A380520
Numbers m such that the sum of squares of nondivisors of m is prime.
Original entry on oeis.org
5, 6, 26, 38, 66, 166, 206, 238, 266, 318, 321, 333, 341, 369, 405, 406, 445, 458, 481, 553, 606, 658, 706, 784, 873, 893, 933, 946, 1125, 1166, 1173, 1273, 1286, 1293, 1353, 1546, 1578, 1606, 1666, 1678, 1705, 1726, 1745, 1773, 1781, 1786, 1858, 1906, 1918, 1941
Offset: 1
Nondivisors of 6 are {4, 5} and their sum of squares 4^2 + 5^2 = 41 is prime.
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filter:= proc(n) local d;
isprime(n*(n+1)*(2*n+1)/6 - add(d^2,d=numtheory:-divisors(n)))
end proc:
select(filter, [$1..2000]); # Robert Israel, Feb 26 2025
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Q1[n_]:=(n > 0) && PrimeQ[n]; Select[Range[2000], Q1[#(#+1)(2#+1)/6 - DivisorSigma[2, #]] &]
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isok(k) = isprime(norml2(setminus([1..k], divisors(k)))); \\ Michel Marcus, Jan 28 2025
A277794
Numbers k such that the sum of proper divisors of k is a prime, and the sum of the numbers less than k that do not divide k is also a prime.
Original entry on oeis.org
4, 21, 85, 129, 201, 237, 324, 325, 517, 549, 669, 837, 865, 1081, 1137, 1161, 1165, 1309, 1389, 1677, 1765, 2169, 2233, 2304, 2305, 2469, 2709, 2737, 2761, 3265, 3297, 3745, 3961, 4161, 4285, 4693, 4705, 4741, 4989, 5061, 5221, 5349, 5817, 5949, 6249, 6457, 6517, 6685, 6789, 6813, 6853, 6921
Offset: 1
21 is in the sequence because 21 has three proper divisors {1, 3, 7}, and therefore seventeen non-divisors {2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, so the sum of proper divisors is 1 + 3 + 7 = 11 (which is prime) and the sum of non-divisors is 2 + 4 + 5 + 6 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 199 (which is also prime).
22 is not in the sequence because its three proper divisors {1, 2, 11} add up to 14, which is composite.
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f:= proc(n) local t; t:= numtheory:-sigma(n) - n; isprime(t) and isprime(n*(n-1)/2 - t) end proc:
select(f, [$1..10^4]); # Robert Israel, Nov 10 2016
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Select[Range[7000], DivisorSigma[0, #1 ((#1 + 1)/2) - DivisorSigma[1, #1]] == 2 && DivisorSigma[0, DivisorSigma[1, #1] - #1] == 2 & ]
A380569
Numbers m such that the sum of cubes of nondivisors of m is prime.
Original entry on oeis.org
22, 82, 130, 144, 154, 178, 226, 274, 309, 322, 325, 514, 562, 565, 586, 670, 778, 1018, 1078, 1081, 1137, 1498, 1618, 1837, 1894, 1906, 1918, 1921, 2182, 2194, 2230, 2254, 2350, 2493, 2497, 2530, 2605, 2686, 2698, 2866, 3130, 3202, 3346, 3370, 3418, 3421, 3502
Offset: 1
Nondivisors of 22 are {3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} and their sum of cubes 3^3 + 4^3 + 5^3 + 6^3 + 7^3 +8^3 + 9^3 + 10^3 + 12^3 + 13^3 + 14^3 + 15^3 + 16^3 + 17^3 + 18^3 + 19^3 + 20^3 + 21^3 = 52021 is prime.
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Q1[n_]:=(n > 0) && PrimeQ[n]; Select[Range[3502], Q1[(#(#+1)/2)^2 - DivisorSigma[3, #]] &]
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isok(m) = isprime(sum(k=1, m-1, if (m%k, k^3))); \\ Michel Marcus, Feb 09 2025
Showing 1-3 of 3 results.
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