A162306 -id:A162306 - cp's OEIS frontend

A243103 Product of numbers m with 2 <= m <= n whose prime divisors all divide n.

1, 2, 3, 8, 5, 144, 7, 64, 27, 3200, 11, 124416, 13, 6272, 2025, 1024, 17, 35831808, 19, 1024000, 3969, 247808, 23, 859963392, 125, 346112, 729, 2809856, 29, 261213880320000000, 31, 32768, 264627, 18939904, 30625, 26748301344768, 37, 23658496, 369603, 32768000000, 41

Offset: 1

Michael De Vlieger, Aug 19 2014

nonn

This sequence is the product of n-regular numbers.
A number m is said to be "regular" to n or "n-regular" if all the prime factors p of m also divide n.
The divisor is a special case of a regular m such that m also divides n in addition to all of its prime factors p | n.
Analogous to A007955 (Product of divisors of n).
If n is 1 or prime, a(n) = n.
If n is a prime power, a(n) = A007955(n).
Note: b-file ends at n = 4619, because a(4620) has more than 1000 decimal digits.
Product of the numbers 1 <= k <= n such that (floor(n^k/k) - floor((n^k - 1)/k)) = 1. - Michael De Vlieger, May 26 2016

			a(12) = 124416 since 1 * 2 * 3 * 4 * 6 * 8 * 9 * 12 = 124416. These numbers are products of prime factors that are the distinct prime divisors of 12 = {2, 3}.
From _David A. Corneth_, Feb 09 2015: (Start)
Let p# be the product of primes up to p, A002110. Then
a(13#) ~= 8.3069582 * 10 ^ 4133
a(17#) ~= 1.3953000 * 10 ^ 22689
a(19#) ~= 3.8258936 * 10 ^ 117373
a(23#) ~= 6.7960327 * 10 ^ 594048
a(29#) ~= 1.3276817 * 10 ^ 2983168
a(31#) ~= 2.8152792 * 10 ^ 14493041
a(37#) ~= 1.9753840 * 10 ^ 69927040
Up to n = 11# already in the table.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..4619
Encyclopedia Britannica, Regular Number (base-neutral definition)
Eric W. Weisstein, Regular Number (decimal definition)
Wikipedia, Regular Number (sexagesimal / Hamming number definition)

Cf. A162306 (irregular triangle of regular numbers of n), A010846 (number of regular numbers of n), A244974 (sum of regular numbers of n), A007955, A244052 (record transform of regular numbers of n).

Cf. A123275, A283990.

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;
  convert(S,`*`)
end proc:
seq(A(n), n=1..100); # Robert Israel, Feb 09 2015

regularQ[m_Integer, n_Integer] := Module[{omega = First /@ FactorInteger @ m }, If[Length[Select[omega, Divisible[n, #] &]] == Length[omega], True, False]]; a20140819[n_Integer] := Times @@ Flatten[Position[Thread[regularQ[Range[1, n], n]], True]]; a20140819 /@ Range[41]
regulars[n_] := Block[{f, a}, f[x_] := First /@ FactorInteger@ x; a = f[n];{1}~Join~Select[Range@ n, SubsetQ[a, f@ #] &]]; Array[Times @@ regulars@ # &, 12] (* Michael De Vlieger, Feb 09 2015 *)
Table[Times @@ Select[Range@ n, (Floor[n^#/#] - Floor[(n^# - 1)/#]) == 1 &], {n, 41}] (* Michael De Vlieger, May 26 2016 *)

lista(nn) = {vf = vector(nn, n, Set(factor(n)[,1])); vector(nn, n, prod(i=1, n, if (setintersect(vf[i], vf[n]) == vf[i], i, 1)));} \\ Michel Marcus, Aug 23 2014

for(n=1, 100, print1(prod(k=1, n, k^(floor(n^k/k) - floor((n^k - 1)/k))),", ")) \\ Indranil Ghosh, Mar 22 2017

from sympy import primefactors
def A243103(n):
    y, pf = 1, set(primefactors(n))
    for m in range(2,n+1):
        if set(primefactors(m)) <= pf:
            y *= m
    return y # Chai Wah Wu, Aug 28 2014

