A007818 -id:A007818 - cp's OEIS frontend

A132775 A007818 * A132774.

1, 3, 3, 5, 10, 5, 7, 21, 21, 7, 9, 36, 54, 36, 9, 11, 55, 110, 110, 55, 11, 13, 78, 195, 260, 195, 78, 13, 15, 105, 315, 525, 525, 315, 105, 15, 17, 136, 476, 952, 1190, 952, 476, 136, 17, 19, 171, 684, 1596, 2394, 2394, 1596, 684, 171, 19

Offset: 0

Gary W. Adamson, Aug 29 2007

Row sums = A014480: (1, 6, 20, 56, 144, 352, 832, ...).

			First few rows of the triangle:
   1;
   3,   3;
   5,  10,   5;
   7,  21,  21,   7;
   9,  36,  54,  36,   9;
  11,  55, 110, 110,  55,  11;
  13,  78, 195, 260, 195,  78,  13;
  15, 105, 315, 525, 525, 315, 105, 15;
  ...
Row 3 = (7, 21, 21, 7) = 7 * (1, 3, 3, 1).

Cf. A132774, A014480.

Binomial transform of A132774. T(n,k) = (2n+1) * A007318(n,k).

A038119 Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).

Original entry on oeis.org

1, 1, 2, 7, 23, 112, 607, 3811, 25413, 178083, 1279537, 9371094, 69513546, 520878101, 3934285874, 29915913663, 228779330204, 1758309223457, 13573319825615, 105192814197984, 818136047201932, 6383528588447574

Offset: 1

N. J. A. Sloane

a(1)-a(12) computed by Achim Flammenkamp.
A000162 but with one copy of each mirror-image deleted.
From R. J. Mathar, Mar 19 2018: (Start)
We can split the numbers into an irregular table which lists in row n how many configurations have c contacts for c >= 0:
  1;
  0 1;
  0 0 2;
  0 0 0 6 1;
  0 0 0 0 21 2;
  0 0 0 0 0 91 19 2;
  0 0 0 0 0 0 484 110 12 1;
  0 0 0 0 0 0 0 2817 852 129 12 0 1;
  0 0 0 0 0 0 0 0 17788 6321 1166 132 5 1;
Row lengths are 1+A007818(n). Row sums are a(n).
(End)
Number of unoriented polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}. For unoriented polyominoes, chiral pairs are counted as one.- Robert A. Russell, Mar 21 2024

S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition (Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972. [See https://books.google.nl/books?id=ja7iBQAAQBAJ&pg=PA101]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Achim Flammenkamp, Home page.
Kevin L. Gong, Polyominoes Home Page.
John Mason, Counting free polycubes.
John Mason, Coordinate sets of examples of polycube symmetry (version 2)

A038119 = (A007743+A000162)/2, A007743 = 2*A038119 - A000162, A000162 = 2*A038119 - A007743.
32nd row of A366766.
Cf. A000162 (oriented), A371397 (chiral), A007743 (achiral), A001931 (fixed).
Cf. for each symmetry: A376964, A376965, A376966, A376967, A376968, A376969, A376970, A376972, A376973, A376974, A376975, A376976, A376977, A376978, A376979, A376980, A376981, A376982, A376983, A377127, A376984, A376985, A376986, A376987, A376988, A376989, A377128, A376990, A376991, A377129, A377130, A377131, A376971

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000162 = A@000162;
A007743 = A@007743;
a[n_] := (A007743[[n]] + A000162[[n]])/2;
a /@ Range[16] (* Jean-François Alcover, Jan 16 2020 *)

a(n) = A000162(n) - A371397(n) = A371397(n) + A007743(n). - Robert A. Russell, Mar 21 2024

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Jan 02 2002
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
More terms from John Mason, Sep 19 2024

A070804 Number of primes not exceeding phi(n).

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 2, 3, 2, 4, 2, 5, 3, 4, 4, 6, 3, 7, 4, 5, 4, 8, 4, 8, 5, 7, 5, 9, 4, 10, 6, 8, 6, 9, 5, 11, 7, 9, 6, 12, 5, 13, 8, 9, 8, 14, 6, 13, 8, 11, 9, 15, 7, 12, 9, 11, 9, 16, 6, 17, 10, 11, 11, 15, 8, 18, 11, 14, 9, 19, 9, 20, 11, 12, 11, 17, 9, 21, 11, 16, 12, 22, 9, 18, 13, 16

Offset: 1

Labos Elemer, May 08 2002

			n=50: phi[50]=20,Pi[20]=8=a(50)

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A007818, A000010, A000720, A070800, A070801, A070802, A070803.

[#PrimesUpTo(EulerPhi(n)): n in [1..90]]; // Vincenzo Librandi, Mar 26 2017

Table[PrimePi[EulerPhi[n]], {n, 1, 256}]

a(n) = A000720(A000010(n)) = pi(phi(n)).

A070803 Number of primes not exceeding sum of divisors of n.

