A136117 -id:A136117 - cp's OEIS frontend

A136118 Least index m>0 such that A136117(n)-A000326(m) is again a pentagonal number.

5, 4, 7, 12, 19, 17, 25, 20, 10, 28, 45, 42, 39, 17, 37, 21, 36, 35, 13, 33, 65, 28, 67, 32, 52, 40, 74, 31, 70, 85, 35, 16, 60, 70, 77, 68, 42, 30, 105, 76, 59, 26, 74, 49, 115, 19, 125, 115, 102, 110, 92, 56, 103, 29, 145, 100, 114, 77, 92, 47, 63, 108, 152, 95, 22, 116

Offset: 1

M. F. Hasler, Dec 25 2007

nonn

			a(1)=5 is the least integer m>0 such that A136117(1)-P(m) is a pentagonal number, namely P(7)-P(5)=70-35=35=P(5).
a(2)=4 is the least integer m>0 such that A136117(2)-P(m) is a pentagonal number, namely P(8)-P(4)=92-22=70=P(7).

Cf. A000326, A136112-A136117.

A136118vect(n,i=-1)=vector(n,k,until(0,for(j=2,#n=sum2sqr((i+=6)^2+1),n[j]%6==[5,5]||next;n=n[j];break(2)));n[1]\6+1) /* This uses sum2sqr(), cf. A133388. Below some simpler but much slower code. */
my(P=A000326(n)=n*(3*n-1)/2,isPent(t)=P(sqrtint(t*2\3)+1)==t); for(i=1,299,for(j=1,(i+1)\sqrt(2),isPent(P(i)-P(j))&print1(j",")||next(2)))

A133215 Hexagonal numbers (A000384) which are sum of 2 other hexagonal numbers > 0.

Original entry on oeis.org

276, 703, 861, 1225, 2850, 3003, 4560, 5151, 8128, 10878, 11781, 12090, 12720, 13366, 14706, 15400, 16110, 18721, 21115, 22366, 24090, 24531, 26796, 29161, 29646, 31125, 32131, 33153, 36315, 38503, 39621, 40186, 42486, 45451, 47895

Offset: 1

Jonathan Vos Post, Dec 18 2007

nonn

This is to A136117 as A000384 is to A000326. Duke and Schulze-Pillot (1990) proved that every sufficiently large integer (and hence every sufficiently large hexagonal number) can be written as the sum of three hexagonal numbers.

			hex(19) = 703 = 378 + 325 = hex(14) + hex(13).
hex(21) = 861 = 630 + 231 = hex(18) + hex(11).
hex(25) = 1225 = 1035 + 190 = hex(23) + hex(10).
hex(38) = 2850 = 2415 + 435 = hex(35) + hex(15).
hex(39) = 3003 = 2850 + 153 = hex(38) + hex(9) = 2415 + 435 + 153 = hex(35) + hex(15) + hex(9).
hex(48) = 4560 = 2415 + 2145 = hex(35) + hex(33).

Donovan Johnson, Table of n, a(n) for n = 1..1000
W. Duke and R. Schulze-Pilot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99(1990), 49-57.
Eric Weisstein's World of Mathematics, Hexagonal Number.

Cf. A000384, A136117.

With[{upto=60000},Select[Union[Total/@Subsets[Table[n(2n-1),{n, Ceiling[ (1+Sqrt[1+8upto])/4]}],{2}]],IntegerQ[(1+Sqrt[1+8#])/4]&&#<=upto&]] (* Harvey P. Dale, Jul 24 2011 *)

{x: x>0 and x in A000384 and x = A000384(i) + A000384(j) for i>0 and j>0}, where A000384 = {n*(2*n-1) for n > 0}.

Added missing term 276 and a(8)-a(35) from Donovan Johnson, Sep 27 2008

A136346 Octagonal numbers which are the sums of exactly two positive octagonal numbers.

Original entry on oeis.org

560, 736, 1541, 3201, 5461, 6816, 7400, 9976, 11041, 11408, 13333, 14981, 15408, 15841, 19521, 21000, 21505, 25761, 28616, 30401, 41536, 45141, 50440, 51221, 52008, 54405, 56856, 61920, 63656, 65416, 69008, 75525, 76480, 81345, 82336, 85345, 87381, 89441

Offset: 1

Jonathan Vos Post, Dec 25 2007

For sums of two positive octagonal numbers, see A136345. This is to octagonal numbers A000567 as A089982 is to triangular numbers A000217, as A009000 is to squares A000290, as A136117 is to pentagonal numbers A000326, as A133215 is to hexagonal numbers A000384, and as A117104 is to heptagonal numbers A000566. If Oc(a) + Oc(b) = Oc(c) then a(3a-2) + b(3b+2) = c(3c+2), so solving the quadratic equations for c we have (when an integer): c = (2 + sqrt(4 + 36a^2 + 36b^2 - 24a - 24b))/6.

