A162306 -id:A162306 - cp's OEIS frontend

A280269 Irregular triangle T(n,m) read by rows: smallest power e of n that is divisible by m = term k in row n of A162306.

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

Offset: 1

Michael De Vlieger, Dec 30 2016

This table eliminates the negative values in row n of A279907.
Let k = A162306(n,m), i.e., the value in column m of row n.
T(n,1) = 0 since 1 | n^0.
T(n,p) = 1 for prime divisors p of n since p | n^1.
T(n,d) = 1 for divisors d > 1 of n since d | n^1.
Row n for prime p have two terms, {0,1}, the maximum value 1, since all k < p are coprime to p, and k | p^1 only when k = p.
Row n for prime power p^i have (i+1) terms, one zero and i ones, since all k that appear in corresponding row n of A162306 are divisors d of p^i.
Values greater than 1 pertain only to composite k of composite n > 4, but not in all cases. T(n,k) = 1 for squarefree kernels k of composite n.
Numbers k > 1 coprime to n and numbers that are products of at least one prime q coprime to n and one prime p | n do not appear in A162306; these do not divide n^e evenly.
T(n,k) is nonnegative for all numbers k for which n^k (mod k) = 0, i.e., all the prime divisors p of k also divide n.
The largest possible value s in row n of T = floor(log_2(n)), since the largest possible multiplicity of any number m <= n pertains to perfect powers of 2, as 2 is the smallest prime. This number s first appears at T(2^s + 2, 2^s) for s > 1.
1/k terminates T(n,k) digits after the radix point in base n for values of k that appear in row n of A162306.
Originally from Robert Israel at A279907: (Start)
T(a*b,c*d) = max(T(a,c),T(b,d)) if GCD(a,b)=1, GCD(b,d)=1,T(a,c)>=0 and T(b,d)>=0.
T(n,a*b) = max(T(n,a),T(n,b)) if GCD(a,b)=1 and T(n,a)>=0 and T(n,b)>=0.
(End)

			Triangle T(n,m) begins:  Triangle A162306(n,k):
1:  0                    1
2:  0  1                 1  2
3:  0  1                 1  3
4:  0  1  1              1  2  4
5:  0  1                 1  5
6:  0  1  1  2  1        1  2  3  4  6
7:  0  1                 1  7
8:  0  1  1  1           1  2  4  8
9:  0  1  1              1  3  9
10: 0  1  2  1  3  1     1  2  4  5  8  10
...

Michael De Vlieger, Table of n, a(n) for n = 1..10202 (rows 1 <= n <= 660)

Cf. A162306, A279907 (T(n,k) with values for all 1 <= k <= n), A280274 (maximum values in row n), A010846 (number of nonnegative k in row n), A051731 (k with e <= 1), A000005 (number of k in row n with e <= 1), A272618 (k with e > 1), A243822 (number of k in row n with e > 1), A007947.

Table[SelectFirst[Range[0, #], PowerMod[n, #, k] == 0 &] /. m_ /; MissingQ@ m -> Nothing &@ Floor@ Log2@ n, {n, 24}, {k, n}] // Flatten (* Version 10.2, or *)
DeleteCases[#, -1] & /@ Table[If[# == {}, -1, First@ #] &@ Select[Range[0, #], PowerMod[n, #, k] == 0 &] &@ Floor@ Log2@ n, {n, 24}, {k, n}] // Flatten (* or *)
DeleteCases[#, -1] & /@ Table[Boole[k == 1] + (Boole[#[[-1, 1]] == 1] (-1 + Length@ #) /. 0 -> -1) &@ NestWhileList[Function[s, {#1/s, s}]@ GCD[#1, #2] & @@ # &, {k, n}, And[First@# != 1, ! CoprimeQ @@ #] &], {n, 24}, {k, n}] // Flatten

A372323 A124652(n) is the a(n)-th term in row A372111(n-1) of irregular triangle A162306.

