct and split between 6 and 21;
the second triple is (153,3,51) and we see that 153 = 3 * 51, the digits of 153 being distinct and split between 3 and 51;
the third triple is (1206,6,201) and we see that 1206 = 6 * 201, the digits of 1206 being distinct and split between 6 and 201.
...
The last triple is (8410593762,9654,871203): we see that 8410593762 = 9654 * 871203, the digits of 8410593762 being distinct and split between 9654 and 871203.

Gilles Esposito-Farèse, Table of n, a(n) for n = 1..50000

Cf. A020342 (Vampire numbers, definition 1).

A094128 Numbers n that can be factorized n=x*y and containing in their ternary representation the same digits the same number of times as x and y together.

Original entry on oeis.org

44, 116, 128, 132, 152, 296, 320, 332, 344, 348, 380, 384, 396, 440, 452, 456, 464, 476, 800, 872, 880, 888, 944, 960, 980, 992, 996, 1024, 1028, 1032, 1044, 1136, 1140, 1152, 1184, 1188, 1196, 1232, 1280, 1304, 1316, 1320, 1352, 1356, 1368

Offset: 1

Reinhard Zumkeller, Jun 01 2004

			320=16*20, 320->'102212' with 1x'0', 2x'1' and 3x'2': 16->'121' and 20->'202' have together the same ternary digits the same number of times, therefore 320 is a term.

J. K. Andersen, Vampire Numbers
Eric Weisstein's World of Mathematics, Vampire Number
Eric Weisstein's World of Mathematics, Ternary

Cf. A020342, A048936, A007089.

A094208 Least vampire number with n fang pairs.

Original entry on oeis.org

1260, 125460, 13078260, 16758243290880, 24959017348650

Offset: 1

Lekraj Beedassy, May 27 2004

			a(2) is the smallest number which can be written in 2 ways as the product of two numbers with half as many digits and not both ending in 0: 125460 = 204*615 = 246*510. - _Jens Kruse Andersen_, Jun 14 2014

J. K. Andersen, Vampire Numbers

Cf. A020342, A014575, A048936, A048937.

Typo in Cf. to A014575 fixed by Jens Kruse Andersen, Jun 14 2014

cp's OEIS Frontend

A124100 Sum_(x^iy^jz^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (8, 15, 17).

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A331401 Table of distinct triples (A,B,C) such that A = B * C with B < C and A's digits being distinct and split between B and C.

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A094128 Numbers n that can be factorized n=x*y and containing in their ternary representation the same digits the same number of times as x and y together.

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Author

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A094208 Least vampire number with n fang pairs.

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cp's OEIS Frontend

A124100 Sum_(x^i*y^j*z^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (8, 15, 17).

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A331401 Table of distinct triples (A,B,C) such that A = B * C with B < C and A's digits being distinct and split between B and C.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A094128 Numbers n that can be factorized n=x*y and containing in their ternary representation the same digits the same number of times as x and y together.

Views

Author

Keywords

Examples

Links

Crossrefs

A094208 Least vampire number with n fang pairs.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A124100 Sum_(x^iy^jz^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (8, 15, 17).