A136112 -id:A136112 - cp's OEIS frontend

A135771 Terms in A136112 which are not in A135768.

5, 23, 51, 71, 72, 99, 123, 239, 263, 311, 359, 479, 599, 699, 743, 863, 911, 1031, 1103, 1151, 1431, 1563, 1583, 1823, 1851, 1863, 2111, 2543, 2663, 3023, 3119, 3191, 3291, 3671, 3719, 3863, 4131, 4203, 4271, 4463, 4671, 4703, 5039, 5231, 5351, 5391, 5399

Offset: 1

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

nonn

Pentagonal-Indices of terms in A136113 which are not in A135769.

A135768 resp. A135769 are subsequences of A136112 resp. A136113; the present sequence gives the indices of the elements of the former which are not in the latter: A136113(A135771(k)), k=1,2,3,... are the pentagonal numbers P(m) which are not the difference of two pentagonal numbers P(n)-P(q) with n,q>m, but only with n>m>q. A136112(A135771(k)) are the corresponding indices of the pentagonal numbers.

			The first terms of this sequence correspond to the following elements of A136113:
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_10.

Donovan Johnson, Table of n, a(n) for n = 1..200

Cf. A000326, A136112-A136118, A135768-A135769.

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

Equals the difference set A136112 \ A135768.

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

Original entry on oeis.org

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)))

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

Original entry on oeis.org

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

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

nonn

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.

			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

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

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

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 *)

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","))

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.

Extended by T. D. Noe, Dec 05 2011

A136113 Pentagonal numbers > 0 which are not the difference of two larger pentagonal numbers.

Original entry on oeis.org

1, 5, 12, 35, 51, 92, 117, 176, 330, 477, 782, 852, 1080, 3876, 4347, 7526, 7740, 9801, 13776, 14652, 22632, 24512, 27270, 39285, 69876, 85562, 88452, 103622, 124272, 137562, 144926, 193142, 220992, 268182, 315792, 343922, 354051, 403782, 523626

Offset: 1

M. F. Hasler, Dec 15 2007, Feb 07 2008

nonn

			a(1..3)=P(1),P(2),P(3) since these cannot be written as difference of 2 other pentagonal numbers > 0.
P(4)=22=P(8)-P(7), therefore P(4) is not in this sequence.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A000326, A136112-A136118.

P(n)=n*(3*n-1)>>1 /* a.k.a. A000326 */
isPent(t)=P(sqrtint((t<<1)\3)+1)==t
for( i=1,999,for( j=i+1,(P(i)-1)\3, isPent(P(i)+P(j))&next(2)); print1(P(i)","))

a(n)=A000326(A136112(n)). A number m is in this sequence iff A136114(m) = 0 iff A136115(m) = 0.

a(34)-a(39) from Donovan Johnson, Sep 05 2008

A136117 Pentagonal numbers (A000326) which are the sum of 2 other positive pentagonal numbers.

Original entry on oeis.org

70, 92, 852, 925, 1247, 1426, 1926, 2625, 3577, 5192, 6305, 6501, 7107, 7740, 7957, 8177, 8626, 9560, 10292, 12927, 13207, 14652, 15555, 16172, 18095, 20475, 20827, 21901, 22265, 22632, 23002, 23751, 24130, 28497, 29330, 31032, 33227, 33675

Offset: 1

M. F. Hasler, Dec 15 2007; corrected Dec 25 2007

nonn

It is conjectured that every integer and hence every pentagonal number, greater than 33066, hence greater than A000326(149) = 33227, can be represented as the sum of three pentagonal numbers. - Jonathan Vos Post, Dec 18 2007

			a(1)=70=P(7) is the least pentagonal number which can be written as sum of two other pentagonal numbers, P(7)=P(5)+P(5).

Cf. A000326, A136112-A136118, A007527.

