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.

1, 3, 5, 6, 7, 11, 13, 14, 17, 19, 20, 22, 23, 27, 29, 31, 37, 38, 41, 43, 44, 45, 46, 47, 49, 53, 54, 56, 59, 61, 62, 67, 68, 71, 73, 79, 83, 86, 89, 92, 94, 96, 97, 99, 101, 103, 107, 109, 113, 116, 118, 125, 126, 127, 131, 134, 137, 139, 142, 149, 150, 151

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).

Links

Table of n, a(n) for n=1..62.

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

Crossrefs

Cf. A000005 (tau), A000203 (sigma).

Cf. A002145 (4k+3 primes), A002476 (6k+1 primes).

Subsequence of A003601.

Subsequences: A065091 (odd primes), A030078 \ {8} (cubes of odd primes).

Cf. A291565 (similar for primitive balanced numbers).

Sequence in context: A072600 A047582 A333217 * A301975 A015814 A227026

Adjacent sequences: A342435 A342436 A342437 * A342439 A342440 A342441

Keyword

nonn

Author

Bernard Schott, Mar 12 2021

Extensions

Name clarified by N. J. A. Sloane, Mar 13 2021

Status

approved