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 1-3 of 3 results.

A138416 a(n) = (p^3 - p^2)/2, where p = prime(n).

Original entry on oeis.org

2, 9, 50, 147, 605, 1014, 2312, 3249, 5819, 11774, 14415, 24642, 33620, 38829, 50807, 73034, 100949, 111630, 148137, 176435, 191844, 243399, 282449, 348524, 451632, 510050, 541059, 606797, 641574, 715064, 1016127, 1115465, 1276292, 1333149
Offset: 1

Views

Author

Artur Jasinski, Mar 19 2008

Keywords

Comments

Differences (p^k - p^m)/q with k > m:
expression OEIS sequence
-------------- -------------
p^2 - p A036689
(p^2 - p)/2 A008837
p^3 - p A127917
(p^3 - p)/2 A127918
(p^3 - p)/3 A127919
(p^3 - p)/6 A127920
p^3 - p^2 A135177
(p^3 - p^2)/2 this sequence
p^4 - p A138401
(p^4 - p)/2 A138417
p^4 - p^2 A138402
(p^4 - p^2)/2 A138418
(p^4 - p^2)/3 A138419
(p^4 - p^2)/4 A138420
(p^4 - p^2)/6 A138421
(p^4 - p^2)/12 A138422
p^4 - p^3 A138403
(p^4 - p^3)/2 A138423
p^5 - p A138404
(p^5 - p)/2 A138424
(p^5 - p)/3 A138425
(p^5 - p)/5 A138426
(p^5 - p)/6 A138427
(p^5 - p)/10 A138428
(p^5 - p)/15 A138429
(p^5 - p)/30 A138430
p^5 - p^2 A138405
(p^5 - p^2)/2 A138431
p^5 - p^3 A138406
(p^5 - p^3)/2 A138432
(p^5 - p^3)/3 A138433
(p^5 - p^3)/4 A138434
(p^5 - p^3)/6 A138435
(p^5 - p^3)/8 A138436
(p^5 - p^3)/12 A138437
(p^5 - p^3)/24 A138438
p^5 - p^4 A138407
(p^5 - p^4)/2 A138439
p^6 - p A138408
(p^6 - p)/2 A138440
p^6 - p^2 A138409
(p^6 - p^2)/2 A138441
(p^6 - p^2)/3 A138442
(p^6 - p^2)/4 A138443
(p^6 - p^2)/5 A138444
(p^6 - p^2)/6 A138445
(p^6 - p^2)/10 A138446
(p^6 - p^2)/12 A138447
(p^6 - p^2)/15 A138448
(p^6 - p^2)/20 A122220
(p^6 - p^2)/30 A138450
(p^6 - p^2)/60 A138451
p^6 - p^3 A138410
(p^6 - p^3)/2 A138452
p^6 - p^4 A138411
(p^6 - p^4)/2 A138453
(p^6 - p^4)/3 A138454
(p^6 - p^4)/4 A138455
(p^6 - p^4)/6 A138456
(p^6 - p^4)/8 A138457
(p^6 - p^4)/12 A138458
(p^6 - p^4)/24 A138459
p^6 - p^5 A138412
(p^6 - p^5)/2 A138460
.
We can prove that for n>1, a(n) is the remainder of the Euclidean division of Sum_{k=0..p-1} k^p by p^3 where p = prime(n). - Pierre Vandaële, Nov 30 2024

Links

Programs

  • Magma
    [(p^3-p^2)/2: p in PrimesUpTo(1000)]; // Vincenzo Librandi, Jun 17 2011
  • Mathematica
    a = {}; Do[p = Prime[n]; AppendTo[a, (p^3 - p^2)/2], {n, 1, 50}]; a
(#^3-#^2)/2&/@Prime[Range[50]] (* Harvey P. Dale, Nov 01 2020 *)
  • PARI
    forprime(p=2,1e3,print1((p^3-p^2)/2", ")) \\ Charles R Greathouse IV, Jun 16 2011

Extensions

Definition corrected by T. D. Noe, Aug 25 2008

A138459 a(n) = ((n-th prime)^6-(n-th prime)^4)/12.

Original entry on oeis.org

4, 54, 1250, 9604, 146410, 399854, 2004504, 3909630, 12313004, 49509670, 73881680, 213654354, 395606540, 526495354, 897861304, 1846372554, 3514034690, 4292210710, 7536519254, 10672906020, 12608819004, 20254042120, 27241076254
Offset: 1

