A014499 -id:A014499 - cp's OEIS frontend

A372689 Positive integers whose binary indices (positions of ones in reversed binary expansion) sum to a prime number.

2, 3, 4, 6, 9, 11, 12, 16, 18, 23, 26, 29, 33, 38, 41, 43, 44, 48, 50, 55, 58, 61, 64, 69, 71, 72, 74, 79, 81, 86, 89, 91, 92, 96, 101, 103, 104, 106, 111, 113, 118, 121, 131, 132, 134, 137, 142, 144, 149, 151, 152, 154, 159, 163, 164, 166, 169, 174, 176, 181

Offset: 1

Gus Wiseman, May 18 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Note the function taking a set s to its binary rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

			The terms together with their binary expansions and binary indices begin:
   2:      10 ~ {2}
   3:      11 ~ {1,2}
   4:     100 ~ {3}
   6:     110 ~ {2,3}
   9:    1001 ~ {1,4}
  11:    1011 ~ {1,2,4}
  12:    1100 ~ {3,4}
  16:   10000 ~ {5}
  18:   10010 ~ {2,5}
  23:   10111 ~ {1,2,3,5}
  26:   11010 ~ {2,4,5}
  29:   11101 ~ {1,3,4,5}
  33:  100001 ~ {1,6}
  38:  100110 ~ {2,3,6}
  41:  101001 ~ {1,4,6}
  43:  101011 ~ {1,2,4,6}
  44:  101100 ~ {3,4,6}
  48:  110000 ~ {5,6}
  50:  110010 ~ {2,5,6}
  55:  110111 ~ {1,2,3,5,6}
  58:  111010 ~ {2,4,5,6}
  61:  111101 ~ {1,3,4,5,6}

Numbers k such that A029931(k) is prime.
Union of prime-indexed rows of A118462.
For even instead of prime we have A158704, odd A158705.
For prime indices instead of binary indices we have A316091.
The prime case is A372885, indices A372886.
A000040 lists the prime numbers, A014499 their binary indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A058698 counts partitions of prime numbers, strict A064688.
A372471 lists binary indices of primes, row-sums A372429.
A372687 counts strict partitions of prime binary rank, counted by A372851.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A035100, A071814, A096111, A023506, A277319, A372688, A372850, A372887.

Select[Range[100],PrimeQ[Total[First /@ Position[Reverse[IntegerDigits[#,2]],1]]]&]

A071600 Numbers k such that k and prime(k) have the same number of 1's in their binary representation.

Original entry on oeis.org

1, 3, 13, 19, 21, 23, 25, 30, 44, 45, 47, 57, 60, 61, 71, 77, 98, 99, 101, 103, 107, 108, 110, 118, 121, 125, 158, 159, 178, 179, 184, 186, 187, 188, 209, 215, 218, 221, 237, 244, 246, 247, 248, 249, 251, 279, 287, 312, 334, 335, 346, 350, 359, 361, 362, 365

Offset: 1

Benoit Cloitre, Jun 01 2002

			221 = 11011101 in base 2, prime(221) = 1381 = 10101100101 in base 2, both have 6 "1's" in their binary representation, hence 221 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000120, A014499, A033549, A049084, A072439, A090455.

Select[Range[400],DigitCount[#,2,1]==DigitCount[Prime[#],2,1]&] (* Harvey P. Dale, Mar 09 2015 *)

for(n=1,1000,s=1; if(sum(i=1,length(binary(n)), component(binary(n),i))==sum(i=1,length(binary(prime(n))), component(binary(prime(n)),i)),print1(n,",")))

is(n)=hammingweight(n)==hammingweight(prime(n)) \\ Charles R Greathouse IV, Mar 07 2013

a(n) = A049084(A072439(n)); A000120(a(n)) = A000120(A072439(n)). - Reinhard Zumkeller, Jun 17 2002

A090455(a(n)) = 0, A000120(a(n)) = A014499(a(n)).

