A006039
Primitive nondeficient numbers.
Original entry on oeis.org
6, 20, 28, 70, 88, 104, 272, 304, 368, 464, 496, 550, 572, 650, 748, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4030
Offset: 1
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n = 1..8671
- L. E. Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors, Amer. J. Math., 35 (1913), 413-426.
- R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
- Jared Duker Lichtman, The reciprocal sum of primitive nondeficient numbers, Journal of Number Theory, Vol. 191 (2018), pp. 104-118.
- Joshua Zelinsky, The Sum of the Reciprocals of the Prime Divisors of an Odd Perfect or Odd Primitive Non-deficient Number, Integers (2025) Vol. 25, Art. No. A59. See p. 2.
- Index entries for sequences where any odd perfect numbers must occur
-
k = 1; lst = {}; While[k < 4050, If[DivisorSigma[1, k] >= 2 k && Min@Mod[k, lst] > 0, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Mar 09 2017 *)
A338133
Primitive nondeficient numbers sorted by largest prime factor then by increasing size. Irregular triangle T(n, k), n >= 2, k >= 1, read by rows, row n listing those with largest prime factor = prime(n).
Original entry on oeis.org
6, 20, 28, 70, 945, 1575, 2205, 88, 550, 3465, 5775, 7425, 8085, 12705, 104, 572, 650, 1430, 2002, 4095, 6435, 6825, 9555, 15015, 78975, 81081, 131625, 189189, 297297, 342225, 351351, 570375, 63126063, 99198099, 117234117, 272, 748, 1870, 2210, 5355, 8415, 8925, 11492
Offset: 2
Row 1 is empty as there exists no primitive nondeficient number of the form prime(1)^k = 2^k.
Row 2 is (6) as 6 is the only primitive nondeficient number of the form prime(1)^k * prime(2)^m = 2^k * 3^m that is a multiple of prime(2) = 3.
Irregular triangle T(n, k) begins:
n prime(n) row n
2 3 6;
3 5 20;
4 7 28, 70, 945, 1575, 2205;
5 11 88, 550, 3465, 5775, 7425, 8085, 12705;
...
See also the factorization of initial terms below:
6 = 2 * 3,
20 = 2^2 * 5,
28 = 2^2 * 7,
70 = 2 * 5 * 7,
945 = 3^3 * 5 * 7,
1575 = 3^2 * 5^2 * 7,
2205 = 3^2 * 5 * 7^2,
88 = 2^3 * 11,
550 = 2 * 5^2 * 11,
3465 = 3^2 * 5 * 7 * 11,
5775 = 3 * 5^2 * 7 * 11,
7425 = 3^3 * 5^2 * 11,
8085 = 3 * 5 * 7^2 * 11,
12705 = 3 * 5 * 7 * 11^2,
104 = 2^3 * 13,
572 = 2^2 * 11 * 13,
650 = 2 * 5^2 * 13,
1430 = 2 * 5 * 11 * 13,
2002 = 2 * 7 * 11 * 13,
4095 = 3^2 * 5 * 7 * 13,
...
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rownupto(n, u) = { my(res = List(), pr = primes(n), e = vector(n, i, logint(u, pr[i]))); vu = vector(n, i, [0, e[i]]); vu[n][1] = 1; forvec(x = vu, c = prod(i = 1, n, pr[i]^x[i]); if(c <= u && isprimitive(c), listput(res, c) ) ); Set(res) }
isprimitive(n) = { my(f = factor(n), c); if(sigma(f) < 2*n, return(0)); for(i = 1, #f~, c = n / f[i,1]; if(sigma(c) >= c * 2, return(0) ) ); 1 }
for(i = 2, 7, print(rownupto(i, 10^9)))
Showing 1-2 of 2 results.
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