A179983 Positive integers m such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90

Offset: 1

Author

Matthew Vandermast, Jan 15 2011

Keywords

• nonn

Comments

Numbers m such that A181819(m) is a term of A025487.

Examples

			The prime signature of 20 = 2^2*5 is (2,1). Since the largest number appearing in 20's prime signature is 2, and 1 appears as many times as 2, 20 is a member of this sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index to sequences related to prime signature.

Crossrefs

Includes all squarefree numbers (A005117); also includes all members of A054753, A085987, A163569, A182862, A182863.

Cf. A025487, A181819.

Programs

Maple

isA179983 := proc(n)
    local es,me,k ;
    # list of exponents in prime signature
    es := [seq(op(2,pe), pe =ifactors(n)[2])] ;
    # maximum exponent
    me := max(op(es)) ;
    for k from me to 2 by -1 do
        if numboccur(es,k-1) < numboccur(es,k) then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 1 to 100 do
    if isA179983(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Mar 21 2023

Mathematica

q[n_] := Module[{t = SortBy[Tally[FactorInteger[n][[;; , 2]]], First], t1, t2}, t1 = t[[;; , 1]]; t2 = t[[;; , 2]]; Sort[t1] == Range[Length[t1]] && Max[Differences[t2]] < 1]; Select[Range[100], q] (* Amiram Eldar, Aug 04 2024 *)