A182944
Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.
Original entry on oeis.org
2, 4, 3, 8, 9, 5, 16, 27, 25, 7, 32, 81, 125, 49, 11, 64, 243, 625, 343, 121, 13, 128, 729, 3125, 2401, 1331, 169, 17, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23
Offset: 1
Square array A(i,j) begins:
i \ j: 1 2 3 4 5 ...
---\-------------------------------------
1: 2, 4, 8, 16, 32, ...
2: 3, 9, 27, 81, 243, ...
3: 5, 25, 125, 625, 3125, ...
4: 7, 49, 343, 2401, 16807, ...
...
The triangle T(n,k) begins:
n\k: 1 2 3 4 5 6 ...
1: 2
2: 4 3
3: 8 9 5
4: 16 27 25 7
5: 32 81 125 49 11
6: 64 243 625 343 121 13
...
- Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
- Michael De Vlieger, Diagram showing row n of triangle in a semicircle as noted, with a color function associated with the magnitude of T(n,k) compared to 2^n in light blue, where prime(n) is the smallest and the prime power indicated in red the largest in the row.
A319075 extends the array with 0th powers.
-
TableForm[Table[Prime[n]^j,{n,1,14},{j,1,8}]]
Clarified in respect of alternate reading as a triangle by
Peter Munn, Aug 28 2022
A319075
Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.
Original entry on oeis.org
1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 7, 1, 32, 81, 125, 49, 11, 1, 64, 243, 625, 343, 121, 13, 1, 128, 729, 3125, 2401, 1331, 169, 17, 1, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 1, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23, 1, 1024, 19683, 390625, 823543, 1771561, 371293
Offset: 0
The corner of the square array is as follows:
A000079 A000244 A000351 A000420 A001020 A001022 A001026
A000012 1, 1, 1, 1, 1, 1, 1, ...
A000040 2, 3, 5, 7, 11, 13, 17, ...
A001248 4, 9, 25, 49, 121, 169, 289, ...
A030078 8, 27, 125, 343, 1331, 2197, 4913, ...
A030514 16, 81, 625, 2401, 14641, 28561, 83521, ...
A050997 32, 243, 3125, 16807, 161051, 371293, 1419857, ...
A030516 64, 729, 15625, 117649, 1771561, 4826809, 24137569, ...
A092759 128, 2187, 78125, 823543, 19487171, 62748517, 410338673, ...
A179645 256, 6561, 390625, 5764801, 214358881, 815730721, 6975757441, ...
...
Rows 0-13:
A000012,
A000040,
A001248,
A030078,
A030514,
A050997,
A030516,
A092759,
A179645,
A179665,
A030629,
A079395,
A030631,
A138031.
Other rows n:
A030635 (n=16),
A030637 (n=18),
A137486 (n=22),
A137492 (n=28),
A139571 (n=30),
A139572 (n=36),
A139573 (n=40),
A139574 (n=42),
A139575 (n=46),
A173533 (n=52),
A183062 (n=58),
A183085 (n=60),
A261700 (n=100).
Columns 1-15:
A000079,
A000244,
A000351,
A000420,
A001020,
A001022,
A001026,
A001029,
A009967,
A009973,
A009975,
A009981,
A009985,
A009987,
A009991.
A332979
Largest integer m satisfying Omega(m) + pi(gpf(m)) - [m<>1] = n.
Original entry on oeis.org
1, 2, 4, 9, 27, 125, 625, 3125, 16807, 161051, 1771561, 19487171, 214358881, 2357947691, 25937424601, 285311670611, 3138428376721, 34522712143931, 582622237229761, 9904578032905937, 168377826559400929, 2862423051509815793, 48661191875666868481
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..461
- Michael De Vlieger, Concise table of n, a(n) for n = 1..10000, where a(n) = prime(k)^e written as "pk^e". (a(0) = 1 is presented as "p1^0" to avoid reconversion errors in some CAS associated with "prime(0)".)
- Michael De Vlieger, Annotated plot of a(n) = prime(k)^e at (x,y) = (e,k) for n = 1..64, showing the first and last terms divisible by prime(k) in red, singleton powers of prime(k) in green, otherwise blue.
- Michael De Vlieger, Plot of a(n) = prime(k)^e at (x,y) = (e,k) for n = 1..10000.
- Michael De Vlieger, Fan style binary tree showing row m = 2..15 of A005940 in concentric semicircles. Terms in light blue appear in row m-1 of A182944, highlighting a(m-1) in red.
- Michael De Vlieger, Fan style binary tree showing row m = 2..15 of A005940 in concentric semicircles. We apply a color function with dark blue the minimum and greens the largest values to show the magnitude of terms in row m compared to 2^(m-1). The row maximum a(m-1) appears in red.
- Wikipedia, Iverson bracket
-
b:= proc(n, i) option remember; `if`(n=0, 1, max(seq(b(n-
`if`(i=0, j, 1), j)*ithprime(j), j=1..`if`(i=0, n, i))))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, Max[Table[
b[n - If[i == 0, j, 1], j] Prime[j], {j, 1, If[i == 0, n, i]}]]];
a[n_] := b[n, 0];
a /@ Range[0, 23] (* Jean-François Alcover, May 03 2021, after Alois P. Heinz *)
(* Second program: extract data from the concise a-file of 10000 terms: *)
With[{nn = 23 (* set nn <= 10000 as desired *)}, Prime[#1]^#2 & @@ # & /@ Map[ToExpression /@ {StringTrim[#1, "p"], #2} & @@ StringSplit[#, "^"] &, Import["https://oeis.org/A332979/a332979.txt", "Data"][[1 ;; nn, -1]] ] ] (* Michael De Vlieger, Aug 22 2022 *)
A356627
Primes whose powers appear in A332979.
Original entry on oeis.org
2, 3, 5, 7, 11, 17, 29, 37, 41, 59, 67, 71, 97, 127, 149, 191, 223, 269, 307, 347, 419, 431, 557, 563, 569, 587, 593, 599, 641, 727, 809, 937, 967, 1009, 1213, 1277, 1423, 1861, 1973, 2237, 2267, 2657, 3163, 3299, 3449, 3457, 3527, 3907, 4001, 4211, 4441, 4637
Offset: 1
5 | A332979(5..7), thus 5 is in the sequence.
7 | A332979(8), thus 7 is in the sequence.
13 does not divide any term in A332979, so it is not a term in this sequence.
-
Prime@ Union@ Table[MaximalBy[Table[{k, n - k}, {k, n}], Prime[#1]^#2 & @@ # &][[1, 1]], {n, 2^10}]
(* or use concise file in A332979 *)
Prime /@ Union@ Rest@ Map[ToExpression@ StringTrim[#, "p"] & @@ StringSplit[#, "^"] &, Import["https://oeis.org/A332979/a332979.txt", "Data"][[All, -1]]]
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