;; A naive implementation, code for A123275bi given under A123275:
(define (A243103 n) (let loop ((k n) (m 1)) (cond ((= 1 k) m) ((= 1 (A123275bi n k)) (loop (- k 1) (* m k))) (else (loop (- k 1) m)))))
;; Antti Karttunen, Mar 22 2017

a(n) = product of terms of n-th row of irregular triangle A162306(n,k).
a(n) = Product_{k=1..n} k^( floor(n^k/k)-floor((n^k -1)/k) ). - Anthony Browne, Jul 06 2016
From Antti Karttunen, Mar 22 2017: (Start)
a(n) = Product_{k=2..n, A123275(n,k)=1} k.
For n >= 1, A046523(a(n)) = A283990(n).
(End)

A244974 Sum of numbers m <= n whose set of prime divisors is a subset of the set of prime divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 16, 8, 15, 13, 30, 12, 45, 14, 36, 33, 31, 18, 79, 20, 66, 41, 64, 24, 103, 31, 70, 40, 80, 30, 235, 32, 63, 84, 114, 73, 198, 38, 120, 92, 163, 42, 310, 44, 140, 130, 132, 48, 246, 57, 213, 108, 154, 54, 300, 97, 217, 116, 150, 60, 600, 62, 156, 180, 127, 109, 540, 68, 246

Offset: 1

Michael De Vlieger, Jul 08 2014

nonn

a(n) = A000203(n) when n is prime or a perfect prime power (A000961). This is because all products of the prime divisor p in such numbers produce divisors.

a(n) > A000203(n) when n is composite and not a perfect prime power.

			For n = 4, A162306(4) = {1, 2, 4} and a(4) = 7.
For n = 5, A162306(5) = {1, 5} and a(5) = 6.
For n = 6, A162306(6) = {1, 2, 3, 4, 6} and a(6) = 16.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

a(n) = sum of terms of n-th row of triangle A162306(n,k).

Cf. A010846, A000005, A243822.

Table[Total@ Union[{1}, Function[d, Select[Range@ n, Union[d, First /@ FactorInteger@ #] == d &]][First /@ FactorInteger@ n]], {n, 68}] (* or *)
Table[Sum[k (Floor[n^k/k] - Floor[(n^k - 1)/k]), {k, n}], {n, 68}] (* Michael De Vlieger, May 26 2016 *)

a(n) = {summ = 0; spn = factor(n)[,1]~; for (m=1, n, spm = factor(m)[,1]~; if (setintersect(spm, spn) == spm, summ += m);); summ;} \\ Michel Marcus, Jul 17 2014

a(n) = Sum_{k=1..n} k*( floor(n^k/k)-floor((n^k - 1)/k) ). - Anthony Browne, May 25 2016

a(n) = Sum_{j=1..n} Sum_{i=j..gcd(n^j,j)} i. - Wesley Ivan Hurt, Apr 05 2021

A376248 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) <= bigomega(n), where rad = A007947 and bigomega = A001222.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 4, 6, 9, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 4, 5, 10, 25, 1, 11, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 13, 1, 2, 4, 7, 14, 49, 1, 3, 5, 9, 15, 25, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 19, 1, 2, 4, 5, 8, 10, 20, 25, 50, 125

Offset: 1

Michael De Vlieger, Oct 09 2024

Analogous to A162306 regarding m such that rad(m) | n, but instead of taking m <= n, we take m such that bigomega(m) <= bigomega(n).
Row n is a finite set of products of prime power factors p^k (i.e., p^k | n) such that Sum_{p|n} k <= bigomega(n).
For prime power n = p^k, k >= 0 (i.e., n in A000961), row p^k of this sequence is the same as row p^k of A027750 and A162306. Therefore, for prime p, row p of this sequence is the same as row p of A027750 and A162306: {1, p}.
For n in A024619, row n of this sequence does not match row n of A162306, since the former contains gpf(n)^bigomega(n) = A006530(n)^A001222(n), which is larger than n.

