cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A136112 Indices of pentagonal numbers > 0 which are not the difference of two larger pentagonal numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 15, 18, 23, 24, 27, 51, 54, 71, 72, 81, 96, 99, 123, 128, 135, 162, 216, 239, 243, 263, 288, 303, 311, 359, 384, 423, 459, 479, 486, 519, 591, 599, 639, 648, 683, 699, 729, 743, 783, 863, 864, 879, 891, 911, 1031, 1103, 1151, 1215, 1431
Offset: 1

Views

Author

M. F. Hasler, Dec 15 2007

Keywords

Examples

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

Links

Crossrefs

Cf. A000326, A136113-A136118.

Programs

  • PARI
    P(n)=n*(3*n-1)>>1
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(i","))

Formula

A number m is in this sequence iff A136114(m) = 0 iff A136115(m) = 0

Extensions

Definition corrected, thanks to a remark from R. J. Mathar, Feb 01 2008
More terms from Donovan Johnson, Apr 22 2008

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

Views

Author

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

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Programs

  • 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))&j!=i&next(2)); print1(P(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 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

Views

Author

M. F. Hasler, Dec 15 2007

Keywords

Examples

			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).

Crossrefs

Cf. A000326, A136112-A136118.

Programs

  • PARI
    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","))

Formula

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

A135771 Terms in A136112 which are not in A135768.

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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)))}

Formula

Equals the difference set A136112 \ A135768.

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

Views

Author

M. F. Hasler, Dec 15 2007

Keywords

Examples

			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).

Crossrefs

Cf. A000326, A136112-A136118.

Programs

  • PARI
    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","))

Formula

a(n)=A000326(A136115(n)). a(n)=0 iff n is in A136112 iff A000326(n) is in A136113.
Showing 1-5 of 5 results.

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