

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.


0



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
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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 nonprimitive terms.
Some subsequences:
 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 6k1 (A007528).


LINKS



EXAMPLE

6 and 17 are arithmetic numbers, gcd(6,17)=1, so 102 is a nonprimitive 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

Subsequences: A065091 (odd primes), A030078 \ {8} (cubes of odd primes).
Cf. A291565 (similar for primitive balanced numbers).


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



