A383177
Sphenic numbers k such that floor(log(k)/log(lpf(k))) = 1+floor(log(k)/log(p)) for all primes p | k such that p > lpf(k), where lpf = A020639(k).
Original entry on oeis.org
1001, 1309, 1547, 1729, 2093, 2261, 3553, 4199, 4301, 4807, 5681, 6061, 6479, 7337, 7843, 8671, 9269, 9361, 9889, 10373, 10879, 11063, 11339, 11687, 11803, 11891, 12121, 12617, 13079, 13717, 13949, 13981, 14911, 15283, 15457, 16211, 16523, 17081, 17329, 17719
Offset: 1
Let s(n) = A010846(a(n)).
Table of a(n) for n = 1..12, showing prime factors of a(n) and
n a(n) facs(a(n)) s(n)
---------------------------
1 1001 7*11*13 15
2 1309 7*11*17 15
3 1547 7*13*17 15
4 1729 7*13*19 15
5 2093 7*13*23 15
6 2261 7*17*19 15
7 3553 11*17*19 15
8 4199 13*17*19 15
9 4301 11*17*23 15
10 4807 11*19*23 15
11 5681 13*19*23 15
12 6061 11*19*29 15
Let f(p,k) = floor(log(k)/log(p)) and let w be the list of f(p,k) across the sorted list of distinct prime factors of k.
30 = 2*3*5 is not in the sequence since f(30,2) = 4, f(30,3) = 3, f(30,5) = 2.
a(1) = 1001 = 7*11*13; f(7,1001) = 3, f(11,1001) = 2, f(13,1001) = 2.
a(2) = 1309 = 7*11*17; w(1309) = {3,2,2}, etc.
Pattern of numbers in row a(n) of A275280:
Level r^0 Level r^1 Level r^2
1, p, p^2, p^3 | r, p*r, p^2*r | r^2
q, p*q, p^2*q | q*r, p*q*r |
q^2, p*q^2; |
Example: k = 1001 = 7*11*13
1, 7, 49, 343 | 13, 91, 637 | 169
11, 77, 539 | 143, 1001 |
121, 847 |
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, Hasse diagram of R(1001) with logarithmic vertical scale. Gray represents the empty product, red represents primes, gold represents proper prime powers, green squarefree composites, and blue numbers that are neither squarefree nor prime powers.
- Michael De Vlieger, Three dimensional diagram of R(a(n)), labeling exponents along axes, showing p^3, q^2, and r^2, and using the color scheme above.
- Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048, 8X vertical exaggeration. The green bar at the bottom of the graph emphasizes the x axis that rides on the top edge of the bar.
-
f[om_, lm_ : 0] := Block[{f, i, j, k, nn, w}, i = Abs[om]; j = 1;
If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]]; w = ConstantArray[1, i];
Union@ Reap[Do[
While[Set[k, Times @@ Map[Prime, Accumulate@w]]; k <= nn,
If[Or[k == 1, Union[#2] == #1 - 1 & @@
TakeDrop[Map[Floor@Log[#, k] &, FactorInteger[k][[All, 1]] ], 1] ],
Sow[k]];
j = 1; w[[-j]]++];
If[j == i, Break[], j++; w[[-j]]++;
w = PadRight[w[[;; -j]], i, 1]], {n, Infinity}] ][[-1, 1]] ];
f[3, 20000]
A382926
Irregular table where row n lists numbers k in row n of A162306 for which there exists a prime p | n such that k*p > n.
Original entry on oeis.org
2, 3, 4, 5, 3, 4, 6, 7, 8, 9, 4, 5, 8, 10, 11, 6, 8, 9, 12, 13, 4, 7, 8, 14, 5, 9, 15, 16, 17, 8, 9, 12, 16, 18, 19, 5, 8, 10, 16, 20, 7, 9, 21, 4, 8, 11, 16, 22, 23, 9, 12, 16, 18, 24, 25, 4, 8, 13, 16, 26, 27, 7, 8, 14, 16, 28, 29, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30
Offset: 2
Let s(n) = A382964(n).