			Triangle begins:
   n    row n of this sequence:
  -------------------------------------------
   1:   1;
   2:   1,  2;
   3:   1,  3;
   4:   1,  2   4;
   5:   1,  5;
   6:   1,  2,  3,  4,  6,  9;
   7:   1,  7;
   8:   1,  2,  4,  8;
   9:   1,  3,  9;
  10:   1,  2,  4,  5, 10, 25;
  11:   1, 11;
  12:   1,  2,  3,  4,  6,  8, 9, 12, 18, 27;
        ...
Row n = 10 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 5^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m and the parenthetic 8 are in row 10 of A162306:
   1   2   4  (8)
   5  10
  25*
Row n = 12 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 3^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m are in row 12 of A162306:
   1   2   4   8
   3   6  12
   9  18*
  27*

Michael De Vlieger, Table of n, a(n) for n = 1..17475 (rows n = 1..1000, flattened)
Michael De Vlieger, Log log scatterplot of rows n = 1..2^16 of this sequence, flattened, (3153752 terms).

Cf. A000961, A001221, A001222, A006530, A007947, A010846, A024619, A027750, A162306, A376567.

Table[Clear[p]; MapIndexed[Set[p[First[#2]], #1] &, FactorInteger[n][[All, 1]]]; k = PrimeOmega[n]; w = PrimeNu[n]; Union@ Map[Times @@ MapIndexed[p[First[#2]]^#1 &, #] &, Select[Tuples[Range[0, k], w], Total[#] <= k &] ], {n, 120}]

Row n of this sequence is { m : rad(m) | n, bigomega(m) <= bigomega(n) }.

A376567(n) = binomial(bigomega(n) + omega(n)) = Length of row n, where omega = A001221.

A286563 Triangular table T(n,k) read by rows: T(n,1) = 1, and for 1 < k <= n, T(n,k) = the highest exponent e such that k^e divides n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 20 2017

T(n,k) > 0 for k in row n of A027750. - Michael De Vlieger, May 20 2017

Compare rows to those of triangle A279907, smallest exponent e of n divisible by k. The values of k > -1 in row n of A279907 pertain to k in row n of A162306 rather than k in row n of A027750. - Michael De Vlieger, May 21 2017

			The first fifteen rows of this triangular table:
  1,
  1, 1,
  1, 0, 1,
  1, 2, 0, 1,
  1, 0, 0, 0, 1,
  1, 1, 1, 0, 0, 1,
  1, 0, 0, 0, 0, 0, 1,
  1, 3, 0, 1, 0, 0, 0, 1,
  1, 0, 2, 0, 0, 0, 0, 0, 1,
  1, 1, 0, 0, 1, 0, 0, 0, 0, 1,
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1,
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1,
  1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of the triangle

Lower triangular region of A286561.
Cf. A286564 (same triangle reversed).
Cf. A169594 (row sums).
Cf. also arrays A051731, A286158, A027750, A279907, A280269.

T := (n, k) -> ifelse(k = 1, 1, padic:-ordp(n, k)):
for n from 1 to 12 do seq(T(n, k), k = 1..n) od;  # Peter Luschny, Apr 07 2025

Table[If[k == 1, 1, IntegerExponent[n, k]], {n, 15}, {k, n}] // Flatten (* Michael De Vlieger, May 20 2017 *)

def T(n, k):
    i=1
    if k==1: return 1
    while n%(k**i)==0:
        i+=1
    return i-1
for n in range(1, 21): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 20 2017

(define (A286563 n) (A286561bi (A002024 n) (A002260 n))) ;; For A286561bi see A286561.

T(n,k) = A286561(n,k) listed row by row for n >= 1, k = 1 .. n.

A138109 Positive integers k whose smallest prime factor is greater than the cube root of k and strictly less than the square root of k.

Original entry on oeis.org

6, 15, 21, 35, 55, 65, 77, 85, 91, 95, 115, 119, 133, 143, 161, 187, 203, 209, 217, 221, 247, 253, 259, 287, 299, 301, 319, 323, 329, 341, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703

Offset: 1

David S. Newman, May 04 2008

nonn

This sequence was suggested by Moshe Shmuel Newman.
A020639(n)^2 < a(n) < A020639(n)^3. - Reinhard Zumkeller, Dec 17 2014
In other words, k = p*q with primes p, q satisfying p < q < p^2. - Charles R Greathouse IV, Apr 03 2017
If "strictly less than" in the definition were changed to "less than or equal to" then this sequence would also include the squares of primes (A001248), resulting in A251728. - Jon E. Schoenfield, Dec 27 2022