Original entry on oeis.org

2, 4, 4, 4, 5, 7, 5, 8, 8, 2, 10, 8, 12, 11, 13, 6, 13, 6, 6, 9, 8, 11, 4, 8, 16, 5, 6, 7, 13, 12, 7, 10, 19, 15, 16, 17, 9, 6, 15, 10, 3, 11, 8, 18, 28, 14, 14, 10, 30, 28, 15, 4, 20, 33, 13, 12, 6, 22, 18, 21, 12, 11, 29, 12, 11, 8, 24, 18, 8, 14, 17, 32, 33

Offset: 3

Michael De Vlieger, May 05 2024

nonn

Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.
Let r(x) = A010846(x), the number of m <= x such that rad(m) | x, where rad = A007947.
Let row k of A162306 contain { m : rad(m) | k, m <= k }. Thus r(k) is the length of row k of A162306.
Let T(k,j) represent the j-th term in row k of irregular triangle A162306.
a(n) = j is the position of b(n) in row s(n-1) of A162306.
b(n) = T(s(n-1), a(n)).
Analogous to A371910, which instead regards A109890 and A109735.

			Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.
a(3) = 2 since b(3) = 3 is the 2nd term in row s(3) = 3 of A162306, {1, [3]}.
a(4) = 4 since b(4) = 4 is the 4th term in row s(4) = 6 of A162306, {1, 2, 3, [4], 6}.
a(5) = 4 since b(5) = 5 is T(s(n-1), 4) = T(10, 4), {1, 2, 4, [5], 8, 10}.
a(6) = 4 since b(6) = 9 is T(s(n-1), 4) = T(15, 4), {1, 3, 5, [9], 15}.
a(7) = 5 since b(7) = 6 is T(s(n-1), 5) = T(24, 5), {1, 2, 3, 4, [6], 8, 9, 12, 16, 18, 24}, etc.
Table relating this sequence to b = A124652, s = A372111, r = A372322, and A162306.
   n b(n) s(n-1) a(n) r(n) row s(n-1) of A162306
  ---------------------------------------------------------------------
   3    3    3    2    2   {1, [3]}
   4    4    6    4    5   {1, 2, 3, [4], 6}
   5    5   10    4    6   {1, 2, 4, [5], 8, 10}
   6    9   15    4    5   {1, 3, 5, [9], 15}
   7    6   24    5   11   {1, 2, 3, 4, [6], ..., 24}
   8    8   30    7   18   {1, 2, 3, 4, 5, 6, [8], ..., 30}
   9   16   38    5    8   {1, 2, 4, 8, [16], 19, 32, 38}
  10   12   54    8   16   {1, 2, 3, 4, 6, 8, 9, [12], ..., 54}
  11   11   66    8   22   {1, 2, 3, 4, 6, 8, 9, [11], ..., 66}
  12    7   77    2    5   {1, [7], 11, 49, 77}
  13   14   84   10   28   {1, 2, 3, 4, ..., 12, [14], ..., 84}
  14   28   98    8   13   {1, 2, 4, 7, ..., 16, [28], ..., 98}

Michael De Vlieger, Table of n, a(n) for n = 3..10000
Michael De Vlieger, Bar chart showing a(n)/A372322(n-1) for n = 3..1024. This chart illustrates the "depth" of A124652(n) among the terms of the A372111(n-1)-th row of A162306.

Cf. A007947, A010846, A124652, A162306, A371910, A372111, A372322.

nn = 75; c[_] := False;
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
f[x_] := Select[Range[x], Divisible[x, rad[#]] &];
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
  Sow[FirstPosition[r, k][[1]]]; c[k] = True;
  s += k, {i, 3, nn}] ][[-1, 1]]

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

Michael De Vlieger, Apr 28 2025

The number n appears in each row. For n in A024619, for all p|n, p^floor(log_p n) is in row n. Thus, the number of terms in row n for n in A024619 is at least 1+omega(n), where omega = A001221 is the number of distinct prime factors of n.

			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]

Michael De Vlieger, Table of n, a(n) for n = 2..11938 (rows n = 2..1000, flattened)

Cf. A000961, A007947, A024619, A162306, A275280, A382964 (row lengths).

(* 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

For n in A000961, row n is {n}.

