A135768 Indices of pentagonal numbers > 0 which are not the difference of 2 other pentagonal numbers > 0.

1, 2, 3, 6, 8, 9, 11, 15, 18, 24, 27, 54, 81, 96, 128, 135, 162, 216, 243, 288, 303, 384, 423, 459, 486, 519, 591, 639, 648, 683, 729, 783, 864, 879, 891, 1215, 1458, 1719, 1944, 2031, 2043, 2048, 2151, 2187, 2463, 2799, 3231, 3456, 3543, 3879, 3903, 4023

Offset: 1

Author

R. J. Mathar and M. F. Hasler, Feb 07 2008

Keywords

• nonn

Comments

A subsequence of A136112, obtained by omitting A136112(A135771(k)), k=1,2,3,... ; i.e. those which are not the difference of two larger pentagonal numbers, but the difference of a larger and a smaller (or equal) pentagonal number. Sequence A135769 has the pentagonal numbers corresponding to these indices.

Examples

			Indices of the following numbers are not here but in A136112:
P_5 = P_7 - P_5
P_23 = P_24 - P_7
P_51 = P_66 - P_42
P_71 = P_74 - P_21
P_72 = P_80 - P_35
P_99 = P_104 - P_32
P_123 = P_144 - P_75
P_239 = P_249 - P_70
P_263 = P_274 - P_77
P_311 = P_324 - P_91
P_359 = P_374 - P_105
		

Links

• Donovan Johnson and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n = 1..200 from Donovan Johnson

Crossrefs

Cf. A000326, A136112-A136118, A135769(n) = A000326(a(n)), A135771 = A136112 \ A135768.

Programs

Mathematica

Select[Range[100], Reduce[# (3 # - 1) == x (3 x - 1) - y (3 y - 1) && x > 0 && y > 0, {x, y}, Integers] == False &] (* T. D. Noe, Dec 05 2011 *)

PARI

P(n)=n*(3*n-1)/2
isPent(t)=P(sqrtint((t*2)\3)+1)==t
for( i=1,999,for( j=1,(P(i)-1)\3, isPent(P(i)+P(j))&next(2)); print1(i","))

Formula

P(n)=n*(3*n-1)/2 <=> n*(n-1/3) = (2/3)*P(n), thus m = P(n) <=> m = P([sqrt(2m/3)]+1) and m = P(n) <=> 24m+1 = (6n-1)^2, useful for investigating the possibility of write P(n)=P(n')+P(n"): this is possible whenever (6n-1)^2 = (6n'-1)^2 + (6n"-1)^2.

Extensions

Extended by T. D. Noe, Dec 05 2011