A355596 Numbers all of whose divisors are alternating numbers (A030141).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 32, 36, 41, 43, 47, 49, 50, 54, 58, 61, 63, 67, 69, 81, 83, 87, 89, 94, 98, 101, 103, 107, 109, 123, 125, 127, 129, 141, 145, 147, 149, 161, 163, 167, 181, 183, 189, 214, 218, 250, 254, 290, 298

Offset: 1

Author

Bernard Schott, Jul 12 2022

Keywords

• nonn
• base

Comments

The smallest alternating number that is not a term is 30, because of 15.

Examples

			32 is a term since all the divisors of 32, i.e., 1, 2, 4, 8, 16 and 32, are alternating numbers
		

Crossrefs

Subsequence of A030141.

Cf. A355593, A355594, A355595.

Similar sequences: A062687, A190217, A329419, A337941.

Programs

Mathematica

q[n_] := AllTrue[Divisors[n], !MemberQ[Differences[Mod[IntegerDigits[#], 2]], 0] &]; Select[Range[300], q] (* Amiram Eldar, Jul 12 2022 *)

PARI

isokd(n, d=digits(n))=for(i=2, #d, if((d[i]-d[i-1])%2==0, return(0))); 1; \\ A030141
isok(m) = sumdiv(m, d, isokd(d)) == numdiv(m); \\ Michel Marcus, Jul 12 2022

Python

from sympy import divisors
def p(d): return 0 if d in "02468" else 1
def c(n):
    if n < 10: return True
    s = str(n)
    return all(p(s[i]) != p(s[i+1]) for i in range(len(s)-1))
def ok(n):
    return c(n) and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(1, 200) if ok(k)]) # Michael S. Branicky, Jul 12 2022

Extensions

a(51) and beyond from Michael S. Branicky, Jul 12 2022