A010846 Number of numbers <= n whose set of prime factors is a subset of the set of prime factors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 6, 2, 8, 2, 6, 5, 5, 2, 10, 2, 8, 5, 7, 2, 11, 3, 7, 4, 8, 2, 18, 2, 6, 6, 8, 5, 14, 2, 8, 6, 11, 2, 19, 2, 9, 8, 8, 2, 15, 3, 12, 6, 9, 2, 16, 5, 11, 6, 8, 2, 26, 2, 8, 8, 7, 5, 22, 2, 10, 6, 20, 2, 18, 2, 9, 9, 10, 5, 23, 2, 14, 5, 9, 2, 28, 5, 9, 7, 11, 2, 32, 5, 10

Offset: 1

Olivier Gérard

This function of n appears in an ABC-conjecture by Andrew Granville. See Goldfeld. - T. D. Noe, Jun 30 2009

			From _Wolfdieter Lang_, Jun 30 2014: (Start)
a(1) = 1 because the empty set is a subset of any set.
a(6) = 5 from the five numbers: 1 with the empty set, 2 with the set {2}, 3 with {3}, 4 with {2} and 6 with {2,3}, which are all subsets of {2,3}. 5 is out because {5} is not a subset of {2,3}. (End)
From _David A. Corneth_, Feb 10 2015: (Start)
Let p# be the product of primes up to p, A002110. Then,
a(13#) = 1161
a(17#) = 4843
a(19#) = 19985
a(23#) = 83074
a(29#) = 349670
a(31#) = 1456458
a(37#) = 6107257
a(41#) = 25547835
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 5000 terms from T. D. Noe)
Dorian Goldfeld, Modular forms, elliptic curves, and the ABC conjecture

Cf. A162306 (numbers for each n).

A:= proc(n) local F, S, s,j,p;
  F:= numtheory:-factorset(n);
  S:= {1};
  for p in F do
    S:= {seq(seq(s*p^j, j=0..floor(log[p](n/s))),s=S)}
  od;
  nops(S)
end proc;
seq(A(n),n=1..1000); # Robert Israel, Jun 27 2014

pf[n_] := If[n==1, {}, Transpose[FactorInteger[n]][[1]]]; SubsetQ[lst1_, lst2_] := Intersection[lst1,lst2]==lst1; Table[pfn=pf[n]; Length[Select[Range[n], SubsetQ[pf[ # ],pfn] &]], {n,100}] (* T. D. Noe, Jun 30 2009 *)
Table[Total[MoebiusMu[#] Floor[n/#] &@ Select[Range@ n, CoprimeQ[#, n] &]], {n, 92}] (* Michael De Vlieger, May 08 2016 *)

a(n,f=factor(n)[,1])=if(#f>1,my(v=f[1..#f-1],p=f[#f],s); while(n>0, s+=a(n,v); n\=p); s, if(#f&&n>0,log(n+.5)\log(f[1])+1,n>0)) \\ Charles R Greathouse IV, Jun 27 2013

a(n) = sum(k=1,n,if(gcd(n,k)-1,0,moebius(k)*(n\k))) \\ Benoit Cloitre, May 07 2016

a(n,f=factor(n)[,1])=if(#f<2, return(if(#f, valuation(n,f[1])+1, 0))); my(v=f[1..#f-1],p=f[#f],s); while(n, s+=a(n,v); n\=p); s \\ Charles R Greathouse IV, Nov 03 2021

def A010846(n): return sum((m:=n**k)//k-(m-1)//k for k in range(1,n+1)) # Chai Wah Wu, Aug 15 2024

from math import gcd
from sympy import mobius
def A010846(n): return sum(mobius(k)*(n//k) for k in range(1,n+1) if gcd(n,k)==1) # Chai Wah Wu, Apr 23 2025

a(n) = |{k<=n, k|n^(tau(k)-1)}|. - Vladeta Jovovic, Sep 13 2006
a(n) = Sum_{j = 1..n} Product_{primes p | j} delta(n mod p,0) where delta is the Kronecker delta. - Robert Israel, Feb 09 2015
a(n) = Sum_{1<=k<=n,(n,k)=1} mu(k)*floor(n/k). - Benoit Cloitre, May 07 2016
a(n) = Sum_{k=1..n} floor(n^k/k)-floor((n^k -1)/k). - Anthony Browne, May 28 2016

Definition made more precise at the suggestion of Wolfdieter Lang

A243822 Number of k < n such that rad(k) | n but k does not divide n, where rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 4, 0, 2, 1, 3, 0, 3, 0, 3, 0, 2, 0, 10, 0, 0, 2, 4, 1, 5, 0, 4, 2, 3, 0, 11, 0, 3, 2, 4, 0, 5, 0, 6, 2, 3, 0, 8, 1, 3, 2, 4, 0, 14, 0, 4, 2, 0, 1, 14, 0, 4, 2, 12, 0, 6, 0, 5, 3, 4, 1, 15, 0, 4, 0, 5, 0, 16, 1, 5, 3, 3, 0, 20, 1, 4, 3, 5, 1, 8, 0, 7, 2, 6

Offset: 1

Michael De Vlieger, Jun 11 2014

nonn

Former name: number of "semidivisors" of n, numbers m < n that do not divide n but divide n^e for some integer e > 1. See ACM Inroads paper.