Table of select rows:
n s(n) row n of this sequence
--------------------------------------------------------
6 3 3, 4, 6;
10 4 4, 5, 8, 10;
12 4 6, 8, 9, 12;
14 4 4, 7, 8, 14;
15 3 5, 9, 15;
18 5 8, 9, 12, 16, 18;
20 5 5, 8, 10, 16, 20;
21 3 7, 9, 21;
22 5 4, 8, 11, 16, 22;
24 5 9, 12, 16, 18, 24;
26 5 4, 8, 13, 16, 26;
28 5 7, 8, 14, 16, 28;
30 12 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30.
In the examples below, we place terms in row n in brackets [] among other terms in row n of A162306, presented in order of row n of A275280.
Row p^m for m > 0 and prime p is {p^m}, since multiplying p^m by p exceeds p^m.
Row 10 = {4, 5, 8, 10}, since numbers k such that rad(k) | 10 contains these numbers, furthermore, we have the following: 2 or 5 times 8 exceeds 10, 5*4 > 10, 2 or 5 times 10 exceeds 10, and 5*5 > 10.
1 2 [4] [8]
[5] [10]
Row 24 = {9, 12, 16, 18, 24}, since numbers k such that rad(k) | 24 contains these numbers, furthermore, we have the following: 2 or 3 times 16 exceeds 24, 2 or 3 times 24 exceeds 24, 3*12 > 24, 2 or 3 times 18 exceeds 24, and 3*9 > 24.
1 2 4 8 [16]
3 6 [12] [24]
[9] [18]
-
(* First, run the "regs" function from A369609, then: *)
Table[Select[regs[n], Function[k, AnyTrue[FactorInteger[n][[All, 1]], #*k > n &]]], {n, 2, 30}] // Flatten
A384875
Irregular triangle T(n,k) = 2^(floor(n/3)-k) * nextprime(2^(n-2*(floor(n/3)-k))), with k = 0..floor(n/3)-1.
Original entry on oeis.org
6, 10, 22, 20, 34, 44, 74, 68, 134, 88, 148, 262, 136, 268, 514, 296, 524, 1042, 272, 536, 1028, 2062, 592, 1048, 2084, 4106, 1072, 2056, 4124, 8198, 1184, 2096, 4168, 8212, 16418, 2144, 4112, 8248, 16396, 32822, 4192, 8336, 16424, 32836, 65542, 4288, 8224, 16496, 32792, 65644, 131074
Offset: 3
Table begins:
n\k 0 1 2 3 4
---------------------------------------
3: 6
4: 10
5: 22
6: 20 34
7: 44 74
8: 68 134
9: 88 148 262
10: 136 268 514
11: 296 524 1042
12: 272 536 1028 2062
13: 592 1048 2084 4106
14: 1072 2056 4124 8198
15: 1184 2096 4168 8212 16418
...
Let S = A010846.
Tables showing terms in row a(n) of A162306, listed in order of row a(n) of A275280.
T(3,1) = 6,
S(6) = 5:
1 2 4
3 6
T(4,1) = 10,
S(10) = 6:
1 2 4 8
5 10
T(5,1) = 22,
S(22) = 7:
1 2 4 8 16
11 22
T(6,1) = 20, T(6,2) = 34,
S(20) = 8: S(34) = 8:
1 2 4 8 16 1 2 4 8 16 32
5 10 20 17 34
T(7,1) = 44, T(7,2) = 74,
S(44) = 9: S(74) = 9:
1 2 4 8 16 32 1 2 4 8 16 32 64
11 22 44 37 74
T(8,1) = 68, T(8,2) = 134,
S(68) = 10: S(134) = 10:
1 2 4 8 16 32 64 1 2 4 8 ... 128
17 34 68 67 134
T(9,1) = 88, T(9,2) = 148, T(9,3) = 262,
S(88) = 11: S(148) = 11: S(262) = 11:
1 2 4 8 16 32 64 1 2 4 8 ... 128 1 2 ... 256
11 22 44 88 37 74 148 131 262
etc.
-
Table[2^k*NextPrime[2^(n - 2*k)], {n, 3, 18}, {k, Floor[n/3], 1, -1}] // TableForm
Showing 1-3 of 3 results.
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