			Where Oc(n) = A000567(n) = n-th octagonal number:
a(1) = 560 = Oc(14) = 280 + 280 = Oc(10) + Oc(10).
a(2) = 736 = Oc(16) = 560 + 176 = Oc(14) + Oc(8).
a(3) = 1541 = Oc(23) = 1408 + 133 = Oc(22) + Oc(7).
a(4) = 3201 = Oc(33) = 2465 + 736 = Oc(29) + Oc(16).
a(5) = 5461 = Oc(43) = 2821 + 2640 = Oc(31) + Oc(30).

B. D. Swan, Table of n, a(n) for n = 1..1800
Eric Weisstein's World of Mathematics, Octagonal Number

Cf. A000567, A009000, A089982, A136117, A136345.

Module[{nn=300,ono},ono=PolygonalNumber[8,Range[nn]];Union[Select[ Total/@ Tuples[ono,2],MemberQ[ono,#]&]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 26 2019 *)

A000567 INTERSECTION {A000567(i) + A000567(j), i, j > 0}. {i*(3*i-2)} INTERSECTION {i*(3*i-2) + j(3*j-2), i > 0}.

Corrected and edited by B. D. Swan (bdswan(AT)gmail.com), Dec 20 2008

A136345 Sums of exactly two positive octagonal numbers A000567.

Original entry on oeis.org

2, 9, 16, 22, 29, 41, 42, 48, 61, 66, 73, 80, 86, 97, 104, 105, 117, 130, 134, 136, 141, 154, 161, 173, 177, 184, 192, 197, 198, 216, 226, 229, 233, 241, 246, 265, 266, 272, 281, 288, 290, 301, 309, 320, 321, 342, 345, 349, 352, 358, 362, 376, 381, 401, 406, 409

Offset: 1

Jonathan Vos Post, Dec 25 2007

For octagonal numbers within this sequence, see: A136346. This is to octagonal numbers A000567 as A051533 is to triangular numbers A000217 and as A000404 is to squares A000290 and as A117104 is to heptagonal numbers A000566.

Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000567, A136117, A136346.

{A000567(i) + A000567(j), i, j > 0}. {i*(3*i-2) + j(3*j-2), i, j > 0}.

Missing term (80) from Giovanni Resta, Jun 19 2016

A133251 Heptagonal numbers A000566 which are the sum of two other heptagonal numbers > 0.

Original entry on oeis.org

697, 3186, 3744, 5221, 7209, 8323, 12496, 12852, 19228, 20566, 21022, 24850, 29539, 35224, 38254, 40768, 44023, 44689, 52345, 53802, 58293, 62173, 63760, 66178, 67815, 78057, 79834, 80730, 82537, 95746, 97713, 101707, 115240, 131905, 135373

Offset: 1

Jonathan Vos Post, Dec 19 2007

nonn

This is to A000566 as A136117 is to A000326.

The sequence contains 12852 and 19751431167846, which are the smallest heptagonal numbers equal to twice another heptagonal number. - R. J. Mathar, Jan 13 2008

			Where hep(k) = k-th heptagonal number = A000566(k):
a(1) = 697 = hep(17) = 616 + 81 = hep(16) + hep(6).
a(2) = 3186 = hep(36) = 1782 + 1404 = hep(27) + hep(24).
a(3) = 3744 = hep(39) = 2673 + 1071 = hep(33) + hep(21).
a(4) = 5221 = hep(46) = 4347 + 874 = hep(42) + hep(19).

Eric Weisstein's World of Mathematics, Heptagonal Number.

Cf. A000566, A136117.

Module[{nn=1000,heps},heps=Table[(n(5n-3))/2,{n,nn}]; Select[ Union[ Total/@ Tuples[Take[heps,nn/2],2]],MemberQ[heps,#]&]] (* Harvey P. Dale, Dec 18 2015 *)

{x such that x in A000566 and x = A000566(i) + A000566(j) for i, j > 0 and where A000566(k) = k*(5*k-3)/2}.

More terms from R. J. Mathar, Jan 13 2008

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A136118 Least index m>0 such that A136117(n)-A000326(m) is again a pentagonal number.

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A133215 Hexagonal numbers (A000384) which are sum of 2 other hexagonal numbers > 0.

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A136346 Octagonal numbers which are the sums of exactly two positive octagonal numbers.

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A136345 Sums of exactly two positive octagonal numbers A000567.

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A133251 Heptagonal numbers A000566 which are the sum of two other heptagonal numbers > 0.

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