			From _Michael De Vlieger_, Aug 11 2024: (Start)
Let S(n) = row n of A162306 and let D(n) = row n of A027750.a(2) = 0 since S(2) \ D(2) = {1, 2} \ {1, 2} is null.
a(10) = 2 since S(10) \ D(10) = {1, 2, 4, 5, 8, 10} \ {1, 2, 5, 10} = {4, 8}.a(16) = 0 since S(16) \ D(16) = {1, 2, 4, 8, 16} \ {1, 2, 4, 8, 16} is null, etc.Table of a(n) and S(n) \ D(n):
   n  a(n)  row n of A272618.
  ---------------------------
   6    1   {4}
  10    2   {4, 8}
  12    2   {8, 9}
  14    2   {4, 8}
  15    1   {9}
  18    4   {4, 8, 12*, 16}
  20    2   {8, 16}
  21    1   {9}
  22    3   {4, 8, 16}
  24    3   {9, 16, 18*}
  26    3   {4, 8, 16}
  28    2   {8, 16}
  30   10   {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}
Terms in A272618 marked with an asterisk are counted by A355432. All other terms are counted by A361235. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.
Michael De Vlieger, Regular and coregular numbers, ResearchGate, 2024.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20

Cf. A000005, A000961, A024619, A027750, A010846, A045763, A162306, A243823, A272618, A304570, A355432, A361235.

Table[Count[Range[n], ?(And[Divisible[n, Times @@ FactorInteger[#][[All, 1]]], ! Divisible[n, #]] &)], {n, 120}] (* _Michael De Vlieger, Aug 11 2024 *)

a(n) = A010846(n) - A000005(n) = card({row n of A162306} \ {row n of A027750}).
a(n) = A045763(n) - A243823(n).
a(n) = (Sum_{1<=k<=n, gcd(n,k)=1} mu(k)*floor(n/k)) - tau(n). - Michael De Vlieger, May 10 2016, after Benoit Cloitre at A010846.
From Michael De Vlieger, Aug 11 2024" (Start)
a(n) = 0 for n in A000961, a(n) > 0 for n in A024619.
a(n) = A051953(n) - A000005(n) + 1 = n - A000010(n) - A000005(n) - A243823(n) + 1.
a(n) = A355432(n) + A361235(n).
a(n) = A355432(n) for n in A360768.
a(n) = A361235(n) for n not in A360768.
a(n) = number of terms in row n of A272618.
a(n) = sum of row n of A304570. (End)

New name from David James Sycamore, Aug 11 2024

A272618 Irregular array read by rows: n-th row contains (in ascending order) the nondivisors 1 <= k < n such that all the prime divisors p of k also divide n.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 8, 0, 8, 9, 0, 4, 8, 9, 0, 0, 4, 8, 12, 16, 0, 8, 16, 9, 4, 8, 16, 0, 9, 16, 18, 0, 4, 8, 16, 0, 8, 16, 0, 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 0, 0, 9, 27, 4, 8, 16, 32, 25, 8, 16, 24, 27, 32, 0, 4, 8, 16, 32, 9, 27, 16, 25, 32, 0, 4, 8, 9, 12, 16, 18, 24, 27, 28, 32

Offset: 1

Michael De Vlieger, May 03 2016

The k are the "semidivisors" or nondivisor regular numbers of n as counted by A243822(n).
All nonzero terms k are composite and pertain to composite rows n. This is because prime k must either divide or be coprime to n, and k = 1 is both a divisor of and coprime to n.
Row n for prime p contains zero, since numbers 1 <= k < p must either divide or be coprime to prime p.
Row n for prime powers p^e contains zero, since there is only one prime divisor p of p^e and every power 1 <= m <= e of p divides p^e.
Row n = 4 is a special case of composite n that contains zero. This is because 4 is the smallest composite number; there are no composites k < n.
Thus rows n for composite n > 4 contain at least 1 nonzero value.
In base n, 1/a(n) has a terminating expansion with at least 2 places.

