Michael De Vlieger, <a href="/A376248/b376248_1.txt">Table of n, a(n) for n = 1..17475</a> (rows n = 1..1000, flattened)

proposed

approved

editing

proposed

allocated for Michael De Vlieger

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

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

1,3

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.

Michael De Vlieger, <a href="/A376248/b376248_1.txt">Table of n, a(n) for n = 1..17475</a> (rows n = 1..1000, flattened)

Michael De Vlieger, <a href="/A376248/a376248.png">Log log scatterplot of rows n = 1..2^16 of this sequence, flattened</a>, (3153752 terms).

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.

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*

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}]

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

allocated

nonn,tabf

Michael De Vlieger, Oct 09 2024

approved

editing

allocated for Michael De Vlieger

recycled

allocated

reviewed

approved

proposed

reviewed

editing

proposed

The smallest nonnegative integer that can be made out of n segments on a seven-segment calculator display. -1 is used when no nonnegative integer is possible.

-1, 1, -1, 4, 2, 0, 8, 10, 18, 22, 20, 28, 68, 88, 108, 188, 200, 208, 288, 688, 888, 1088, 1888, 2008, 2088, 2888, 6888, 8888, 10888, 18888, 20088, 20888, 28888, 68888, 88888, 108888, 188888, 200888, 208888, 288888, 688888, 888888, 1088888, 1888888, 2008888

1,4

Typically, on a seven-segment digital display, the digit one is made out of 2 segments, two out of 5 segments, three out of 5 segments, four out of 4 segments, five out of 5 segments, six out of 6 segments, seven out of 4 segments, eight out of 7 segments, nine out of 6 segments, and zero out of 6 segments.

a(n) = min {k such that A010371(k) = n}, or -1 if the set is empty. - Jason Yuen, Sep 24 2024

a(8) = 10 because the smallest nonnegative integer that can be made out of 8 segments is 10 (2 + 6 segments).

Cf. A010371.

sign,base,changed

recycled

Dave Rutt, Sep 16 2024

sign,base,dumb,uned,changed

proposed

editing