A090706 -id:A090706 - cp's OEIS frontend

A014312 Numbers with exactly 4 ones in binary expansion.

15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 177, 178, 180, 184, 195, 197

Offset: 1

Al Black (gblack(AT)nol.net)

T. D. Noe and Ivan Neretin, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

Cf. A090706.
Cf. A000079, A018900, A014311, A014313, A023688, A023689, A023690, A023691 (Hamming weight = 1, 2, ..., 9), A057168.
Cf. A194882, A194883, A194884, A127324.

Select[ Range[ 180 ], (Count[ IntegerDigits[ #, 2 ], 1 ]==4)& ] (* Olivier Gérard *)

for(n=0,10^3,if(hammingweight(n)==4,print1(n,", "))); \\ Joerg Arndt, Mar 04 2014

print1(t=15); for(i=2, 50, print1(", "t=A057168(t))) \\ M. F. Hasler, Aug 27 2014

$N = 4;
my $vector = 2 ** $N - 1;  # first key (15)
for (1..100) {
  print "$vector, ";
  my ($v, $d) = ($vector, 0);
  until ($v & 1 or !$v) { $d = ($d << 1)|1; $v >>= 1 }
  $vector += $d + 1 + (($v ^ ($v + 1)) >> 2);  # next key
} # Ruud H.G. van Tol, Mar 02 2014

A014312_list = [2**a+2**b+2**c+2**d for a in range(3,6) for b in range(2,a) for c in range(1,b) for d in range(c)] # Chai Wah Wu, Jan 24 2021

from itertools import islice
def A014312_gen(): # generator of terms
    yield (n:=15)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A014312_list = list(islice(A014312_gen(),20)) # Chai Wah Wu, Mar 10 2025

pub const fn next_choice(value: usize) -> usize {
  // Passing a term will return the next number in the sequence
  let zeros = value.trailing_zeros();
  let ones = (value >> zeros).trailing_ones();
  value + (1 << zeros) + (1 << (ones - 1)) - 1
} // Andrew Bennett, Jan 07 2022

a(n+1) = A057168(a(n)). - M. F. Hasler, Aug 27 2014
a(n) = 2^A194882(n-1) + 2^A194883(n-1) + 2^A194884(n-1) + 2^A127324(n-1). - Ridouane Oudra, Sep 06 2020
Sum_{n>=1} 1/a(n) = 1.399770961748474333075618147113153558623203796657745865012742162098738541849... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Extension by Olivier Gérard

A091708 Number of primes less than 10^n having at least one digit 7.

Original entry on oeis.org

1, 9, 68, 549, 4819, 43506, 397756, 3681943, 34344296, 322199867, 3036544958, 28722439343, 272501681533, 2591959274190

Offset: 1

Enoch Haga, Jan 30 2004

			a(1) = 1 because of the 4 primes less than 10^1, 1 has at least one digit 7.

a(n) + A091641(n) = A006880(n).

Cf. A091644, A091645, A091646, A091647, A090705, A090706, A091707, A091709, A091710.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; c = 0; p = 1; Do[ While[ p = NextPrim[p]; p < 10^n, If[ Position[ IntegerDigits[p], 7] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)
Table[Count[Prime[Range[PrimePi[10^n]]],?(DigitCount[#,10,7]>0&)],{n,12}] (* _Harvey P. Dale, Mar 03 2013 *)

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 21 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A091709 Number of primes less than 10^n having at least one digit 8.

Original entry on oeis.org

0, 2, 27, 314, 3217, 31699, 308774, 2987107, 28824402, 277779084, 2674980022, 25752370493, 247919235555, 2387154761520

Offset: 1

Enoch Haga, Jan 30 2004

			a(2) = 2 because of the 25 primes less than 10^2, 2 have at least one digit 8.

a(n) + A091642(n) = A006880(n).

Cf. A091644, A091645, A091646, A091647, A090705, A090706, A091707, A091708, A091710.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; c = 0; p = 1; Do[ While[ p = NextPrim[p]; p < 10^n, If[ Position[ IntegerDigits[p], 8] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)

from sympy import sieve # use slower primerange for larger terms
def a(n): return sum('8' in str(p) for p in sieve.primerange(2, 10**n))
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Apr 23 2021

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 22 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A091710 Number of primes less than 10^n having at least one digit 9.

Original entry on oeis.org

0, 6, 60, 542, 4826, 43359, 397093, 3677641, 34316162, 321993007, 3035059323, 28710966351, 272413818120, 2591276923548

Offset: 1

Enoch Haga, Jan 30 2004

			a(2) = 6 because of the 25 primes less than 10^2, 6 have at least one digit 9.

a(n) + A091643(n) = A006880(n).

Cf. A091644, A091645, A091646, A091647, A090705, A090706, A091707, A091708, A091709.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; c = 0; p = 1; Do[ While[ p = NextPrim[p]; p < 10^n, If[ Position[ IntegerDigits[p], 9] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 22 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A187786 Table read by rows, where n-th row contains all numbers having in binary representation as many zeros and ones as n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 5, 6, 7, 8, 9, 10, 12, 9, 10, 12, 11, 13, 14, 9, 10, 12, 11, 13, 14, 11, 13, 14, 15, 16, 17, 18, 20, 24, 17, 18, 20, 24, 19, 21, 22, 25, 26, 28, 17, 18, 20, 24, 19, 21, 22, 25, 26, 28, 19, 21, 22, 25, 26, 28, 23, 27, 29, 30, 17, 18, 20

Offset: 0

Reinhard Zumkeller, Jan 06 2013

For k = 0..A090706(n)-1: A023416(T(n,k))=A023416(n); A000120(T(n,k))=A000120(n); A053644(n)<=T(n,k)<=A003817(n);
T(n,k) = n for some k;
A187769 contains all rows without repetitions.

