A231176
Let A={1,3,4,7,8,10,13,15,...} be the sequence of numbers k>=1 such that k+2 is evil (A001969), let B be the complement of A. The sequence lists numbers for which the number of A-divisors equals the number of B-divisors.
Original entry on oeis.org
1, 4, 25, 36, 100, 121, 289, 361, 529, 625, 841, 1156, 1764, 2116, 2209, 2500, 2809, 3249, 3364, 3481, 4489, 5041, 5929, 6241, 7225, 7396, 7921, 10201, 11236, 11449, 12769, 12996, 15625, 17161, 20164, 21025, 22201, 27556, 28900, 30276, 30625, 31329, 31684
Offset: 1
n=100 has 8 proper divisors {1,2,4,5,10,20,25,50} from which 4 from A {1,4,10,25} and 4 from B {2,5,20,50}. So 100 is in the sequence.
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odiousQ[n_]:=OddQ[DigitCount[n,2][[1]]];
Select[Range[100],0==Length[#]-2Length[Select[#,odiousQ[#+2]&]]&[Most[Divisors[#^2]]]&]^2 (* Peter J. C. Moses, Nov 08 2013 *)
A231177
Let A = {1,4,5,8,10,11,13,...} be the sequence of numbers k>=1 such that k+3 is odious (A000069), and let B be the complement of A. The sequence lists the numbers for which the number of A-divisors equals the number of B-divisors.
Original entry on oeis.org
1, 4, 9, 49, 196, 289, 961, 1156, 1369, 1849, 3249, 3844, 5476, 6889, 7921, 8281, 10609, 12769, 12996, 14161, 15129, 16129, 17689, 19321, 22801, 24649, 25281, 26569, 27889, 28561, 29584, 31329, 31684, 32761, 39601, 42436, 44944, 45369, 49729, 51076, 52441
Offset: 1
n=196 has 8 proper divisors {1,2,4,7,14,28,49,98} from which 4 from A {1,4,28,49} and 4 from B {2,7,14,98}. So 196 is in the sequence.
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odiousQ[n_]:=OddQ[DigitCount[n,2][[1]]];
Select[Range[200],0==Length[#]-2Length[Select[#,odiousQ[#+3]&]]&[Most[Divisors[#^2]]]&]^2 (* Peter J. C. Moses, Nov 08 2013 *)
A231178
Let A={1,2,5,6,8,11,13,...} be the sequence of numbers k>=1 such that k+4 is evil (A001969), and let B be the complement of A. The sequence lists numbers for which the number of A-divisors equals number of B-divisors.
Original entry on oeis.org
1, 9, 49, 289, 324, 676, 961, 1369, 1849, 3249, 4356, 6084, 6889, 7921, 8281, 8836, 10609, 11236, 12769, 14161, 14884, 15129, 16129, 17689, 19321, 21316, 22500, 22801, 24649, 25281, 26569, 27889, 28561, 30276, 31329, 32761, 39601, 44944, 45369, 45796, 47524
Offset: 1
324 has 14 proper divisors {1,2,3,4,6,9,12,18,27,36,54,81,108,162} from which 7 from A {1,2,6,36,54,81,162} and 7 from B {3,4,9,12,18,27,108}. So 324 is in the sequence.
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odiousQ[n_]:=OddQ[DigitCount[n,2][[1]]];
Select[Range[200],0==Length[#]-2Length[Select[#,odiousQ[#+4]&]]&[Most[Divisors[#^2]]]&]^2 (* Peter J. C. Moses, Nov 08 2013 *)
A231180
Let A={2,3,6,8,9,11,14,...} be the sequence of numbers k>=1 such that k+5 is odious (A000069). Let B be the complement of A. The sequence lists numbers for which the number of A-divisors equals the number of B-divisors.
Original entry on oeis.org
1, 4, 9, 16, 36, 121, 144, 289, 441, 484, 529, 1156, 1369, 1600, 1764, 2025, 2116, 2209, 3249, 3481, 4624, 5041, 5476, 6241, 6889, 7056, 7569, 7921, 8100, 8464, 8649, 8836, 11449, 12321, 12769, 12996, 13924, 14641, 15129, 16641, 20164, 24336, 24649, 24964
Offset: 1
n=16 has 4 proper divisors {1,2,4,8} from which 2 from A {2,8} and 2 from B {1,4}. So 16 is in the sequence.
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odiousQ[n_]:=OddQ[DigitCount[n,2][[1]]];
Select[Range[200],0==Length[#]-2Length[Select[#,odiousQ[#+5]&]]&[Most[Divisors[#^2]]]&]^2 (* Peter J. C. Moses, Nov 08 2013 *)
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