A072084 In prime factorization of n replace all primes with their numbers of 1's in binary representation.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 3, 3, 4, 1, 2, 4, 3, 2, 6, 3, 4, 2, 4, 3, 8, 3, 4, 4, 5, 1, 6, 2, 6, 4, 3, 3, 6, 2, 3, 6, 4, 3, 8, 4, 5, 2, 9, 4, 4, 3, 4, 8, 6, 3, 6, 4, 5, 4, 5, 5, 12, 1, 6, 6, 3, 2, 8, 6, 4, 4, 3, 3, 8, 3, 9, 6, 5, 2, 16, 3, 4, 6, 4, 4, 8, 3, 4, 8, 9, 4, 10, 5, 6

Offset: 1

Reinhard Zumkeller, Jun 14 2002

a(n)1.

Multiplicative with a(p^e) = A000120(p)^e. - Mitch Harris, Apr 19 2005

			a(30) = a(2*3*5) = a(2)*a(3)*a(5) = 1*2*2 = 4,
as a(2)=a('10')=1, a(3)=a('11')= 2 and a(5)=a('101')=2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000120, A072085, A072087, A027746, A014499.

a072084 = product . map a000120 . a027746_row
-- Reinhard Zumkeller, Feb 10 2013

A072084 := proc(n) local a,c; readlib(ifactors):
a := n -> add(i,i=convert(n, base, 2));
mul(a(c[1])^c[2],c=ifactors(n)[2]) end:
# Peter Luschny, Jan 16 2010

a[n_] := Times @@ Power @@@ (FactorInteger[n] /. {p_Integer, e_} :> {DigitCount[p, 2, 1], e}); Array[a, 100] (* Jean-François Alcover, Feb 09 2018 *)

a(n)=my(f=factor(n)); f[,1]=apply(hammingweight, f[,1]); factorback(f) \\ Charles R Greathouse IV, Aug 06 2015

A072084 = lambda n: prod(p.digits(base=2).count(1)**m for p,m in factor(n)) # D. S. McNeil, Jan 17 2011

Completely multiplicative with a(p) = number of 1's in binary representation of prime p.

Multiplicative with a(p^e) = A000120(p)^e

A080085 Number of factors of 2 in the factorial of the n-th prime, counted with multiplicity.

Original entry on oeis.org

1, 1, 3, 4, 8, 10, 15, 16, 19, 25, 26, 34, 38, 39, 42, 49, 54, 56, 64, 67, 70, 74, 79, 85, 94, 97, 98, 102, 104, 109, 120, 128, 134, 135, 145, 146, 152, 159, 162, 168, 174, 176, 184, 190, 193, 194, 206, 216, 222, 224, 228, 232, 236, 244, 255, 259, 265, 266, 273, 277

Offset: 1

Paul D. Hanna, Jan 26 2003

nonn

n-th prime minus number of 1's in binary representation of n-th prime. [Juri-Stepan Gerasimov, May 17 2010]

Antti Karttunen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A276133 (first differences).
Cf. A000040, A080084, A080086, A080087, A294898, A371151.
Column 1 of array A379008, incremented by one.

lst={};Do[p=Prime[n];s=0;While[p>1,p=IntegerPart[p/2];s+=p;];AppendTo[lst,s],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 28 2009 *)

vector(58, n, valuation(prime(n)!, 2)) \\ Arkadiusz Wesolowski, Feb 22 2014

a(n) = prime(n) - hammingweight(prime(n)); \\ Joerg Arndt, Feb 22 2014

a(n) = Sum_{k=1..L} floor( p_n /2^k ), where L = log(p_n)/log(2), where p_n is the n-th prime.
a(n) = A000040(n) - A014499(n). - Juri-Stepan Gerasimov, May 17 2010
a(n) = 1+A294898(A000040(n)). - Antti Karttunen, Dec 14 2024

A177718 a(n) = |(number of 1's in binary representation of prime(n)) - (number of 0's in binary representation of prime(n))|.