			6 is a term because the smallest prime factor of 6 is 2 and 6^(1/3) = 1.817... < 2 < 2.449... = sqrt(6).
From _Michael De Vlieger_, Apr 27 2024: (Start):
Table of p*q where p = prime(n) and q = prime(n+k):
n\k   1     2     3     4     5     6     7     8     9    10    11
-------------------------------------------------------------------
1:    6;
2:   15,   21;
3:   35,   55,   65,   85,   95,  115;
4:   77,   91,  119,  133,  161,  203,  217,  259,  287,  301,  329;
     ... (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of A251728 and of A006881. A006094 is a proper subset.

Cf. A020639, A010846, A079047, A162306.

a138109 n = a138109_list !! (n-1)
a138109_list = filter f [1..] where
   f x = p ^ 2 < x && x < p ^ 3 where p = a020639 x
-- Reinhard Zumkeller, Dec 17 2014

s = {}; Do[f = FactorInteger[i]; test = f[[1]][[1]]; If [test < N[i^(1/2)] && test > N[i^(1/3)], s = Union[s, {i}]], {i, 2, 2000}]; Print[s]
Select[Range[1000],Surd[#,3]Harvey P. Dale, May 10 2015 *)

is(n)=my(f=factor(n)); f[,2]==[1,1]~ && f[1,1]^3 > n \\ Charles R Greathouse IV, Mar 28 2017

list(lim)=if(lim<6, return([])); my(v=List([6])); forprime(p=3,sqrtint(1+lim\=1)-1, forprime(q=p+2, min(p^2-2,lim\p), listput(v,p*q))); Set(v) \\ Charles R Greathouse IV, Mar 28 2017

from math import isqrt
from sympy import primepi, primerange
def A138109(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(min(x//p,p**2)) for p in primerange(s+1)))
    return bisection(f,n,n) # Chai Wah Wu, Mar 05 2025

From Michael De Vlieger, Apr 27 2024: (Start)
Let k = a(n); row k of A162306 = {1, p, q, p^2, p*q}, therefore A010846(k) = 5.
A079047(n) = card({ q : p < q < p^2 }), p and q primes. (End)

A361373 Number of prime powers p^m <= n such that p | n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 4, 3, 4, 1, 6, 1, 5, 3, 5, 1, 6, 2, 5, 3, 5, 1, 9, 1, 5, 4, 6, 3, 8, 1, 6, 4, 7, 1, 9, 1, 6, 5, 6, 1, 8, 2, 7, 4, 6, 1, 8, 3, 7, 4, 6, 1, 10, 1, 6, 5, 6, 3, 10, 1, 7, 4, 10, 1, 9, 1, 7, 5, 7, 3, 10, 1, 8, 4, 7, 1, 12, 3, 7

Offset: 1

Michael De Vlieger, Jun 17 2024

Let p be prime. The term "prime power" p^m, m > 0, used here is that of A246655 = A000040 U A246547, the union of primes and perfect prime powers. Essentially, 1 is not considered a prime power.

			Let S = {k <= n : rad(k) | n} = row n of A162306
a(1) = 0 since S = {1} has 0 prime powers.
a(2) = 1 since S = {1, [2]} has 1 prime power.
a(4) = 2 since S = {1, [2, 4]} has 2 prime powers.
a(6) = 3 since S = {1, [2, 3, 4], 6} has 3 prime powers.
a(10) = 4 since S = {1, [2, 4, 5, 8], 10} has 4 prime powers.
a(12) = 5 since S = {1, [2, 3, 4], 6, [8, 9], 12} has 5 prime powers, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Diagram illustrating a(1440) = 20. Terms are arranged according to prime decomposition and sorted vertically. This sequence counts primes (red) and perfect prime powers (gold).