P(n)=n*(3*n-1)>>1 /* a.k.a. A000326 */
isPent(t)=P(sqrtint(t<<1\3)+1)==t
for(i=1,299,for(j=1,(i+1)\sqrt(2),isPent(P(i)-P(j)) && print1(P(i)",") || next(2)))
/* The following is much faster, at the cost of implementing sum2sqr(), cf. A133388*/
A136117next(i)=i=sqrtint(i\3*2)*6+5; until(0, for(j=2,#t=sum2sqr((i+=6)^2+1), t[j]%6==[5,5] && break(2)));i^2\24
A136117vect(n,i)=vector(n,j,i=A136117next(i)) /* 2nd arg =0 by default but allows one to start elsewhere */
A136117(n,i)=until(!n--,i=A136117next(i));i \\ M. F. Hasler, Dec 25 2007

a(n) = A000326(A136116(n)) = A000326(m)+A136114(m) where m is the index of the n-th nonzero term in A136114 or A136115.

A135769 Pentagonal numbers > 0 which are not the difference of two other pentagonal numbers > 0.

Original entry on oeis.org

1, 5, 12, 51, 92, 117, 176, 330, 477, 852, 1080, 4347, 9801, 13776, 24512, 27270, 39285, 69876, 88452, 124272, 137562, 220992, 268182, 315792, 354051, 403782, 523626, 612162, 629532, 699392, 796797, 919242, 1119312, 1158522, 1190376

Offset: 1

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

nonn

A subsequence of A136113, obtained by omitting A136113(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 pentagonal number.

The definition ("...two other...") excludes the case P(n) = P(m)-P(n), cf. comment by R. J. Mathar in A000326.

			See A135768 for a list of P(n) which are in A136113 but not in A135769.

Donovan Johnson, Table of n, a(n) for n = 1..200

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

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))&j!=i&next(2)); print1(P(i)","))

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 writing P(n)=P(n')+P(n"): this is possible whenever (6n-1)^2=(6n'-1)^2+(6n"-1)^2-1.

A136115 Index m of least pentagonal number P(m) > P(n) such that P(m)+P(n) is again a pentagonal number, 0 if no such m exists.

Original entry on oeis.org

0, 0, 0, 7, 0, 0, 23, 0, 0, 48, 0, 22, 82, 47, 0, 125, 26, 0, 22, 37, 71, 238, 0, 0, 26, 166, 0, 52, 207, 147, 117, 99, 87, 572, 72, 67, 63, 357, 57, 110, 416, 51, 917, 82, 47, 1050, 217, 380, 167, 246, 0, 97, 697, 0, 374, 191, 537, 1672, 152, 112, 136, 380, 215, 2037, 68

Offset: 1

M. F. Hasler, Dec 15 2007

nonn

			a(1..3)=0 since P(1),P(2),P(3) cannot be written as difference of 2 other pentagonal numbers > 0.
a(4)=7 since P(7)=70 is the least pentagonal number > P(4)=22 such that their sum is again a pentagonal number, P(8).

Cf. A000326, A136112-A136118.

P(n)=n*(3*n-1)>>1 /* a.k.a. A000326 */ /* newline */ isPent(t)=P(sqrtint(t<<1\3)+1)==t /* newline */ for(i=1,99,for(j=i+1,(P(i)-1)\3,isPent(P(i)+P(j))&print1(j",")|next(2));print1(0","))

a(n)=0 iff n is in A136112 iff A000326(n) is in A136113.

A136114 Least pentagonal number P(m) > P(n) such that P(m)+P(n) is again a pentagonal number, 0 if no such m exists.

Original entry on oeis.org

0, 0, 0, 70, 0, 0, 782, 0, 0, 3432, 0, 715, 10045, 3290, 0, 23375, 1001, 0, 715, 2035, 7526, 84847, 0, 0, 1001, 41251, 0, 4030, 64170, 32340, 20475, 14652, 11310, 490490, 7740, 6700, 5922, 190995, 4845, 18095, 259376, 3876, 1260875, 10045, 3290

Offset: 1

M. F. Hasler, Dec 15 2007

nonn

			a(1..3)=0 since P(1),P(2),P(3) cannot be written as difference of 2 other pentagonal numbers > 0.
a(4)=70=P(7) is the least pentagonal number > P(4)=22 such that their sum is again a pentagonal number, P(8).

Cf. A000326, A136112-A136118.

