A000069 -id:A000069 - cp's OEIS frontend

A248138 Consider the partition of consecutive evil numbers (A001969) into minimal blocks such that concatenation of numbers in each block is an odious number (A000069). Sequence lists numbers of evil numbers in each block.

3, 2, 2, 3, 2, 3, 4, 2, 2, 5, 2, 5, 3, 2, 2, 2, 2, 2, 6, 4, 3, 6, 4, 7, 4, 5, 3, 4, 3, 2, 3, 3, 4, 2, 2, 2, 2, 2, 2, 2, 4, 3, 3, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 3, 5, 2, 3, 3, 6, 2, 4, 5, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 4, 3, 2, 2, 2, 3

Offset: 1

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

nonn

The blocks of consecutive evil numbers of the partition are

0,3,5| 6,9| 10,12| 15,17,18| 20,23| 24,27,29| 30,33,34,36| 39,40| 43,45| 46,48,51,53,54| 57,58| 60,63,65,66,68|, etc.

Cf. A000069, A001969, A248140, A248145, A248146.

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.

Original entry on oeis.org

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)).

A355968 a(n) is the smallest number that has exactly n odious divisors (A000069).

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 64, 56, 84, 112, 1024, 168, 4096, 448, 336, 728, 36309, 672, 57057, 1456, 1344, 7168, 105105, 2184, 6384, 24150, 5376, 5208, 405405, 4368, 389025, 11648, 20020, 72618, 10416, 8736, 927675, 114114, 48300, 24024, 855855, 17472, 1426425, 40040

Offset: 1

Bernard Schott, Jul 21 2022

a(n) <= 2^(n-1) with equality for n = 1, 2, 3, 4, 5, 7, 11, 13 up to a(44).

			a(6) = 28 since 28 has 6 divisors {1, 2, 4, 7, 14, 28} that have all an odd number of 1's in their binary expansion: 1, 10, 100, 111, 1110 and 11100; also, no positive integer smaller than 28 has six divisors that are odious.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000069, A093696, A227872, A330289, A355969.

Similar sequences: A087997, A333456, A355303, A355699.

f[n_] := DivisorSum[n, 1 &, OddQ[DigitCount[#, 2, 1]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[20, 10^6] (* Amiram Eldar, Jul 21 2022 *)

isod(n) = hammingweight(n) % 2; \\ A000069
a(n) = my(k=1); while (sumdiv(k, d, isod(d)) != n, k++); k; \\ Michel Marcus, Jul 22 2022

from sympy import divisors
from itertools import count, islice
def c(n): return bin(n).count("1")&1
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 1, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 36))) # Michael S. Branicky, Jul 25 2022

More terms from Amiram Eldar, Jul 21 2022

A129676 Permutation sequence generated by the "odious numbers" (A000069), by swapping n-th natural number by the (n-g)-th sequentially, where g=min(odious(n+1)-odious(n)-1,n-1).

Original entry on oeis.org

3, 1, 5, 4, 6, 2, 9, 7, 10, 12, 11, 8, 15, 13, 17, 16, 18, 20, 19, 14, 23, 21, 24, 22, 27, 25, 29, 28, 30, 26, 33, 31, 34, 36, 35, 32, 39, 37, 40, 38, 43, 41, 45, 44, 46, 48, 47, 42, 51, 49, 53, 52, 54, 50, 57, 55, 58, 60, 59, 56, 63, 61, 65, 64, 66, 68, 67, 62, 71, 69, 72, 70

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. It is also notable that the difference of the number of fixed points and the number of cycles never differs by more than 3, 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 A129677, Cf. A128754, A128756, A129674, A129675, A129680 and A000069.

{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(A000069)

A223024 Numbers k such that 3^k is odious (A000069).

Original entry on oeis.org

0, 4, 7, 10, 11, 13, 15, 16, 19, 20, 22, 28, 29, 30, 34, 36, 41, 43, 46, 48, 49, 50, 53, 54, 56, 62, 63, 65, 66, 67, 68, 69, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 100, 101, 103, 107, 108, 110, 111, 113, 114, 115, 117, 118, 119, 120

Offset: 1

Vladimir Shevelev, Mar 29 2013

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A000069, A001969, A000244, A237545.

Complement of A371970.

select(t -> convert(convert(3^t,base,2),`+`)::odd, [$0..200]); # Robert Israel, Oct 10 2016

Select[Range[0,200], OddQ[DigitCount[3^#,2][[1]]]&] (* Peter J. C. Moses, Mar 30 2013 *)

isok(n) = (hammingweight(3^n) % 2) == 1; \\ Michel Marcus, Oct 11 2016

A230226 Odd odious numbers (A000069) which can be written as a product of two odious numbers > 1.

Original entry on oeis.org

49, 91, 121, 133, 143, 217, 247, 259, 273, 341, 361, 385, 403, 451, 475, 481, 511, 517, 539, 589, 611, 625, 637, 651, 665, 671, 721, 737, 749, 767, 775, 779, 793, 803, 805, 847, 861, 871, 875, 889, 925, 949, 959, 961, 1001, 1015, 1027, 1029, 1053, 1057, 1067

Offset: 1

Vladimir Shevelev, Oct 12 2013

			From _Bernard Schott_, Sep 23 2019: (Start)
49 = 7 * 7 with 7 = 111_2 and 49 = 110001_2 hence 49 is a term.
91 = 7 * 13 with 7 = 111_2, 13 = 1101_2 and 91 = 1011011_2, hence 91 is a term. (End)

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

Cf. A000069, A230213.

odiousQ[n_] := OddQ[DigitCount[n, 2][[1]]]; fQ[n_] := Module[{f, i}, If[PrimeQ[n], False, f = Select[Divisors[n], # > 1 && # <= Sqrt[n] &]; i = 1; While[i <= Length[f] && ! (odiousQ[f[[i]]] && odiousQ[n/f[[i]]]), i++]; i <= Length[f]]]; Select[Range[1, 1000, 2], odiousQ[#] && fQ[#] &] (* T. D. Noe, Oct 16 2013 *)

Extended by T. D. Noe, Oct 16 2013

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

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.

cp's OEIS Frontend

A248138 Consider the partition of consecutive evil numbers (A001969) into minimal blocks such that concatenation of numbers in each block is an odious number (A000069). Sequence lists numbers of evil numbers in each block.

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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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A355968 a(n) is the smallest number that has exactly n odious divisors (A000069).

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A129676 Permutation sequence generated by the "odious numbers" (A000069), by swapping n-th natural number by the (n-g)-th sequentially, where g=min(odious(n+1)-odious(n)-1,n-1).

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A223024 Numbers k such that 3^k is odious (A000069).

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A230226 Odd odious numbers (A000069) which can be written as a product of two odious numbers > 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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