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 1-4 of 4 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

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

Views

Author

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

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000326, A136112-A136118.

Programs

  • PARI
    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)","))

Formula

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

Extensions

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

Views

Author

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

Keywords

Comments

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

Examples

			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).

Crossrefs

Cf. A000326, A136112-A136118, A007527.

Programs

  • PARI
    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

Formula

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

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

Views

Author

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

Keywords

Examples

			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).

Crossrefs

Cf. A000326, A136112-A136118.

Programs

  • PARI
    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

Formula

A000326(a(n))=A000326(m)+A136114(m) where m is the index of the n-th nonzero term in A136114 or A136115.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.