A121057 -id:A121057 - cp's OEIS frontend

A223556 T(n,k)=Petersen graph (3,1) coloring a rectangular array: number of nXk 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0.

1, 3, 6, 9, 27, 36, 27, 171, 243, 216, 81, 1089, 3249, 2187, 1296, 243, 6939, 44217, 61731, 19683, 7776, 729, 44217, 609309, 1795473, 1172889, 177147, 46656, 2187, 281763, 8410671, 53599905, 72906921, 22284891, 1594323, 279936, 6561, 1795473

Offset: 1

R. H. Hardin Mar 22 2013

Table starts
........1..........3.............9...............27...................81
........6.........27...........171.............1089.................6939
.......36........243..........3249............44217...............609309
......216.......2187.........61731..........1795473.............53599905
.....1296......19683.......1172889.........72906921...........4715559621
.....7776.....177147......22284891.......2960456193.........414863325945
....46656....1594323.....423412929.....120212193177.......36498667573629
...279936...14348907....8044845651....4881332621169.....3211064180380305
..1679616..129140163..152852067369..198211242377097...282501632829717621
.10077696.1162261467.2904189280011.8048559615522273.24853807982558115945

			Some solutions for n=3 k=4
..0..3..5..2....0..1..4..1....0..2..5..2....0..2..5..2....0..1..2..0
..5..3..0..1....0..1..0..3....1..2..5..2....5..2..1..4....2..0..1..2
..4..3..4..3....2..1..4..5....0..2..0..1....1..2..1..2....1..4..5..3

R. H. Hardin, Table of n, a(n) for n = 1..219

Column 1 is A000400(n-1)
Column 2 is A013708(n-1)
Column 3 = 9*19^(n-1) is row 8 of A223556 with T(2+,3) = A121057(8,1+)
Row 1 is A000244(n-1)

Empirical for column k:
k=1: a(n) = 6*a(n-1)
k=2: a(n) = 9*a(n-1)
k=3: a(n) = 19*a(n-1)
k=4: a(n) = 41*a(n-1) -16*a(n-2)
k=5: a(n) = 95*a(n-1) -626*a(n-2) +720*a(n-3) for n>4
k=6: [order 8] for n>9
k=7: [order 13] for n>15
Empirical for row n:
n=1: a(n) = 3*a(n-1)
n=2: a(n) = 7*a(n-1) -4*a(n-2) for n>3
n=3: a(n) = 17*a(n-1) -47*a(n-2) +41*a(n-3) -10*a(n-4) for n>6
n=4: [order 13] for n>16
n=5: [order 41] for n>45

A217983 If n = floor(p/2) * p^e, for some (by necessity unique) prime p and exponent e > 0, then a(n) = p, otherwise a(n) = 1.

Original entry on oeis.org

1, 2, 3, 2, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Johannes W. Meijer, Oct 25 2012

nonn

a(A130290(n) * A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n) = 1 elsewhere. - The original name of the sequence.
The a(n) are related to the prime numbers A000040 and the number of nonzero quadratic residues modulo the n-th prime A130290, see the first formula and the Maple program.
This sequence resembles the exponential of the von Mangoldt function A014963; for the latter sequence a(A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n) = 1 elsewhere.
Positions of the first occurrence of each successive noncomposite number (and also the records) is given by the union of {2} and A008837. - Antti Karttunen, Jan 17 2025

Antti Karttunen, Table of n, a(n) for n = 1..100949

Cf. A000040, A014963, A128060, A130290, A160479.
Cf. A000079, A000244 (after their initial 1's, the positions of 2's and 3's respectively), A020699 (positions of 5's from its third term 10 onward), A169634 (positions of 7's from the second term onward), A379956 (positions of terms > 1).
Cf. A008837, A121057.

nmax := 78: A000040 := proc(n): ithprime(n) end: A130290 := proc(n): if n =1 then 1 else (A000040(n)-1)/2 fi: end: for n from 1 to nmax do A217983(n) := 1 od: for n from 1 to nmax do for n1 from 1 to floor(log[A000040(n)](nmax)) do A217983(A130290(n) * A000040(n)^n1) := A000040(n) od: od: seq(A217983(n), n=1..nmax);

A217983(n) = { my(f=factor(n)); for(i=1,#f~,if((n/(f[i,1]^f[i,2])) == (f[i,1]\2), return(f[i,1]))); (1); }; \\ Antti Karttunen, Jan 16 2025

a(A130290(n) * A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n)= 1 elsewhere.

a(n) = (A160479(n+1) * A128060(n+1))/(2*n+1) for n >= 2.

Definition simplified, original definition moved to comments; more terms added by Antti Karttunen, Jan 16 2025

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A217983 If n = floor(p/2) * p^e, for some (by necessity unique) prime p and exponent e > 0, then a(n) = p, otherwise a(n) = 1.

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