A273130 Numbers which have only positive entries in the difference table of their divisors.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 79, 81, 83, 85, 87, 89, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 128, 129, 131, 133

Offset: 1

Author

Peter Luschny, May 16 2016

Keywords

• nonn

Comments

Primes and powers of primes are in the sequence.

Examples

			85 is in the sequence because the difference table of the divisors of 85 has only entries greater than 0:
[1, 5, 17, 85]
[4, 12, 68]
[8, 56]
[48]
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A014567, A187202, A273102, A273103, A273109, A273157 (complement).

Programs

Mathematica

Select[Range@ 1000, {} == NestWhile[ Differences, Divisors @ #, # != {} && Min[#] > 0 &] &] (* Giovanni Resta, May 16 2016 *)

PARI

has(v)=if(#v<2, v[1]>0, if(vecmin(v)<1, 0, has(vector(#v-1,i,v[i+1]-v[i]))))
is(n)=has(divisors(n)) \\ Charles R Greathouse IV, May 16 2016

Sage

def sf(z):
    D = divisors(z)
    T = matrix(ZZ, len(D))
    for m, d in enumerate(D):
        T[0, m] = d
        for k in range(m-1, -1, -1) :
            T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
            if T[m-k, k] <= 0: return False
    return True
print([z for z in range(1,100) if sf(z)])