Cf. A000005, A000040, A007947, A010846, A020639, A126706, A138109, A138511, A162306, A246655, A246547.

a := n -> add(ilog[p](n), p in NumberTheory:-PrimeFactors(n)):
seq(a(n), n = 1..92); # Peter Luschny, Jun 20 2024

{0}~Join~Table[Total@ Map[Floor@ Log[#, n] &, FactorInteger[n][[All, 1]]], {n, 2, 120}]

a(n) = if (n==1, 0, my(f=factor(n)[,1]); sum(k=1, #f, logint(n, f[k]))); \\ Michel Marcus, Jun 20 2024

from sympy import integer_log, primefactors
def A361373(n): return sum(integer_log(n,p)[0] for p in primefactors(n)) # Chai Wah Wu, Sep 20 2024

a(n) = Sum_{p | n} floor(log n / log p).
a(n) = number of prime powers in row n of A162306.
a(n) < A000005(n), since A000005 counts 1.
a(n) < A010846(n), since A010846 counts 1.
Let tau = A000005, rad = A007947, rcf = A010846, and lpf = A020639.
a(p) = tau(p) - 1 = rcf(p) - 1 = 1 since S = row p of both A027750 and A162306 = {1, p} contains the prime power p.
a(p^m) = tau(p^m) - 1 = rcf(p^m) = 1 = m since S = row p^m of both A027750 and A162306 = {1, p, p^2, ..., p^m} contains the prime powers {p, p^2, ..., p^m}.
a(k) = tau(k) - 1 = 3 for squarefree composite k = p*q, p < q < p^2 in A138109 since S = row k of A162306 = {1, p, q, p^2, p*q} contains 3 prime powers {p, q, p^2}.
a(k) < tau(k) for k in A138511 and k in A126706 since m = lpf(k)^(-1 + floor(log k / log lpf(k))) is such that m < k but m does not divide k.

A381801 Irregular triangle read by rows: row n lists the residues r mod n of numbers k such that rad(k) | n, where rad = A007947.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 4, 0, 1, 0, 1, 2, 4, 0, 1, 3, 0, 1, 2, 4, 5, 6, 8, 0, 1, 0, 1, 2, 3, 4, 6, 8, 9, 0, 1, 0, 1, 2, 4, 7, 8, 0, 1, 3, 5, 6, 9, 10, 12, 0, 1, 2, 4, 8, 0, 1, 0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16, 0, 1, 0, 1, 2, 4, 5, 8, 10, 12, 16

Offset: 1

Michael De Vlieger, Mar 07 2025

Define S(p,n) to be the set of residues r (mod n) taken by the power range of prime divisor p, i.e., {p^m, m >= 1}.
Define T(n) to be the union of the tensor product of distinct terms in S(p,n) for all p|n, where the products are expressed mod n.
Row n of this triangle is T(n), a superset of row n of A381799.
For n > 1, the intersection of row n of this triangle and row n of A038566 is {1}.

			Table of c(n) = A381800(n) and T(n) for select n:
 n  c(n)  T(n)
-----------------------------------------------------------------------------
 1    1   {0}
 2    2   {0, 1}
 3    2   {0, 1}
 4    3   {0, 1, 2}
 5    2   {0, 1}
 6    5   {0, 1, 2, 3, 4}
 8    4   {0, 1, 2, 4}
 9    3   {0, 1, 3}
10    7   {0, 1, 2, 4, 5, 6, 8}
11    2   {0, 1}
12    8   {0, 1, 2, 3, 4, 6, 8, 9}
14    6   {0, 1, 2, 4, 7, 8}
15    8   {0, 1, 3, 5, 6, 9, 10, 12}
16    5   {0, 1, 2, 4, 8}
18   12   {0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16}
20    9   {0, 1, 2, 4, 5, 8, 10, 12, 16}
24   11   {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18}
28    9   {0, 1, 2, 4, 7, 8, 14, 16, 21}
30   19   {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27}
36   16   {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32}
For n = 10, we have S(2,10) = {1, 2, 4, 6, 8} and S(5,10) = {1, 5}. Therefore we have the following distinct products:
   1  2  4  8  6
   5  0
Hence T(10) = {0, 1, 2, 4, 5, 6, 8}; terms in A003592 belong to these residues (mod 10).
For n = 12, we have S(2,12) = {1, 2, 4, 8} and S(3,12) = {1, 3, 9}. Therefore we have the following distinct products:
   1  2  4  8
   3  6  0
   9
Thus T(12) = {0, 1, 2, 3, 4, 6, 8, 9}, terms in A003586 belong to these residues (mod 12).
For n = 30, we have {1, 2, 4, 8, 16}, {1, 3, 9, 21, 27}, and {1, 5, 25}. Therefore we have the following distinct products:
   1  2  4  8  16         5  10  20         25
   3  6 12 24            15   0
   9 18
  27
Thus T(30) = {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27}; terms in A051037 belong to these residues (mod 30).