Original entry on oeis.org

0, 2, 2, 4, 3, 5, 4, 6, 6, 7, 5, 9, 6, 9, 9, 11, 7, 12, 8, 13, 11, 11, 9, 17, 11, 13, 12, 16, 10, 20, 11, 18, 15, 16, 15, 24, 12, 17, 16, 24, 13, 24, 14, 23, 21, 20, 15, 30, 16, 24, 20, 25, 16, 30, 20, 30, 22, 24, 17, 39, 18, 24, 27, 31, 23, 34, 19, 30, 24, 34, 20, 44, 21, 30, 30

Offset: 1

Labos Elemer, May 08 2002

			n=50: sigma(50) = 93, pi(93) = 24 = a(50).

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A007818, A000203, A000720, A070800-A070804.

[#PrimesUpTo(SumOfDivisors(n)): n in [1..100]]; // Vincenzo Librandi, Feb 06 2017

Table[PrimePi[DivisorSigma[1, n]], {n, 1, 256}]

A070803(n) = primepi(sigma(n)) \\ Michael B. Porter, Jan 28 2010

[prime_pi(sigma(n,1)) for n in range(1, 76)] # - Zerinvary Lajos, Jun 06 2009

a(n) = A000720(A000203(n)) = pi(sigma(n)).

A193416 Minimum surface area of polycubes with volume n.

Original entry on oeis.org

6, 10, 14, 16, 20, 22, 24, 24, 28, 30, 32, 32, 36, 38, 40, 40, 42, 42, 46, 48, 50, 50, 52, 52, 54, 54, 54, 58, 60, 62, 62, 64, 64, 66, 66, 66, 70, 72, 74, 74, 76, 76, 78, 78, 78, 80, 80, 80, 84, 86, 88, 88, 90, 90, 92, 92, 92, 94, 94, 94, 96, 96, 96, 96, 100

Offset: 1

Juan Barajas Martin, Jul 25 2011

nonn

First differences are 0, 2, or 4. - Charles R Greathouse IV, Aug 25 2011
From Juan Barajas Martin, Aug 27 2011 - Sep 03 2011, Sep 26 2001: (Start)
Initial cubic shape polycube edge = e; volume = e^3; minimum surface area = 6e^2.
Target cubic shape polycube edge = e+1; volume = (e+1)^3; minimum surface area = 6(e+1)^2.
The target polycube is achieved by the successive addition of 3 orthogonal layers of unit cubes to the faces of the initial polycube. The number of unit cubes added in each successive layer take values e^2, e*(e+1) and (e+1)^2, with every layer fully completed before starting the following layer.
This is so because the first unit cube added on every layer will add a surface area of 4 to the previous polycube, which is greater than keeping the new unit cube in the current layer, where we ensure that area will increase by 2 at maximum.
Every layer is constructed as an Ulam spiral, starting at the center and until the completion of the layer.
Except for the first cube placed on every layer which will increase the surface area of the previous polycube by 4, we can count 2 surface area increments for the unit cubes placed at the positions given by the function of the Quarter-squares + 1 sequence (i.e., A033638 = A002620 + 1, which set the locations of right angle turns in Ulam square spiral).
All other positions in the Ulam spiral will increase no area to the previous polycube.
An infinite sequence of polycubes with minimal surface area is approached between pairs of successive polycubes with cubic shape and edge values (e) and (e+1), having respectively polycube volumes e^3 and (e+1)^3 and minimum surface areas 6e^2 and 6(e+1)^2.
(End)

			The unique polycube of volume 1 is a cube with surface area 6, so a(1) = 6. There are eight polycubes of volume 4, of which seven have surface area 18 and one has surface area 16, so a(4) = 16. - _Charles R Greathouse IV_, Aug 25 2011

Eric Weisstein's World of Mathematics, Polycube
Eric Weisstein's World of Mathematics, Ulam Spiral

Cf. A033638 (A002620 + 1) identify the unit cubes which will increase the minimum area by 2 (locations of right angle turns in Ulam square spiral), A007818.

vals=100;
az[n_]:=Floor[(n-1)^(1/3)];
kz[n_]:=n-az[n]^3;
av[n_]:=6*az[n]^2;
bv[n_]:=If[n==1,4,If[kz[n]>az[n]^2+(az[n]+1)*az[n],12,If[kz[n]>az[n]^2,8,4]]];
pz[n_]:=If[kz[n]1,2*(c1[n]+c2[n]+c3[n]),2];
smin[n_]:=av[n]+bv[n]+cv[n];
Table[smin[n], {n,vals}] (* Juan Barajas Martin, Sep 01 2011 *)

a(n^3) = 6n^2, a(n) ~ 6n^(2/3). - Charles R Greathouse IV, Aug 25 2011
From Juan Barajas Martin, Aug 28 2011: (Start)
The following formula is derived from the Mathematica program below:
smin[n]={6 Floor[(-1+n)^(1/3)]^2+If[n==1,4,If[kz[n]>az[n]^2+az[n] (az[n]+1),12,If[kz[n]>az[n]^2,8,4]]]+If[n>1,2 (c1[n]+c2[n]+c3[n]),2]}
az[n_]:Floor[Power[n-1, (3)^-1]]
kz[n_]:=n-az(n)^3
c1-c3: number of unit cubes increasing the surface area by 2 in every layer (see Comments above). (End)
a(n) = 6*n - 2*A007818(n). - Mohammed Yaseen, Aug 08 2021

