A380989 Position of first appearance of n in A380958 (number of prime factors minus sum of distinct prime exponents).

1, 6, 30, 210, 900, 7776, 27000, 279936, 810000, 9261000, 24300000, 362797056, 729000000, 13060694016, 21870000000, 408410100000, 656100000000, 16926659444736, 19683000000000, 609359740010496, 590490000000000, 18010885410000000, 17714700000000000

Offset

0,2

Comments

Is this sequence strictly increasing?

From David Consiglio, Jr., Feb 20 2025: (Start)

The answer to the question above is: no, a(21) < a(20). And all subsequent odd indexed terms are lower than their even predecessors.

All terms must be a product of x primes (with multiplicity) to the y power where x-y = n and x mod y = 0. There are very few combinations of numbers that meet these criteria, so checking all of them to find the minimum outcome is quite fast.

Example --> n=5

6 primes to the 1 power --> 6 distinct primes

2*3*5*7*11*13 = 30030

7 primes to the 2 power -- disallowed (5 mod 2 = 1)

8 primes to the 3 power -- disallowed (4 mod 3 = 1)

9 primes to the 4 power -- disallowed (9 mod 4 = 1)

10 primes to the 5 power --> 2 distinct primes

2*2*2*2*2*3*3*3*3*3 = 7776

The minimum value is 7776 and thus a(5) = 7776. (End)

Links

David Consiglio, Jr., Table of n, a(n) for n = 0..100

David Consiglio, Jr., Python program

Example

The terms together with their prime indices begin:
        1: {}
        6: {1,2}
       30: {1,2,3}
      210: {1,2,3,4}
      900: {1,1,2,2,3,3}
     7776: {1,1,1,1,1,2,2,2,2,2}
    27000: {1,1,1,2,2,2,3,3,3}
   279936: {1,1,1,1,1,1,1,2,2,2,2,2,2,2}
   810000: {1,1,1,1,2,2,2,2,3,3,3,3}
  9261000: {1,1,1,2,2,2,3,3,3,4,4,4}

Mathematica

prisig[n_]:=If[n==1, {}, Last/@FactorInteger[n]];
q=Table[Total[prisig[n]]-Total[Union[prisig[n]]], {n, 10000}];
mnrm[s_]:=If[Min@@s==1, mnrm[DeleteCases[s-1, 0]]+1, 0];
Table[Position[q, k][[1, 1]], {k, 0, mnrm[q+1]-1}]

Crossrefs

Position of first appearance of n in A001222 - A136565.

For factors instead of exponents we have A280286 (sorted A381075), firsts of A280292.

For indices instead of exponents we have A380956 (sorted A380957), firsts of A380955.

A000040 lists the primes, differences A001223.

A005361 gives product of prime exponents.

A055396 gives least prime index, greatest A061395.

A056239 (reverse A296150) adds up prime indices, row sums of A112798.

A124010 lists prime exponents (signature); A001221, A051903, A051904.

Cf. A046660, A071625, A075254, A075255, A076694, A130091, A130092, A290106, A380986.

Sequence in context: A057896 A336509 A147779 * A054721 A374660 A074111

Adjacent sequences: A380986 A380987 A380988 * A380990 A380991 A380992

Keyword

nonn

Author

Gus Wiseman, Feb 18 2025

Extensions

a(10)-a(11) from Michel Marcus, Feb 20 2025

a(12) and beyond from David Consiglio, Jr., Feb 20 2025

Status

approved