Michael De Vlieger, Table of n, a(n) for n = 1..22906 (rows n = 1..500, flattened)
Michael De Vlieger, Plot k in row n at (x,y) = (k,-n), n = 1..36, showing reduced residues mod n in gray and labeling terms in row n. The number n appears on the left in red italic, and row length A381800(n) in blue at right.
Michael De Vlieger, Plot k in row n at (x,y) = (k,-n), n = 1..5000.

Cf. A007947, A038566, A121998, A162306, A381799, A381800.

Table[Union@ Flatten@ Mod[TensorProduct @@ Map[(p = #; NestWhileList[Mod[p*#, n] &, 1, UnsameQ, All]) &, FactorInteger[n][[All, 1]] ], n], {n, 30}]

Row 1 is {0} since 1 is the empty product and the only number that has zero prime factors is 1, congruent to 0 (mod 1).
Row n begins with {0,1} for n > 1.
For prime p, row p = {0,1}.
For prime power p^m, m > 0, row p = union of {0} and {p^i, i < m}.
Row n is a subset of row n of A121998, considering n in A121998 instead as n mod n = 0.
Row n is a superset of row n of A162306, considering n in A162306 instead as n mod n = 0.

A299991 Numbers n for which A243822(n) > A000005(n).

Original entry on oeis.org

30, 42, 60, 66, 70, 74, 78, 82, 84, 86, 90, 94, 98, 102, 106, 110, 114, 118, 120, 122, 126, 130, 132, 134, 138, 140, 142, 146, 150, 154, 156, 158, 162, 165, 166, 168, 170, 174, 178, 180, 182, 186, 190, 194, 195, 198, 202, 204, 206, 210, 214, 218, 220, 222, 226

Offset: 1

Michael De Vlieger, Feb 25 2018

nonn

Composite numbers m have nondivisors k in the cototient such that k | n^e with e > 1. These k appear in row n of A272618 and are enumerated by A243822(n). These nondivisors k are a kind of "regular" number along with divisors d of n; both are listed in row n of A162306 and are together enumerated by A010846(n).
This sequence lists numbers that have more nondivisors k in the cototient of n than divisors d.
This sequence contains all n for which A299990(n) is positive.
The smallest odd term is 165.
For m >= 3, A002110(m) is in a(n).
For m >= 9, A244052(m) is in a(n).

			30 is the first term since it is the smallest number for which A243822(n) > A000005(n), alternatively, for which A010846(n) > 2*A000005(n).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A002110, A010846, A162306, A243822, A272618, A299990.

Select[Range@ 226, Function[n, Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] > 2 DivisorSigma[0, n]]]

A372720 a(n) = A000005(n) - A008479(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 4, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 4, 1, 2, 3, 4, 1, 1, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 3, 4, 3, 7, 1, 5, 1, 3, 1, 10, 3, 3, 3

Offset: 1

Michael De Vlieger, May 13 2024

sign

A095960(50) = 3, a(50) = 2.

a(162) = -2 is the first negative term.

