OFFSET
1,2
COMMENTS
A positive integer m is an arithmetic number (A003601) if sigma(m) (A000203) is a multiple of tau(m) (A000005). Since sigma and tau are multiplicative, if m and q are arithmetic numbers and gcd(m,q)=1, m*q is also an arithmetic number. This sequence eliminates these non-primitive terms.
Some subsequences:
- odd primes (A065091),
- squares of primes of the form 6m+1 (A002476),
- cubes of odd primes (A030078 \ {8}),
- semiprimes 2*p where prime p is of the form 4k+3 (A002145),
- Integers equal to 4*p where p is a prime of the form 6k-1 (A007528).
EXAMPLE
6 and 17 are arithmetic numbers, gcd(6,17)=1, so 102 is a non-primitive arithmetic number while 6 and 17 are primitive arithmetic numbers.
7 is an arithmetic number; gcd(7,7) = 7; as sigma(49) = 57 and tau(49) = 3, sigma(49)/tau(49) = 19, so 7*7 = 49 is a primitive term because gcd(7,7) <> 1.
MATHEMATICA
arithQ[n_] := arithQ[n] = Divisible[DivisorSigma[1, n], DivisorSigma[0, n]]; primArithQ[n_] := primArithQ[n] = (n == 1) || (arithQ[n] && !AnyTrue[Most @ Rest @ Divisors[n], CoprimeQ[#, n/#] && arithQ[#] && arithQ[n/#] &]); Select[Range[150], primArithQ] (* Amiram Eldar, Mar 12 2021 *)
PROG
(PARI) isar(n) = !(sigma(n)%numdiv(n)); \\ A003601
isok(n) = {if (isar(n), fordiv(n, d, if ((d>1) && (d<n) && (gcd(d, n/d)==1) && isar(d) && isar(n/d), return(0))); return (1); ); } \\ Michel Marcus, Mar 12 2021
KEYWORD
nonn
AUTHOR
Bernard Schott, Mar 12 2021
EXTENSIONS
Name clarified by N. J. A. Sloane, Mar 13 2021
STATUS
approved