			For n = 12, the numbers 1 <= k < n such that the prime divisors p of k also divide n are {2, 3, 4, 6, 8, 9}; {2, 3, 4, 6} divide n = 12, thus row n = 12 is {8, 9}.
n: k
1: 0
2: 0
3: 0
4: 0
5: 0
6: 4
7: 0
8: 0
9: 0
10: 4 8
11: 0
12: 8 9
13: 0
14: 4 8
15: 9
16: 0
17: 0
18: 4 8 12 16
19: 0
20: 8 16

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, pp. 144-145, Theorem 136.

Michael De Vlieger, Table of n, a(n) for n = 1..10814 (rows 1 to 1000, flattened).
M. De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.
M. De Vlieger, Neutral Numbers.
M. De Vlieger, Sequence page.

Union of A027750 and nonzero terms of a(n) = A162306, thus A000005(n) + A243822(n) = A010846(n).

The union of nonzero terms of a(n) and A272619 = A133995, thus A243822(n) + A243823(n) = A045763(n).

Table[With[{r = First /@ FactorInteger@ n}, Select[Range@ n,
And[SubsetQ[r, Map[First, FactorInteger@ #]], ! Divisible[n, #]] &]], {n, 30}] /. {} -> 0 // Flatten (* Michael De Vlieger, May 03 2016 *)

A355432 a(n) = number of k < n such that rad(k) = rad(n) and k does not divide n, where rad(k) = A007947(k).

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Feb 22 2023

nonn

a(n) = 0 for prime powers and squarefree numbers.

			a(1) = 18, since 18/6 >= 3. We note that rad(12) = rad(18) = 6, yet 12 does not divide 18.
a(2) = 24, since 24/6 >= 3. rad(18) = rad(24) = 6 and 24 mod 18 = 6.
a(3) = 36, since 36/6 >= 3. rad(24) = rad(36) = 6 and 36 mod 24 = 12.
a(6) = 54, since 54/6 >= 3. m in {12, 24, 36, 48} are such that rad(m) = rad(54) = 6, but none divides 54, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Plot (k, n) at (x, -y), k = 1..n, n = 1..54, showing k in A126706 in dark blue, n in A360768 in dark red, and for n and nondivisor k such that rad(k) = rad(n), we highlight in large black dots. This sequence counts the number of black dots in row n.
Michael De Vlieger, Condensation of the above plot, showing k = 1..n and only n in A360768 and n <= 54.

Cf. A005361, A007947, A008479, A010846, A013929, A020639, A024619, A027750, A126706, A162306, A243822, A272618, A360589, A360768.

rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; Table[Which[PrimePowerQ[n], 0, SquareFreeQ[n], 0, True, r = rad[n]; Count[Select[Range[n], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], _?(And[rad[#] == r, Mod[n, #] != 0] &)]], {n, 120}]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = my(rn=rad(n)); sum(k=1, n-1, if (n % k, rad(k)==rn)); \\ Michel Marcus, Feb 23 2023

a(n) > 0 for n in A360768.
a(n) < A243822(n) < A010846(n).
a(n) = A008479(n) - A005361(n). - Amiram Eldar, Oct 25 2024

A079277 Largest integer k < n such that any prime factor of k is also a prime factor of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 3, 8, 1, 9, 1, 8, 9, 8, 1, 16, 1, 16, 9, 16, 1, 18, 5, 16, 9, 16, 1, 27, 1, 16, 27, 32, 25, 32, 1, 32, 27, 32, 1, 36, 1, 32, 27, 32, 1, 36, 7, 40, 27, 32, 1, 48, 25, 49, 27, 32, 1, 54, 1, 32, 49, 32, 25, 64, 1, 64, 27, 64, 1, 64, 1, 64, 45, 64, 49, 72, 1, 64, 27