Views

Author

Artur Jasinski, Mar 22 2008

Keywords

Comments

Differences (p^k-p^m)/q such that k > m:
p^2-p is given in A036689
(p^2-p)/2 is given in A008837
p^3-p is given in A127917
(p^3-p)/2 is given in A127918
(p^3-p)/3 is given in A127919
(p^3-p)/6 is given in A127920
p^3-p^2 is given in A135177
(p^3-p^2)/2 is given in A138416
p^4-p is given in A138401
(p^4-p)/2 is given in A138417
p^4-p^2 is given in A138402
(p^4-p^2)/2 is given in A138418
(p^4-p^2)/3 is given in A138419
(p^4-p^2)/4 is given in A138420
(p^4-p^2)/6 is given in A138421
(p^4-p^2)/12 is given in A138422
p^4-p^3 is given in A138403
(p^4-p^3)/2 is given in A138423
p^5-p is given in A138404
(p^5-p)/2 is given in A138424
(p^5-p)/3 is given in A138425
(p^5-p)/5 is given in A138426
(p^5-p)/6 is given in A138427
(p^5-p)/10 is given in A138428
(p^5-p)/15 is given in A138429
(p^5-p)/30 is given in A138430
p^5-p^2 is given in A138405
(p^5-p^2)/2 is given in A138431
p^5-p^3 is given in A138406
(p^5-p^3)/2 is given in A138432
(p^5-p^3)/3 is given in A138433
(p^5-p^3)/4 is given in A138434
(p^5-p^3)/6 is given in A138435
(p^5-p^3)/8 is given in A138436
(p^5-p^3)/12 is given in A138437
(p^5-p^3)/24 is given in A138438
p^5-p^4 is given in A138407
(p^5-p^4)/2 is given in A138439
p^6-p is given in A138408
(p^6-p)/2 is given in A138440
p^6-p^2 is given in A138409
(p^6-p^2)/2 is given in A138441
(p^6-p^2)/3 is given in A138442
(p^6-p^2)/4 is given in A138443
(p^6-p^2)/5 is given in A138444
(p^6-p^2)/6 is given in A138445
(p^6-p^2)/10 is given in A138446
(p^6-p^2)/12 is given in A138447
(p^6-p^2)/15 is given in A138448
(p^6-p^2)/20 is given in A122220
(p^6-p^2)/30 is given in A138450
(p^6-p^2)/60 is given in A138451
p^6-p^3 is given in A138410
(p^6-p^3)/2 is given in A138452
p^6-p^4 is given in A138411
(p^6-p^4)/2 is given in A138453
(p^6-p^4)/3 is given in A138454
(p^6-p^4)/4 is given in A138455
(p^6-p^4)/6 is given in A138456
(p^6-p^4)/8 is given in A138457
(p^6-p^4)/12 is given in A138458
(p^6-p^4)/24 is given in A138459
p^6-p^5 is given in A138412
(p^6-p^5)/2 is given in A138460

Programs

  • Mathematica
    a = {}; Do[p = Prime[n]; AppendTo[a, (p^6 - p^4)/12], {n, 1, 24}]; a
  • PARI
    forprime(p=2,1e3,print1((p^6-p^4)/12", ")) \\ Charles R Greathouse IV, Jul 15 2011

A138562 Number of "squashed-tree" graphs with n central nodes, the labeled case, allowing the direct link between L and R.

Original entry on oeis.org

1, 4, 38, 616, 14744, 479364, 20021768, 1031673164, 63597989864, 4579513525216, 377953469391584, 35211153592004064, 3657198048669038384, 419166387797337858500, 52561549979435515611488, 7158828855330149502246076, 1052478318277669232896492064, 166132533639153074372662711680
Offset: 0

Views

Author

Nadia Heninger and N. J. A. Sloane, May 10 2008

Keywords

Comments

These are simple connected graphs with n+2 nodes labeled L, R, 1, 2, ..., n. The subgraph on nodes 1..n is a forest (no loops). Nodes L and R are both connected to some subset of 1..n and perhaps to each other.
These are the graphs that can arise when one starts with a tree with m >= n+2 labeled nodes, some of which are colored blue, some are colored red and the remaining n nodes are uncolored. Then all the blue nodes are coalesced into a single node L and all the red nodes into a single node R.

Examples

			a(0) = 1: L--R.
a(1) = 4: L--1--R, 1--L--R, L--R--1 and the 3-cycle L--1--R--L.
a(2) = 38: the 14 examples shown in A138460 plus the same set with an edge joining L and R: 28 in all, plus the following 10 graphs, for a total of 38.
=====
. 1
./..
L---R (number = 2)
.\..
. 2
=====
. 1
./..
L---R (number = 2)
.../
. 2
=====
. 1
./|.
L-|-R (number = 2)
.\|.
. 2
=====
. 1
./|.
L-|-R (number = 4)
..|.
. 2
=====

Crossrefs

Cf. A138560. A001187(n+2) is an upper bound.

Programs

  • PARI
    { a(n) = local(p,q,m); p=partitions(n); sum(j=1,#p, q=p[j]; m=vector(n); for(i=1,#q,m[q[i]]++); n! * prod(i=1,#q,q[i]^(q[i]-2)/q[i]!) / prod(i=1,#m,m[i]!) * (prod(i=1,#q,4^q[i]-1)*2 - 2^#q*prod(i=1,#q,2^q[i]-1) ) ) } \\ Max Alekseyev, May 10 2009

Formula

Although we have not written out all the details of the proof, it appears that a(n) ~ 2^n*n^(n-2).

Extensions

Edited and extended by Max Alekseyev, May 10 2009
Showing 1-3 of 3 results.

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

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