cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A231253 Terms of A227891 of the form (p*q)^2, where p <= q are odious primes (A027697).

Original entry on oeis.org

582169, 797449, 1874161, 1934881, 2007889, 2181529, 3024121, 3171961, 4879681, 5387041, 6775609, 9174841, 11771761, 16072081, 18653761, 19070689, 20894041, 22762441, 25694761, 26635921, 29953729, 31214569, 33166081, 40081561, 42081169, 45873529, 48177481, 49463089, 50367409
Offset: 1

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Nov 06 2013

Keywords

Comments

a(1) = 582169 is the smallest term > 1 of A227891 which does not have an evil prime divisor.
The sequence lists numbers of the form (p*q)^2 such that either q > p are both odious primes and among the numbers {p^2, p*q, q^2, p^2*q, q^2*p} there is exactly one odious, or q=p is an odious prime, while p^2 and p^3 are both evil.

Crossrefs

Cf. A000069, A001969, A027697, A227891.

A231175 Let A={2,4,5,8,9,11,14,...} be the sequence of numbers k>=1 such that k+1 is evil (A001969), and let B be the complement of A. The sequence lists numbers for which number of A-divisors equals number of B-divisors.

Original entry on oeis.org

1, 4, 25, 100, 121, 289, 361, 529, 625, 841, 1156, 2116, 2209, 2500, 2809, 3249, 3364, 3481, 4489, 5041, 5929, 6241, 7225, 7921, 10201, 11236, 11449, 12769, 12996, 15625, 17161, 20164, 21025, 22201, 28900, 29584, 30625, 31329, 31684, 32041, 36481, 38809, 40804
Offset: 1

Views

Author

Vladimir Shevelev, Nov 05 2013

Keywords

Comments

This is an analog of A227891. All terms are perfect squares.

Examples

			n=100 has 8 proper divisors {1,2,4,5,10,20,25,50} from which 4 from A, {2,4,5,50} and 4 from B, {1,10,20,25}. So 100 is in the sequence.

Links

Crossrefs

Cf. A227891, A000005, A001969.

Programs

  • Mathematica
    evilQ[n_] := EvenQ[DigitCount[n, 2] // First]; selQ[n_] := Length[Select[d = Most[Divisors[n]], evilQ[#+1]&]] == Length[d]/2; Select[Range[200]^2, selQ] (* Jean-François Alcover, Nov 05 2013 *)

Extensions

More terms from Peter J. C. Moses

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

Views

Author

Vladimir Shevelev, Nov 05 2013

Keywords

Comments

An analog of A227891. All terms are perfect squares.

Examples

			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.

Links

Crossrefs

Cf. A000005, A001969, A227891, A231175.

Programs

  • Mathematica
    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 *)

Extensions

More terms from Peter J. C. Moses, Nov 05 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

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Nov 05 2013

Keywords

Comments

All terms are perfect squares.

Examples

			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.

Links

Crossrefs

Cf. A227891, A231175, A231176, A000005, A000069.

Programs

  • Mathematica
    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

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Nov 05 2013

Keywords

Comments

All terms are perfect squares.

Examples

			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.

Links

Crossrefs

Cf. A227891, A231175, A231176, A231177, A000005, A001969.

Programs

  • Mathematica
    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

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Nov 05 2013

Keywords

Comments

All terms are perfect squares.

Examples

			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.

Links

Crossrefs

Cf. A227891, A231175, A231176, A092498, A231178, A000005, A000069.

Programs

  • Mathematica
    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 *)

A231254 Odious primes p (A027697) such that p^2 and p^3 are evil (A027699).

Original entry on oeis.org

37, 47, 107, 137, 233, 331, 463, 491, 557, 587, 607, 631, 653, 733, 823, 829, 883, 947, 971, 997, 1153, 1187, 1193, 1231, 1249, 1321, 1327, 1493, 1543, 1567, 1663, 1667, 1669, 1709, 1787, 1801, 1933, 1987, 2011, 2027, 2087, 2143, 2161, 2213, 2269, 2273, 2311
Offset: 1

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Nov 06 2013

Keywords

Comments

Sequence {a(n)^4} is a subsequence of A227891 such that a(1)^4 = 1874161 is the smallest power of an odious prime that is in A227891.

Links

Crossrefs

Cf. A001969, A027697, A027699, A227891.

Programs

  • Mathematica
    evilQ[n_]:=EvenQ[DigitCount[n,2][[1]]];
odiousQ[n_]:=OddQ[DigitCount[n, 2][[1]]];
Select[Range[2000],PrimeQ[#]&&odiousQ[#]&&evilQ[#^2]&&evilQ[#^3]&] (* Peter J. C. Moses, Nov 08 2013 *)
Showing 1-7 of 7 results.

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