			Table of a(n), b(n) = A000005(n), and c(n) = A008479(n) for n <= 12:
  n  b(n) c(n) a(n)
 ------------------
  1    1    1    0
  2    2    1    1
  3    2    1    1
  4    3    2    1
  5    2    1    1
  6    4    1    3
  7    2    1    1
  8    4    3    1
  9    3    2    1
 10    4    1    3
 11    2    1    1
 12    6    2    4
a(12) = 4 since 12 has 6 divisors {1, 2, 3, 4, 6, 12}, and row 12 of A369609 has 2 terms {6, 12}.
a(18) = 3 since 18 has 6 divisors {1, 2, 3, 6, 9, 18}, and row 18 of A369609 has 3 terms {6, 12, 18}.
a(50) = 2 since 50 has 6 divisors {1, 2, 5, 10, 25, 50}, and row 50 of A369609 has 4 terms {10, 20, 40, 50}
a(162) = -2 since 162 has 10 divisors {1,2,3,6,9,18,27,54,81,162} but row 162 of A369609 has 12 terms {6,12,18,24,36,48,54,72,96,108,144,162}.
a(500) = 0 since 500 has as many divisors {1,2,4,5,10,20,25,50,100,125,250,500} as terms in row 500 of A369609 {10,20,40,50,80,100,160,200,250,320,400,500}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A003557, A008479, A095960, A119288, A162306, A183093, A303554, A355432, A360767, A369609.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], _?(Divisible[r, rad[#]] &)], {n, 120}]

a(n) = my(f=factor(n)[, 1], s); forvec(v=vector(#f, i, [1, logint(n, f[i])]), if(prod(i=1, #f, f[i]^v[i])<=n, s++)); numdiv(n) - s; \\ after A008479 \\ Michel Marcus, Jun 03 2024

a(n) = A095960(n) for n in A303554, i.e., for squarefree n or prime powers n.
a(n) = A095960(n) for n in A360767, i.e., for nonsquarefree composite n such that omega(n) > 1 and A003557(n) < A119288(n), since A008479(n) is the number of terms k in row n of A010846 such that k <= A003557(n).
a(n) = A183093(n) - A355432(n).

A376846 Number of m <= n such that rad(m) | n and Omega(m) > Omega(n), where rad = A007947 and Omega = A001222.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 2, 0, 0, 1, 3, 0, 1, 0, 3, 1, 1, 0, 4, 0, 2, 0, 3, 0, 0, 0, 3, 1, 2, 0, 2, 0, 1, 1, 3, 0, 2, 0, 3, 0, 0, 0, 7, 0, 3, 1, 5, 0, 1, 0, 4, 0, 3, 0, 8, 0, 1, 0, 4, 0, 4, 0, 4, 2

Offset: 1

Michael De Vlieger, Oct 06 2024

nonn

Number of m not exceeding n such that the squarefree kernel of m divides n, and m has more prime factors with repetition than does n.

Number of m in row n of A162306 such that Omega(m) > Omega(n).

			Table of select n such that a(n) > 0:
   n  a(n)  List of m such that Omega(m) > Omega(n).
  -------------------------------------------------
  10   1    {8}
  14   1    {8}
  18   1    {16}
  20   1    {16}
  22   2    {8, 16}
  26   2    {8, 16}
  28   1    {16}
  30   2    {16, 24}
  33   1    {27}
  34   3    {8, 16, 32}
  36   1    {32}
  38   3    {8, 16, 32}
  39   1    {27}
  40   1    {32}
  42   4    {16, 24, 32, 36}

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Hasse diagrams of m in select rows n of A162306 indicating in red those m such that Omega(m) > Omega(n).
Michael De Vlieger, Numbers k for which floor(log k / log lpf(k)) <= bigomega(k), 2024, about zeros in this sequence.

Cf. A000961, A001222, A007947, A010846, A162306.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
{0}~Join~Table[With[{k = PrimeOmega[n]}, Count[Range[n], _?(And[Divisible[n, rad[#]], PrimeOmega[#] > k] &)]], {n, 2, 120}]

a(n) = card({m <= n : rad(m) | n, Omega(m) > Omega(n) }).
a(n) = 0 for prime power n (in A000961).
a(n) < A010846(n).

cp's OEIS Frontend

A243103 Product of numbers m with 2 <= m <= n whose prime divisors all divide n.

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A244974 Sum of numbers m <= n whose set of prime divisors is a subset of the set of prime divisors of n.

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A376248 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) <= bigomega(n), where rad = A007947 and bigomega = A001222.

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A286563 Triangular table T(n,k) read by rows: T(n,1) = 1, and for 1 < k <= n, T(n,k) = the highest exponent e such that k^e divides n.

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A138109 Positive integers k whose smallest prime factor is greater than the cube root of k and strictly less than the square root of k.

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A361373 Number of prime powers p^m <= n such that p | n.

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