A377071 -id:A377071 - cp's OEIS frontend

A378631 Numbers that set records in A377071.

1, 6, 12, 24, 30, 60, 120, 210, 420, 840, 1680, 2310, 3360, 4620, 6720, 9240, 18480, 36960, 60060, 73920, 120120, 147840, 240240, 480480, 960960, 1921920, 3843840, 7687680, 8168160, 15375360, 16336320, 30750720, 32672640, 65345280, 130690560, 261381120, 310390080

Offset: 1

Michael De Vlieger, Dec 02 2024

Numbers n that set records for binomial(bigomega(n)+omega(n)-1, omega(n)-1), where bigomega = A001222 and omega = A001221.
a(n) is of the form 2^k * P(i), k >= 0, where primorial P = A002110.
Proper subset of A070175.

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

Cf. A000079, A001221, A001222, A002110, A007947, A070175, A377071.

f[x_] := Block[{i, k, m, nn, p}, nn = Product[Prime[j], {j, x}]; Set[{k, i, p}, Range[0, 2]]; {1}~Join~Union@ Reap[Until[i > x, While[Set[m, 2^k*p] <= nn, Sow[m]; k++]; k = 0; i++; p *= Prime[i] ] ][[-1, 1]] ] (* generate A070175 *);
r = 0; Reap[Do[If[# > r, r = #; Sow[n]] &@ Binomial[#2 + #1 - 1, #1 - 1] & @@ {PrimeNu[n], PrimeOmega[n]}, {n, f[10]}] ][[-1, 1]]

A377070 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, 2, 3, 4, 5, 4, 6, 9, 7, 8, 9, 4, 10, 25, 11, 8, 12, 18, 27, 13, 4, 14, 49, 9, 15, 25, 16, 17, 8, 12, 18, 27, 19, 8, 20, 50, 125, 9, 21, 49, 4, 22, 121, 23, 16, 24, 36, 54, 81, 25, 4, 26, 169, 27, 8, 28, 98, 343, 29, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 31

Offset: 1

Michael De Vlieger, Oct 25 2024

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), that is, numbers m such that rad(m) | n and m has the same number of prime factors with repetition than does n.

			Triangle begins:
    n    row n of this sequence:
   -------------------------------------------
    1:   {1}
    2:   {2}
    3:   {3}
    4:   {4}
    5:   {5}
    6:   {4, 6, 9}
    7:   {7}
    8:   {8}
    9:   {9}
   10:   {4, 10, 25}
   ...                       (Select rows appear below)
   12:   {8, 12, 18, 27}
   14:   {4, 14, 49}
   15:   {9, 15, 25}
   18:   {8, 12, 18, 27}
   20:   {8, 20, 50, 125}
   24:   {16, 24, 36, 54, 81}
   30:   {8, 12, 18, 20, 27, 30, 45, 50, 75, 125}
   42:   {8, 12, 18, 27, 28, 42, 63, 98, 147, 343}
   60:   {16, 24, 36, 40, 54, 60, 81, 90, 100, 135, 150, 225, 250, 375, 625}.
.
Diagrams of the rank omega(n)-1 simplexes created by row n of this sequence for select n, ordering k in row n by prime decomposition. The number k = n appears in brackets:
Rank 3:
   n = 30:                    n = 42:
             8                         8
           /  \                      /  \
         12 -- 20                  12 -- 28
        /  \  /  \                /  \  /  \
      18 --[30]-- 50            18 --[42]-- 98
     /  \  /  \  /  \          /  \  /  \  /  \
   27 -- 45 -- 75 -- 125     27 -- 63 --147 -- 343
.
   n = 60:     16
              /  \
            24 -- 40
           /  \  /  \
         36 --[60]-- 50
        /  \  /  \  /  \
      54 -- 90 -- 75 --125
     /  \  /  \  /  \  /  \
   81 --150 --135 --375 --625
.
Rank 4:
   n = 210:
   16
        40
   24   56
             100
        60   140
   36   84   196
                   250
             150   350
        90  [210]  490
   54  126   294   686
                            625
                     375    875
              225    525   1225
        135   315    735   1715
   81   189   441   1029   2401

Michael De Vlieger, Table of n, a(n) for n = 1..12021, (rows n = 1..1500, flattened)
Michael De Vlieger, Diagrams of select a(n) illustrating rank omega(n)-1 simplexes formed by the arrangement of terms in row n by prime power decomposition.
Michael De Vlieger, Log log scatterplot of a(n), rows n = 1..65536 (1278755 terms).

Cf. A001221, A001222, A007947, A024619, A376248, A377071.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[k = PrimeOmega[n]; Select[Range[n^PrimeNu[n]], Divisible[n, rad[#]] && PrimeOmega[#] == k &], {n, 30}]

Row n of this sequence is { m : rad(m) | n, bigomega(m) = bigomega(n) }.
For prime p, row p of this sequence is {p}, generally for prime power p^k, row p^k of this sequence is {p^k}.
For n in A024619, row n of this sequence has more than 1 term.
A377071(n) = length of row n of this sequence.

A378182 Sum of row n of A378180.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 25, 1, 10, 9, 15, 1, 25, 1, 47, 11, 14, 1, 90, 6, 16, 13, 77, 1, 80, 1, 31, 15, 20, 13, 90, 1, 22, 17, 250, 1, 116, 1, 161, 58, 26, 1, 301, 8, 47, 21, 215, 1, 90, 17, 554, 23, 32, 1, 490, 1, 34, 90, 63, 19, 212, 1, 347, 27, 152

Offset: 1

Michael De Vlieger, Nov 20 2024

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

Cf. A000203, A007947, A024619, A244974, A376567, A377071, A378180.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Block[{k}, Table[k = PrimeOmega[n];
  Total@ Select[Range[n^PrimeNu[n]],
    Divisible[n, rad[#]] && PrimeOmega[#] < k &], {n, 60}]]

a(n) = A376567(n) - A377071(n).
For prime p, a(p) = 1.
For prime power p^k, a(p^k) = A244974(p^k)-p^k = A000203(p^k)-p^k.
a(2^k) = 2^k - 1.
For n in A024619, a(n) != A244974(n).

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A378631 Numbers that set records in A377071.

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A377070 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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A378182 Sum of row n of A378180.

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