A339884 Triangle read by rows: T(n, m) gives the number of partitions of n with m parts and parts from {1, 2, 3}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 0, 2, 2, 2, 1, 1, 0, 0, 1, 3, 2, 2, 1, 1, 0, 0, 1, 2, 3, 2, 2, 1, 1, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 0, 0, 0, 1, 3, 3, 3, 2, 2, 1, 1, 0, 0, 0, 1, 2, 4, 3, 3, 2, 2, 1, 1

Offset: 1

Wolfdieter Lang, Jan 31 2021

Row sums give A001399(n), for n >= 1.

One could add the column [1,repeat 0] for m = 0 starting with n >= 0.

			The triangle T(n,m) begins:
  n\m  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
  1:   1
  2:   1 1
  3:   1 1 1
  4:   0 2 1 1
  5:   0 1 2 1 1
  6:   0 1 2 2 1 1
  7:   0 0 2 2 2 1 1
  8:   0 0 1 3 2 2 1 1
  9:   0 0 1 2 3 2 2 1 1
  10:  0 0 0 2 3 3 2 2 1  1
  11:  0 0 0 1 3 3 3 2 2  1  1
  12:  0 0 0 1 2 4 3 3 2  2  1  1
  13:  0 0 0 0 2 3 4 3 3  2  2  1  1
  14:  0 0 0 0 1 3 4 4 3  3  2  2  1  1
  15:  0 0 0 0 1 2 4 4 4  3  3  2  2  1  1
  16:  0 0 0 0 0 2 3 5 4  4  3  3  2  2  1  1
  17:  0 0 0 0 0 1 3 4 5  4  4  3  3  2  2  1  1
  18:  0 0 0 0 0 1 2 4 5  5  4  4  3  3  2  2  1  1
  19:  0 0 0 0 0 0 2 3 5  5  5  4  4  3  3  2  2  1  1
  20:  0 0 0 0 0 0 1 3 4  6  5  5  4  4  3  3  2  2  1  1
  ...
Row n = 6: the partitions of 6 with number of parts m = 1,2, ...., 6, and parts from {1,2,3} are (in Abramowitz-Stegun order): [] | [],[],[3,3] | [],[1,2,3],[2^3] | [1^3,3],[1^2,2^2] | [1^4,2] | 1^6, giving 0, 1, 2, 2, 1, 1.

Louis Comtet, Advanced Combinatorics, Reidel (1974)

Cf. A001399, A008284 (all parts), A145362 (parts 1, 2), A232539 (parts <=4), A291983.

Compositions: A007818, A030528 (parts 1, 2), A078803 (parts 1, 2, 3).

Sum_{k=0..n} (-1)^k * T(n,k) = A291983(n). - Alois P. Heinz, Feb 01 2021

G.f.: 1/((1-u*t)*(1-u*t^2)*(1-u*t^3)). [Comtet page 97 [2c]]. - R. J. Mathar, May 27 2025

A070802 a(n)=prevprime[sigma(n)]-nextprime[phi(n)]=A070801(n)-A070800(n).

Original entry on oeis.org

1, 0, 4, 0, 8, 0, 8, 6, 12, 0, 18, 0, 16, 12, 20, 0, 30, 0, 30, 18, 20, 0, 48, 8, 28, 18, 40, 0, 60, 0, 44, 24, 36, 18, 76, 0, 40, 24, 72, 0, 76, 0, 60, 44, 48, 0, 96, 10, 66, 34, 68, 0, 94, 30, 84, 42, 60, 0, 150, 0, 58, 66, 90, 30, 116, 0, 76, 42, 110, 0, 164, 0, 76, 72, 102, 28

Offset: 2

Labos Elemer, May 08 2002

			n=100:sigma[100]=217,prevprime[217]=211, phi[100]=40,nextprime[40]=41,a(100)=211-41=170. The difference is 0 for odd primes.

Cf. A007917, A007818, A000010, A000203, A000720, A000040, A070800, A070801.

Table[Prime[PrimePi[DivisorSigma[1, w]]]- Prime[1+PrimePi[EulerPhi[w]]], {w, 2, 128}]

a(n)=p[Pi(sigma[n])]-p[1+Pi(phi[n])]

cp's OEIS Frontend

A132775 A007818 * A132774.

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A038119 Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).

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A070804 Number of primes not exceeding phi(n).

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Magma

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A070803 Number of primes not exceeding sum of divisors of n.

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Magma

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A193416 Minimum surface area of polycubes with volume n.

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A339884 Triangle read by rows: T(n, m) gives the number of partitions of n with m parts and parts from {1, 2, 3}.

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A070802 a(n)=prevprime[sigma(n)]-nextprime[phi(n)]=A070801(n)-A070800(n).

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