P(n)=n*(3*n-1)>>1 /* a.k.a. A000326 */ /* newline */ isPent(t)=P(sqrtint(t<<1\3)+1)==t /* newline */ for( i=1,99,for( j=i+1,(P(i)-1)\3, isPent(P(i)+P(j))&print1(P(j)",")|next(2));print1(0","))

a(n)=A000326(A136115(n)). a(n)=0 iff n is in A136112 iff A000326(n) is in A136113.

A136116 Indices of pentagonal numbers (A000326) which are the sum of 2 other positive pentagonal numbers.

Original entry on oeis.org

7, 8, 24, 25, 29, 31, 36, 42, 49, 59, 65, 66, 69, 72, 73, 74, 76, 80, 83, 93, 94, 99, 102, 104, 110, 117, 118, 121, 122, 123, 124, 126, 127, 138, 140, 144, 149, 150, 152, 161, 163, 168, 169, 174, 175, 178, 181, 185, 188, 190, 195, 199, 203, 209, 210, 212, 213

Offset: 1

M. F. Hasler, Dec 15 2007; corrected Dec 25 2007

nonn

			a(1)=7 since P(7)=70 is the least pentagonal number which can be written as sum of two other pentagonal numbers, P(7)=P(5)+P(5).

Cf. A000326, A136112-A136118.

P(n)=n*(3*n-1)>>1 /* a.k.a. A000326 */
isPent(t)=P(sqrtint(t<<1\3)+1)==t
for(i=1,999,for(j=1,(i+1)\sqrt(2),isPent(P(i)-P(j))&print1(i",") || next(2)))
/* The following are much faster, at the cost of implementing sum2sqr(), cf. A133388. */
A136116next(i)=i=6*i-1;until(0,for(j=2,#t=sum2sqr((i+=6)^2+1),t[j]%6==[5,5] && break(2))); i\6+1
A136116vect(n,i=0)=vector(n,j,i=A136116next(i))
A136116(n,i=0)=until(!n--,i=A136116next(i));i \\ M. F. Hasler, Dec 25 2007

A000326(a(n))=A000326(m)+A136114(m) where m is the index of the n-th nonzero term in A136114 or A136115.

A137694 Numbers k such that 6k^2-2k = 3n^2-n for some integer n>0.

Original entry on oeis.org

5, 5577, 6435661, 7426747025, 8570459630997, 9890302987423321, 11413401077026881245, 13171054952586033533217, 15199386001883205670450981, 17540078275118266757666898665, 20241235130100477955141930608237, 23358367800057676441967030255006641

Offset: 1

M. F. Hasler, Feb 08 2008

Also indices of pentagonal numbers which are half of some other pentagonal number: see A137693 for more details, comments and links.

Colin Barker, Table of n, a(n) for n = 1..250
Dario Alpern, Quadratic two integer variable equation solver
Index entries for linear recurrences with constant coefficients, signature (1155,-1155,1).

Cf. A000326, A136112-A136118, A135768-A135769, A137693.

vector(20,i, (v=if(i>1,[577,408;816,577]*v-[164;232], [5;7]))[1,1])

a(n) = f^{2n-2}(5,7)[1], where f(x,y) = (577x + 408y - 164, 816x + 577y - 232).
a(n) = (5,7,1,5,7,1,...) (mod 10).
G.f.: -x*(5-198*x+x^2) / ( (x-1)*(x^2-1154*x+1) ). - R. J. Mathar, Apr 17 2011

cp's OEIS Frontend

A135771 Terms in A136112 which are not in A135768.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A136113 Pentagonal numbers > 0 which are not the difference of two larger pentagonal numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A136117 Pentagonal numbers (A000326) which are the sum of 2 other positive pentagonal numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A135769 Pentagonal numbers > 0 which are not the difference of two other pentagonal numbers > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A136115 Index m of least pentagonal number P(m) > P(n) such that P(m)+P(n) is again a pentagonal number, 0 if no such m exists.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A136114 Least pentagonal number P(m) > P(n) such that P(m)+P(n) is again a pentagonal number, 0 if no such m exists.

Views

Author

Keywords

Examples

Crossrefs