A001969 -id:A001969 - cp's OEIS frontend

A276443 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = A000069(1+a(n-1)), a(A088359(n)) = A001969(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 15, 13, 14, 16, 17, 18, 20, 24, 19, 23, 30, 21, 27, 25, 22, 29, 31, 26, 28, 32, 33, 34, 36, 40, 48, 35, 39, 46, 60, 37, 43, 54, 41, 51, 49, 38, 45, 58, 47, 63, 61, 42, 53, 55, 50, 44, 57, 59, 62, 52, 56, 64, 65, 66, 68, 72, 80, 96, 67, 71, 78, 92, 120, 69, 75, 86, 108, 73, 83, 102, 81

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276444.
Cf. A000069, A001969, A004001, A080677, A087686, A088359, A093879.
Similar or related permutations: A003188, A276441, A276445 (compare the scatter plots).

(definec (A276443 n) (cond ((< n 2) n) ((zero? (A093879 (- n 1))) (A000069 (+ 1 (A276443 (+ -1 (A080677 n)))))) (else (A001969 (+ 1 (A276443 (+ -1 (A004001 n))))))))

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = A000069(1+a(A080677(n)-1)), otherwise [when n is in A088359], a(n) = A001969(1+a(A004001(n)-1)).
As a composition of other permutations:
a(n) = A003188(A276441(n)).

A129674 Permutation sequence generated by the "evil numbers" (A001969), by swapping n-th natural number by the (n-g)-th sequentially (iteratively), where g=min(evil(n+1)-evil(n)-1,n-1).

Original entry on oeis.org

3, 2, 1, 6, 4, 7, 5, 10, 8, 12, 11, 13, 15, 14, 9, 18, 16, 20, 19, 21, 17, 24, 22, 25, 27, 26, 23, 30, 28, 31, 29, 34, 32, 36, 35, 37, 33, 40, 38, 41, 43, 42, 39, 46, 44, 48, 47, 49, 51, 50, 45, 54, 52, 55, 53, 58, 56, 60, 59, 61, 63, 62, 57, 66, 64, 68, 67, 69, 65, 72, 70, 73

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), May 01 2007

In contrast to A128754 and A128756 (which are generated analogously from the primes and the lucky numbers, respectively), this sequence seems consisting solely of fixed points and cycles of length 4, 5 and 6 (apart from the initial cycle of length 2). It is also notable that the difference of the number of fixed points and the number of cycles never differs by more than 4, up to index 10000, according to numerical tests. Thus the ratio of the number of fixed points to the number of cycles seems to be asymptotically equal to unity.

Inverse of A129675, Cf. A128754, A128756, A129676, A129677, A129678 and A001969.

{vperm(z)=local(n,m,q,v,x,j,g);
/* Permutation of positive integers so that starting with the sequence of positive integers, sequentially swap the i-th term with max(i-g(i),1)-th term, where g(i)=z[i+1]-z[i]-1. */
j=matsize(z)[2]-1;n=j-z[j]+z[j-6];v=vector(j);x=vector(n);for(i=1,j,v[i]=i);
for(i=1,j,g=min(z[i+1]-z[i]-1,i-1);q=v[i];v[i]=v[i-g];v[i-g]=q);for(i=1,n,x[i]=v[i]);return(x)}
a=vperm(A001969)

A230385 Table read by rows: Least set of n evil numbers (A001969) such that any two or more add up to an odious number (A000069); ordered by total sum of the elements, then by the size of the largest element(s).

Original entry on oeis.org

0, 3, 5, 9, 10, 12, 5, 9, 17, 33, 33, 34, 36, 40, 48, 257, 264, 278, 288, 326, 384

Offset: 1

Vladimir Shevelev and M. F. Hasler, Oct 17 2013

Row sums are given in A230386. See A230384 for a "dual" version.

Is this sequence finite, or is there for any n at least one admissible set of n evil numbers, i.e., such that any sum of two or more elements add up to an odious number?

			The table reads
n=1: {0} with sum = 0,
n=2: {3,5} with sum = 8,
n=3: {9, 10, 12} with sum = 31 (the set {5, 9, 17} having the same sum but a larger maximum),
n=4: {5, 9, 17, 33} with sum = 64,
n=5: {33, 34, 36, 40, 48 } with sum = 191.
n=6: {257, 264, 278, 288, 326, 384} with sum = 1797.
For example, for n=4, all 11 numbers 5+9=14,5+17=22,5+33=38,9+17=26, 9+33=42, 17+33=50, 5+9+17=31, 5+9+33=47, 5+17+33=55, 9+17+33=59, 5+9+17+33=64 are odious.