			.  n  n-th row              binary                          row length
. --  --------------------- ------------------------------- ----------
.  0  {0}                   {0}                                      1
.  1  {1}                   {1}                                      1
.  2  {2}                   {10}                                     1
.  3  {3}                   {11}                                     1
.  4  {4}                   {100}                                    1
.  5  {5,6}                 {101,110}                                2
.  6  {5,6}                 {101,110}                                2
.  7  {7}                   {111}                                    1
.  8  {8}                   {1000}                                   1
.  9  {9,10,12}             {1001,1010,1100}                         3
. 10  {9,10,12}             {1001,1010,1100}                         3
. 11  {11,13,14}            {1011,1101,1110}                         3
. 12  {9,10,12}             {1001,1010,1100}                         3
. 13  {11,13,14}            {1011,1101,1110}                         3
. 14  {11,13,14}            {1011,1101,1110}                         3
. 15  {15}                  {1111}                                   1
. 16  {16}                  {10000}                                  1
. 17  {17,18,20,24}         {10001,10010,10100,11000}                4
. 18  {17,18,20,24}         {10001,10010,10100,11000}                4
. 19  {19,21,22,25,26,28}   {10011,10101,10110,11001,11010,11100}    6
. 20  {17,18,20,24}         {10001,10010,10100,11000}                4 .

Reinhard Zumkeller, Rows n = 0..255 of triangle, flattened
Index entries for sequences related to binary expansion of n

import List (find)
import Maybe (fromJust)
a187786 n k = a187786_tabf !! n !! k
a187786_row n = fromJust $ find (elem n) a187769_tabf
a187786_tabf = map a187786_row [0..]

A361477 a(n) is the number of integers whose binary expansions have the same multiset of run-lengths as that of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 3, 1, 3, 2, 1, 2, 3, 4, 3, 4, 1, 4, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 3, 4, 6, 6, 5, 6, 6, 4, 5, 1, 5, 6, 5, 4, 3, 2, 6, 6, 1, 6, 5, 6, 6, 1, 6, 4, 6, 2, 3, 2, 1, 2, 3, 4, 6, 12, 5, 12, 3, 12, 10, 6, 10, 4, 10, 12, 6, 4, 5, 6, 10

Offset: 0

Rémy Sigrist, Mar 13 2023

This sequence has similarities with A090706; here we consider multisets of run-lengths, there multisets of digits in binary expansions.

			For n = 18:
- the binary expansion of 18 is "10010",
- the corresponding multiset of run-lengths is m = (1, 2, 1, 1),
- m has 4 terms: 3 times "1" and once "2",
- so a(18) = 4! / (3! * 1!) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A090706, A101211, A140690.

a(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); my (s=Set(r), f=vector(#s)); for (k=1, #r, f[setsearch(s, r[k])]++); (#r)! / prod(k=1, #f, f[k]!) }

from math import factorial, prod
from itertools import groupby
from collections import Counter
def A361477(n): return factorial(len(c:=[len(list(g)) for k, g in groupby(bin(n)[2:])]))//prod(map(factorial,Counter(c).values())) # Chai Wah Wu, Mar 16 2023

a(n) = 1 iff n = 0 or n belongs to A140690.

A378959 Sum, in base 10, of the permutations of the digits of n, written in base 2.

Original entry on oeis.org

0, 1, 3, 3, 7, 14, 14, 7, 15, 45, 45, 45, 45, 45, 45, 15, 31, 124, 124, 186, 124, 186, 186, 124, 124, 186, 186, 124, 186, 124, 124, 31, 63, 315, 315, 630, 315, 630, 630, 630, 315, 630, 630, 630, 630, 630, 630, 315, 315, 630, 630, 630, 630, 630, 630, 315, 630

Offset: 0

Paolo P. Lava, Dec 12 2024

Fixed points are in A000225.

			a(5) = 14 because 5 in base 2 is 101 and the permutations of the digits are 101, 110, 011 that correspond to 5, 6, 3 and 5 + 6 + 3 = 14.

Alois P. Heinz, Table of n, a(n) for n = 0..16384

Cf. A000120, A000225, A003817, A090706, A134346.

a:= proc(n) local k, l;
      l:= convert(n, base, 2); k:= nops(l);
      binomial(k-1, add(i, i=l)-1)*(2^k-1)
    end:
seq(a(n), n=0..56);  # Alois P. Heinz, Dec 12 2024

A378959[n_] := (2^# - 1)*Binomial[# - 1, DigitCount[n, 2, 0]] & [BitLength[n]];
Array[A378959, 100, 0] (* Paolo Xausa, Jan 29 2025 *)

a(n) = A003817(n) * A090706(n). - Alois P. Heinz, Dec 12 2024

cp's OEIS Frontend

A014312 Numbers with exactly 4 ones in binary expansion.

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A091708 Number of primes less than 10^n having at least one digit 7.

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A091709 Number of primes less than 10^n having at least one digit 8.

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A091710 Number of primes less than 10^n having at least one digit 9.

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A187786 Table read by rows, where n-th row contains all numbers having in binary representation as many zeros and ones as n.

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A361477 a(n) is the number of integers whose binary expansions have the same multiset of run-lengths as that of n.

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A378959 Sum, in base 10, of the permutations of the digits of n, written in base 2.

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