Offset: 2

Istvan Beck (istbe(AT)online.no), Feb 07 2003

The function a(n) complements Euler's phi-function: 1) a(n)+phi(n) = n if n is a power of a prime (actually, in A285710). 2) It seems also that a(n)+phi(n) >= n for "almost all numbers" (see A285709, A208815). 3) a(2n) = n+1 if and only if n is a Mersenne prime. 4) Lim a(n^k)/n^k =1 if n has at least two prime factors and k goes to infinity.
From Michael De Vlieger, Apr 26 2017: (Start)
In other words, largest integer k < n such that k | n^e with integer e >= 0.
Penultimate term of row n in A162306. (The last term of row n in A162306 is n.)
For prime p, a(p) = 1. More generally, for n with omega(n) = 1, that is, a prime power p^e with e > 0, a(p^e) = p^(e - 1).
For n with omega(n) > 1, a(n) does not divide n. If n = pq with q = p + 2, then p^2 < n though p^2 does not divide n, yet p^2 | n^e with e > 1. If n has more than 2 distinct prime divisors p, powers p^m of these divisors will appear in the range (1..n-1) such that p^m > n/lpf(n) (lpf(n) = A020639(n)). Since a(n) is the largest of these, a(n) is not a divisor of n.
If a(n) does not divide n, then a(n) appears last in row n of A272618.
(End)

			a(10)=8 since 8 is the largest integer< 10 that can be written using only the primes 2 and 5. a(78)=72 since 72 is the largest number less than 78 that can be written using only the primes 2, 3 and 13. (78=2*3*13).

David W. Wilson, Table of n, a(n) for n = 2..10000
Aled Walker and Alexander Walker, Arithmetic Progressions with Restricted Digits, arXiv:1809.02430 [math.NT], 2018.

Cf. A000010, A007947, A051953, A162306, A208815, A272618, A285328, A285699, A285707, A285709, A285710, A285711.

Table[If[n == 2, 1, Module[{k = n - 2, e = Floor@ Log2@ n}, While[PowerMod[n, e, k] != 0, k--]; k]], {n, 2, 81}] (* Michael De Vlieger, Apr 26 2017 *)

a(n) = {forstep(k = n - 1, 2, -1, f = factor(k); okk = 1; for (i=1, #f~, if ((n % f[i,1]) != 0, okk = 0; break;)); if (okk, return (k));); return (1);} \\ Michel Marcus, Jun 11 2013

A007947(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
A079277(n) = { my(r); if((n > 1 && !bitand(n,(n-1))), (n/2), r=A007947(n); if(1==n,0,k = n-1; while(A007947(k*n) <> r, k = k-1); k)); }; \\ Antti Karttunen, Apr 26 2017

from sympy import divisors
from sympy.ntheory.factor_ import core
def a007947(n): return max(d for d in divisors(n) if core(d) == d)
def a(n):
    k=n - 1
    while True:
        if a007947(k*n) == a007947(n): return k
        else: k-=1
print([a(n) for n in range(2, 101)]) # Indranil Ghosh, Apr 26 2017

Largest k < n with rad(kn) = rad(n), where rad = A007947.

A369609 Irregular triangle read by rows where row n lists k <= n such that A007947(k) = A007947(n).

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 2, 4, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 8, 16, 17, 6, 12, 18, 19, 10, 20, 21, 22, 23, 6, 12, 18, 24, 5, 25, 26, 3, 9, 27, 14, 28, 29, 30, 31, 2, 4, 8, 16, 32, 33, 34, 35, 6, 12, 18, 24, 36, 37, 38, 39, 10, 20, 40, 41, 42, 43, 22, 44

Offset: 1

Michael De Vlieger, May 09 2024

Differs from A284318 after 27 terms.
Let rad(x) = A007947(x).
Let T(n,k) be the k-th term of row n in this sequence.
Define S(n,k) to be the k-th term in row n of A162306.
T(n,k) = rad(n) * S(n,k), k <= A008479(n).
The number n appears as the last term in row n.

			First rows of the triangle:
  1;
  2;
  3;
  2, 4;
  5;
  6;
  7;
  2, 4, 8;
  3, 9;
  10;
  11;
  6, 12;
  13;
  14;
  15;
  2, 4, 8, 16;
  17;
  6, 12, 18;
  etc.

Michael De Vlieger, Table of n, a(n) for n = 1..12946 (rows n = 1..5000, flattened)
Michael De Vlieger, Fast Mathematica programs that construct A008479, A010846, A162306, and A369609

Cf. A007947, A008479, A162306, A284318.

f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]]; Flatten@ Table[r = f[n]; Select[Range[n], f[#] == r &], {n, 44}]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
row(n) = my(r=rad(n)); select(x->(rad(x) == r), [1..n]); \\ Michel Marcus, May 11 2024

Row n of this sequence contains row n of A284318.
Length of row n is A008479(n).
For squarefree n, row n = {n}.
For prime power n = p^m, row n = { p^j : j = 1..m }.

