A257689 -id:A257689 - cp's OEIS frontend

A257691 Numbers that are not A000120-abundant.

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 101, 103, 107, 109, 111, 113, 119, 121, 123, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 211, 219, 221, 223, 227, 229, 233, 239, 241, 247, 251, 257

Offset: 1

Antti Karttunen, May 11 2015

nonn

A000120-nonabundant numbers: Numbers n for which A192895(n) <= 0.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000120, A192895.
Complement of A175526 (A000120-abundant numbers).
Disjoint union of A175522 (A000120-perfect numbers) and A175524 (A000120-deficient numbers).
Differs from A206074(n-1), A186891(n) and A257688(n) for the first time at n=19, where a(19) = 59, while A206074(18) = A186891(19) = A257688(19) = 55, a term missing from here.
Differs from A257689 for the first time at n=24, where a(24) = 79, while A257689(24) = 77, a term missing from here.

A192895[n_] := DivisorSum[n, Total[IntegerDigits[#, 2]]*(-1)^Boole[# == n]& ]; Reap[For[n=1, n<300, n++, If[A192895[n] <= 0, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Dec 05 2015 *)

A192895(n) = sumdiv(n, d, hammingweight(d)*(-1)^(d==n));
isA257691(n) = (A192895(n) <= 0);
n=i=0; while(i < 10000, n++; if(isA257691(n), i++; write("b257691.txt", i, " ", n))); \\ Charles R Greathouse IV, Feb 07 2013

A257688 After 1, all numbers that are either primes in Z or whose binary representation encodes a polynomial irreducible over GF(2).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 113, 115, 117, 127, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 171, 173, 179, 181, 185, 191, 193, 197, 199, 203, 211, 213, 223, 227, 229, 233, 239, 241, 247, 251, 253, 257, 263, 269

Offset: 1

Antti Karttunen, May 07 2015

nonn

"Encoded in binary representation" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where each coefficient a(k) = 0 or 1).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to polynomials over GF(2)

Union of A008578 and A014580.
Complement of A091212 (Numbers that are composite in Z and reducible in ring GF(2)[X]).
After the initial 1, a subsequence of A206074 (n-th irreducible polynomial over Q (with coefficients 0 or 1) evaluated at x=2), from which this differs for the first time at n=23, where a(23)=71, while A206074(22) = 69, the first term missing from here.
Differs from A186891 for the first time at n=22, where a(22) = 67, while A186891(22) = 65.
Differs from A257689 and A257691 for the first time at n=19, where a(19) = 55, while 55 is missing from both A257689 and A257691.

isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1,2)); \\ From Charles R Greathouse IV
isA257688(n) = ((1 == n) || isprime(n) || isA014580(n));
n = 0; i = 0; while(i < 10000, n++; if(isA257688(n), i++; write("b257688.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A257688 (MATCHING-POS 1 1 (lambda (n) (or (= 1 n) (= 1 (A091225 n)) (= 1 (A010051 n))))))

A192506 Numbers that are neither ludic nor prime.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94

Offset: 1

Reinhard Zumkeller, Jul 05 2011

nonn

Intersection of A002808 and A192607; (1-A010051(a(n)))*(1-A192490(a(n)))=1;
a(n) = A091212(n) for n <= 60.
a(n) = A175526(n) for n <= 53. - Reinhard Zumkeller, Jul 12 2011
In other words, composite numbers that are nonludic. - Antti Karttunen, May 11 2015

Reinhard Zumkeller (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A257689 (complement, either ludic or prime), A192503 (ludic and prime), A192504 (ludic and nonprime), A192505 (nonludic and prime).
Cf. A010051, A192490, A002808, A192607.
a(n) differs from A091212(n) and A205783(n+1) for the first time at n=37, where a(37) = 55, while 55 is missing from both A091212 and A205783.
Differs from A175526 for the first time at n=54, where a(54) = 78, while A175526(54) = 77, a term which is missing from here.

a192506 n = a192506_list !! (n-1)
a192506_list = filter ((== 0) . a010051) a192607_list
(Scheme, with Antti Karttunen's IntSeq-library)
(define A192506 (MATCHING-POS 1 1 (lambda (n) (and (zero? (A192490 n)) (zero? (A010051 n))))))
;; Antti Karttunen, May 07 2015

a3309[nmax_] := a3309[nmax] = Module[{t = Range[2, nmax], k, r = {1}}, While[Length[t] > 0, k = First[t]; AppendTo[r, k]; t = Drop[t, {1, -1, k}]]; r];
ludicQ[n_, nmax_] /; 1 <= n <= nmax := MemberQ[a3309[nmax], n];
terms = 1000;
f[nmax_] := f[nmax] = Select[Range[nmax], !ludicQ[#, nmax] && !PrimeQ[#]&] // PadRight[#, terms]&;
f[nmax = terms];
f[nmax = 2 nmax];
While[f[nmax] != f[nmax/2], nmax = 2 nmax];
seq = f[nmax] (* Jean-François Alcover, Dec 10 2021, after Ray Chandler in A003309 *)

cp's OEIS Frontend

A257691 Numbers that are not A000120-abundant.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A257688 After 1, all numbers that are either primes in Z or whose binary representation encodes a polynomial irreducible over GF(2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

A192506 Numbers that are neither ludic nor prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica