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A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors. 6

%I #24 Dec 24 2020 04:48:12

%S 117,775,10309,56347,88723,2896363,9597529,12326221,12654079,25774633,

%T 29817121,63455131,105100903,203822581,261019543,296765173,422857021,

%U 573332713,782481673,900952687,1129152721,3350861677,3703086229,7395290407,9347001661,9350506057

%N Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

%C Nineteen initial terms factored:

%C n a(n) factorization A060681(a(n))/A318505(a(n))

%C 1: 117 = 3^2 * 13, (3)

%C 2: 775 = 5^2 * 31, (10)

%C 3: 10309 = 13^2 * 61, (39)

%C 4: 56347 = 29^2 * 67, (58)

%C 5: 88723 = 17^2 * 307, (136)

%C 6: 2896363 = 41^2 * 1723, (820)

%C 7: 9597529 = 73^2 * 1801, (1314)

%C 8: 12326221 = 59^2 * 3541, (1711)

%C 9: 12654079 = 113^2 * 991, (904)

%C 10: 25774633 = 71^2 * 5113, (2485)

%C 11: 29817121 = 97^2 * 3169, (2328)

%C 12: 63455131 = 89^2 * 8011, (3916)

%C 13: 105100903 = 101^2 * 10303, (5050)

%C 14: 203822581 = 157^2 * 8269, (6123)

%C 15: 261019543 = 349^2 * 2143, (2094)

%C 16: 296765173 = 131^2 * 17293, (8515)

%C 17: 422857021 = 233^2 * 7789, (6757)

%C 18: 573332713 = 331^2 * 5233, (4965)

%C 19: 782481673 = 167^2 * 28057, (13861).

%C Note how the quotient (in the rightmost column) seems always to be a multiple of non-unitary prime factor and less than the unitary prime factor.

%C For p, q prime, if p^2+p+1 = kq and k+1|p-1, then p^2*q is in this sequence. - _Charlie Neder_, Jun 09 2019

%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%t Select[Range[15, 10^6 + 1, 2], And[! PrimePowerQ@ #1, Mod[#1 - #2, #2 - #3] == 0] & @@ {#, DivisorSigma[1, #] - #, Divisors[#][[-2]]} &] (* _Michael De Vlieger_, Jun 22 2019 *)

%o (PARI)

%o A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));

%o A060681(n) = (n-A032742(n));

%o A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);

%o isA326064(n) = if((n%2)&&(2!=isprimepower(n)), my(s=A032742(n), t=sigma(n)-s); (gcd(t-n, n-A032742(n)) == t-n), 0);

%Y Subsequence of A326063.

%Y Cf. A032742, A060681, A246282, A318505.

%Y Cf. also A228058, A325981, A326131, A326141.

%K nonn

%O 1,1

%A _Antti Karttunen_, Jun 06 2019

%E More terms from _Amiram Eldar_, Dec 24 2020

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Last modified March 28 12:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)