A284311 Array T(n,k) read by antidiagonals (downward): T(1,k) = A005117(k+1) (squarefree numbers > 1); for n > 1, columns are nonsquarefree numbers (in ascending order) with exactly the same prime factors as T(1,k).

Original entry on oeis.org

2, 3, 4, 5, 9, 8, 6, 25, 27, 16, 7, 12, 125, 81, 32, 10, 49, 18, 625, 243, 64, 11, 20, 343, 24, 3125, 729, 128, 13, 121, 40, 2401, 36, 15625, 2187, 256, 14, 169, 1331, 50, 16807, 48, 78125, 6561, 512, 15, 28, 2197, 14641, 80, 117649, 54, 390625, 19683, 1024

Offset: 1

Bob Selcoe, Mar 24 2017

A permutation of the natural numbers > 1.

T(1,k)= A005117(m) with m > 1; terms in column k = T(1,k) * A162306(T(1,k)) only not bounded by T(1,k). Let T(1,k) = b. Terms in column k are multiples of b and numbers c such that c | b^e with e >= 0. Alternatively, terms in column k are multiples bc with c those numbers whose prime divisors p also divide b. - Michael De Vlieger, Mar 25 2017

			Array starts:
    2    3     5  6      7  10       11        13  14  15
    4    9    25 12     49  20      121       169  28  45
    8   27   125 18    343  40     1331      2197  56  75
   16   81   625 24   2401  50    14641    371293  98 135
   32  243  3125 36  16807  80   161051   4826809 112 225
   64  729 15625 48 117649 100  1771561  62748517 196 375
  128 2187 78125 54 823543 160 19487171 815730721 224 405
Column 6 is: T(1,6) = 2*5; T(2,6) = 2^2*5; T(3,6) = 2^3*5; T(4,6) = 2*5^2; T(5,6) = 2^4*5, etc.

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A005117 (squarefree numbers), A033845 (column 4), columns 1,2,3,5 are powers of primes, A033846 (column 6), A033847 (column 9), A033849 (column 10).
The columns that are powers of primes have indices A071403(n) - 1. - Michel Marcus, Mar 24 2017
See also A007947; the k-th column of the array corresponds to the numbers with radical A005117(k+1). - Rémy Sigrist, Mar 24 2017
Cf. A284457 (this sequence read by antidiagonals upwards), A285321 (a similar array, but columns come in different order).
Cf. A065642.
Cf. A008479 (index of the row where n is located), A285329 (of the column).

f[n_, k_: 1] := Block[{c = 0, sgn = Sign[k], sf}, sf = n + sgn; While[c < Abs[k], While[! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[sgn < 0, sf--, sf++]; c++]; sf + If[sgn < 0, 1, -1]] (* after Robert G. Wilson v at A005117 *); T[n_, k_] := T[n, k] = Which[And[n == 1, k == 1], 2, k == 1, f@ T[n - 1, k], PrimeQ@ T[n, 1], T[n, 1]^k, True, Module[{j = T[n, k - 1]/T[n, 1] + 1}, While[PowerMod[T[n, 1], j, j] != 0, j++]; j T[n, 1]]]; Table[T[n - k + 1, k], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Mar 25 2017 *)

(define (A284311 n) (A284311bi  (A002260 n) (A004736 n)))
(define (A284311bi row col) (if (= 1 row) (A005117 (+ 1 col)) (A065642 (A284311bi (- row 1) col))))
;; Antti Karttunen, Apr 17 2017

From Antti Karttunen, Apr 17 2017: (Start)
A(1,k) = A005117(1+k), A(n,k) = A065642(A(n-1,k)).
A(A008479(n), A285329(n)) = n for all n >= 2.
(End)

cp's OEIS Frontend

A280269 Irregular triangle T(n,m) read by rows: smallest power e of n that is divisible by m = term k in row n of A162306.

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A372323 A124652(n) is the a(n)-th term in row A372111(n-1) of irregular triangle A162306.

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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.

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A010846 Number of numbers <= n whose set of prime factors is a subset of the set of prime factors of n.

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A243822 Number of k < n such that rad(k) | n but k does not divide n, where rad = A007947.

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A272618 Irregular array read by rows: n-th row contains (in ascending order) the nondivisors 1 <= k < n such that all the prime divisors p of k also divide n.

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A355432 a(n) = number of k < n such that rad(k) = rad(n) and k does not divide n, where rad(k) = A007947(k).

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