M. F. Hasler, in reply to V. Shevelev, Peculiar sets of evil numbers (Cf. A001969), SeqFan list, Oct 17 2013

a(16)-a(21) by M. F. Hasler, Oct 18 2013

A230387 Least sum of a set of n odious numbers (A000069) such that the sum of two or more is an evil number (A001969).

Original entry on oeis.org

1, 3, 17, 139, 795, 3903, 28575

Offset: 1

M. F. Hasler, Oct 17 2013

Is this sequence finite, or is there for any n at least one admissible set of n odious numbers, i.e., such that any sum of two or more elements add up to an evil number?

			For n=1 to 4, we have the sets
n=1: {1} with sum = 1,
n=2: {1, 2} with sum = 3
n=3: {2, 7, 8} with sum = 17,
n=4: {4, 19, 49, 67} with sum = 139.
E.g., for n=3, the numbers 2, 7 and 8 have an odd bit sum, but 2+7, 2+8, 7+8 and 2+7+8 all have an odd bit sum.
For n=4, we also have the admissible set {14, 31, 44, 61} which has a smaller maximal element, but a larger total sum.
n=5: {42, 84, 138, 174, 357} with sum = 795.
n=6: {168, 348, 372, 702, 906, 1407} with sum = 3903.
n=7: {2273, 2274, 2276, 2280, 2288, 3296, 13888} with sum = 28575.

M. F. Hasler, in reply to V. Shevelev, Peculiar sets of evil numbers (Cf. A001969), SeqFan list, Oct 17 2013

Cf. A230386, A230385.

A69=select(is_A69=n->bittest(hammingweight(n),0),vector(700,n,n)); A230387(n,m=9e9)={ local(v=vector(n,i,i), ve=vector(n,i,A69[i]), t=0, s=vector(n,i,if(i>1,A230387(i-1))), ok(e)=!forstep(i=3,2^#e-1,2, is_A69( sum( j=1,#t=vecextract(e,i),t[j] )) && return), inc(i)=for(j=1,n-i,v[j]=j); for(j=n-i+1,n-1, v[j]++

Row sums of A230384.

a(5)-a(6) from Charles R Greathouse IV, Oct 18 2013

a(7) from Donovan Johnson, Oct 27 2013

A230851 Numbers with divisors which are half odious (A000069) and half evil (A001969).

Original entry on oeis.org

3, 5, 6, 10, 12, 17, 20, 23, 24, 29, 33, 34, 39, 40, 43, 46, 48, 53, 57, 58, 63, 65, 66, 68, 69, 71, 78, 80, 83, 86, 87, 89, 92, 95, 96, 101, 105, 106, 111, 113, 114, 115, 116, 117, 119, 123, 125, 126, 130, 132, 136, 138, 139, 141, 142, 145, 149, 156, 160, 163, 166, 171, 172, 174, 177, 178, 183

Offset: 1

Juri-Stepan Gerasimov, Oct 31 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000069, A001969, A167171, A227872.

aQ[n_] := DivisorSum[n, (-1)^DigitCount[#, 2][[1]] &] == 0; Select[Range[200], aQ] (* Amiram Eldar, Sep 23 2019 *)

is(n)=!sumdiv(n,d,(-1)^hammingweight(d)) \\ Charles R Greathouse IV, Oct 31 2013

Numbers n such that d(n) = 2*A227872(n) where A227872(n) is number of odious divisors of n.

Corrected by Charles R Greathouse IV, Oct 31 2013

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

Vladimir Shevelev, Nov 05 2013

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

			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.

Vladimir Shevelev, A set of sequences of perfect squares

Cf. A227891, A000005, A001969.

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 *)

More terms from Peter J. C. Moses

A248009 Partition of the positive odd integers into minimal blocks such that the concatenation of the numbers in each block is an evil number (A001969). Sequence lists the evil numbers obtained in this way.

Original entry on oeis.org

135, 7911, 131517, 19212325272931, 33, 35373941, 43, 45, 4749, 51, 53, 5557, 596163, 65, 676971, 737577, 798183, 85, 8789, 9193, 95, 9799101103105, 107109, 111, 113, 115117119, 121123125127129, 131133, 135, 137139, 141, 143145147, 149, 151153155157, 159

Offset: 1

Vladimir Shevelev, Oct 05 2014

The numbers of the consecutive positive odd integers in the blocks of the partition are 3,3,3,7,1,4,1,1,2,1,1,2,3,1,3,3,3,1,2,2,...

			a(1)=135, since 1 and 13 are odious numbers, while 135 is evil.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A248138, A248140, A248145, A248146, A248171, A248172.

lista(nn) = {s = ""; forstep(n=1, nn, 2, ns = concat(s, Str(n)); if ((hammingweight(eval(ns)) % 2) == 0, print1(ns, ", "); s = "", s = ns););} \\ Michel Marcus, Oct 09 2014

More terms from Peter J. C. Moses, Oct 09 2014

A248140 Consider the partition of the consecutive odious numbers (A000069) into minimal blocks such that concatenation of numbers in each block is an evil number (A001969). Sequence gives the number of odious numbers in each block.

