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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. 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 (list; graph; refs; listen; history; text; internal format)
 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, 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

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Last modified November 30 18:31 EST 2023. Contains 367461 sequences. (Running on oeis4.)