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A342438 Primitive arithmetic numbers: terms of A003601 not of the form m*q where m, q > 1 are both terms of A003601 with gcd(m,q) = 1.

%I A342438 #48 Apr 15 2021 23:49:06
%S A342438 1,3,5,6,7,11,13,14,17,19,20,22,23,27,29,31,37,38,41,43,44,45,46,47,
%T A342438 49,53,54,56,59,61,62,67,68,71,73,79,83,86,89,92,94,96,97,99,101,103,
%U A342438 107,109,113,116,118,125,126,127,131,134,137,139,142,149,150,151
%N A342438 Primitive arithmetic numbers: terms of A003601 not of the form m*q where m, q > 1 are both terms of A003601 with gcd(m,q) = 1.
%C A342438 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.
%C A342438 Some subsequences:
%C A342438   - odd primes (A065091),
%C A342438   - squares of primes of the form 6m+1 (A002476),
%C A342438   - cubes of odd primes (A030078 \ {8}),
%C A342438   - semiprimes 2*p where prime p is of the form 4k+3 (A002145),
%C A342438   - Integers equal to 4*p where p is a prime of the form 6k-1 (A007528).
%e A342438 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.
%e A342438 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.
%t A342438 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 *)
%o A342438 (PARI) isar(n) = !(sigma(n)%numdiv(n)); \\ A003601
%o A342438 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
%Y A342438 Cf. A000005 (tau), A000203 (sigma).
%Y A342438 Cf. A002145 (4k+3 primes), A002476 (6k+1 primes).
%Y A342438 Subsequence of A003601.
%Y A342438 Subsequences: A065091 (odd primes), A030078 \ {8} (cubes of odd primes).
%Y A342438 Cf. A291565 (similar for primitive balanced numbers).
%K A342438 nonn
%O A342438 1,2
%A A342438 _Bernard Schott_, Mar 12 2021
%E A342438 Name clarified by _N. J. A. Sloane_, Mar 13 2021