Original entry on oeis.org

2, 7, 3, 2, 3, 3, 4, 5, 5, 2, 2, 2, 4, 2, 2, 2, 2, 2, 3, 5, 2, 4, 3, 4, 2, 3, 3, 3, 3, 3, 3, 4, 9, 2, 2, 2, 2, 7, 5, 3, 2, 3, 2, 4, 4, 4, 2, 3, 4, 2, 3, 4, 3, 4, 2, 2, 3, 2, 2, 2, 9, 2, 5, 2, 5, 4, 4, 2, 4, 4, 2, 3, 3, 8, 3, 2, 2, 3, 2, 3, 2, 2, 2, 4, 2, 4, 3

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Oct 02 2014

nonn

The blocks of consecutive odious numbers of the partition are

1,2| 4,7,8,11,13,14,16| 19,21,22|25,26| 28,31,32| 35,37,38| 41,42,44,47| 49,50,52,55,56| 59,61,62,64,67| 69,70| 73,74| 76,79|, etc.

Cf. A000069, A001969, A248138, A248145, A248146.

A248171 Partition of the positive integers on minimal blocks such that concatenation of numbers in each block is an evil number (A001969). Sequence lists the evil numbers obtained in this way.

Original entry on oeis.org

12, 3, 45, 6, 78, 9, 10, 1112, 1314, 15, 161718, 1920, 2122, 23, 24, 2526, 27, 2829, 30, 3132, 33, 34, 3536, 37383940, 414243, 4445464748, 495051, 52535455, 5657, 58, 5960, 6162, 63, 646566, 6768, 6970, 71, 72, 7374, 75, 7677787980, 818283, 8485868788, 89, 90

Offset: 1

Vladimir Shevelev, Oct 03 2014

The numbers of the consecutive positive integers over blocks of the partition are 2,1,2,1,2,1,1,2,2,1,3,2,2,1,1,2,1,...

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000069 (odious), A001969 (evil), A248009, A248138, A248140, A248172 (similar, with odious).

from itertools import count
def evil(n): return bin(n)[2:].count('1') % 2 == 0
def aupton(terms):
    alst, t = [], 0
    for k in count(1):
        t = int(str(t) + str(k))
        if evil(t):
            alst.append(t)
            t = 0
            if len(alst) >= terms: return alst
print(aupton(45)) # Michael S. Branicky, Dec 03 2021

More terms from Peter J. C. Moses, Oct 04 2014

A248478 Evil numbers (A001969) becoming odious (A000069) if any digit is deleted (zeros allowed).

Original entry on oeis.org

12, 17, 18, 24, 27, 48, 71, 72, 77, 78, 111, 113, 114, 116, 119, 141, 144, 149, 169, 216, 221, 222, 225, 226, 228, 252, 255, 281, 282, 288, 311, 325, 387, 411, 414, 441, 442, 444, 447, 449, 474, 479, 497, 525, 526, 550, 556, 559, 562, 619, 621, 622, 649, 674

Offset: 1

Vladimir Shevelev, Oct 07 2014

			149 is in the sequence, since 149 = 2^7 + 2^4 + 2^2 + 1 is evil, while 49, 19 and 14 are odious.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000069, A001969, A248642, A248644, A248659.

More terms from Peter J. C. Moses, Oct 11 2014

cp's OEIS Frontend

A276443 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = A000069(1+a(n-1)), a(A088359(n)) = A001969(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

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A129674 Permutation sequence generated by the "evil numbers" (A001969), by swapping n-th natural number by the (n-g)-th sequentially (iteratively), where g=min(evil(n+1)-evil(n)-1,n-1).

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A230385 Table read by rows: Least set of n evil numbers (A001969) such that any two or more add up to an odious number (A000069); ordered by total sum of the elements, then by the size of the largest element(s).

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A230387 Least sum of a set of n odious numbers (A000069) such that the sum of two or more is an evil number (A001969).

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A230851 Numbers with divisors which are half odious (A000069) and half evil (A001969).

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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.

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A248009 Partition of the positive odd integers into minimal blocks such that the concatenation of the numbers in each block is an evil number (A001969). Sequence lists the evil numbers obtained in this way.

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A248140 Consider the partition of the consecutive odious numbers (A000069) into minimal blocks such that concatenation of numbers in each block is an evil number (A001969). Sequence gives the number of odious numbers in each block.

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A248171 Partition of the positive integers on minimal blocks such that concatenation of numbers in each block is an evil number (A001969). Sequence lists the evil numbers obtained in this way.

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