Original entry on oeis.org

0, 2, 1, 3, 2, 2, 1, 1, 3, 3, 5, 0, 0, 2, 4, 2, 4, 4, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 3, 1, 7, 2, 2, 0, 0, 2, 2, 0, 2, 2, 2, 2, 6, 2, 0, 2, 2, 6, 2, 2, 2, 6, 2, 6, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 3, 3, 1, 3, 5, 3, 5, 7, 1, 1, 1, 1, 1, 1, 5, 1, 5, 5, 1, 1, 3, 5, 3, 7, 5, 5, 5, 7, 7, 4, 2, 0, 2, 0, 0, 0, 2

Offset: 1

Juri-Stepan Gerasimov, May 12 2010, May 18 2010

			a(1)=0 because 2 = 10_2 and abs(1-1) = 0;
a(2)=2 because 3 = 11_2 and abs(0-2) = 2;
a(3)=1 because 5 = 101_2 and abs(1-2) = 1.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A000040, A000120, A004676.

A023416 := proc(n) a := 0 ; for d in convert(n,base,2) do if d = 0 then a := a+1 ; end if; end do; a ; end proc:
A000120 := proc(n) a := 0 ; for d in convert(n,base,2) do if d = 1 then a := a+1 ; end if; end do; a ; end proc:
A037861 := proc(n) A023416(n)-A000120(n) ; end proc:
A177718 := proc(n) abs(A037861(ithprime(n))) ; end proc: seq(A177718(n),n=1..120) ; # R. J. Mathar, May 15 2010
# second Maple program:
a:= n-> abs(add(2*i-1, i=Bits[Split](ithprime(n)))):
seq(a(n), n=1..105);  # Alois P. Heinz, Jan 18 2022

nzmnu[n_]:=Module[{z=DigitCount[n,2,0]},Abs[2z-IntegerLength[n,2]]]; nzmnu/@ Prime[Range[110]] (* Harvey P. Dale, Feb 15 2015 *)

from sympy import isprime
print([abs(bin(n)[2:].count("1") - bin(n)[2:].count("0")) for n in range (0,1000) if isprime(n)]) # Karl-Heinz Hofmann, Jan 18 2022

a(n) = abs(A014499(n) - A035103(n)).

a(n) = abs(A037861(prime(n))). - R. J. Mathar, May 15 2010

Corrected at three or more places by R. J. Mathar, May 15 2010

A372438 Least binary index equals greatest prime index.

Original entry on oeis.org

6, 18, 20, 54, 56, 60, 100, 162, 168, 176, 180, 280, 300, 392, 416, 486, 500, 504, 528, 540, 840, 880, 900, 1088, 1176, 1232, 1248, 1400, 1458, 1500, 1512, 1584, 1620, 1936, 1960, 2080, 2432, 2500, 2520, 2640, 2700, 2744, 2912, 3264, 3528, 3696, 3744, 4200

Offset: 1

Gus Wiseman, May 04 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Are there any squarefree terms > 6?

			The binary indices of 60 are {3,4,5,6}, the prime indices are {1,1,2,3}, and 3 = 3, so 60 is in the sequence.
The terms together with their prime indices begin:
     6: {1,2}
    18: {1,2,2}
    20: {1,1,3}
    54: {1,2,2,2}
    56: {1,1,1,4}
    60: {1,1,2,3}
   100: {1,1,3,3}
   162: {1,2,2,2,2}
   168: {1,1,1,2,4}
   176: {1,1,1,1,5}
   180: {1,1,2,2,3}
   280: {1,1,1,3,4}
   300: {1,1,2,3,3}
The terms together with their binary expansions and binary indices begin:
     6:            110 ~ {2,3}
    18:          10010 ~ {2,5}
    20:          10100 ~ {3,5}
    54:         110110 ~ {2,3,5,6}
    56:         111000 ~ {4,5,6}
    60:         111100 ~ {3,4,5,6}
   100:        1100100 ~ {3,6,7}
   162:       10100010 ~ {2,6,8}
   168:       10101000 ~ {4,6,8}
   176:       10110000 ~ {5,6,8}
   180:       10110100 ~ {3,5,6,8}
   280:      100011000 ~ {4,5,9}
   300:      100101100 ~ {3,4,6,9}

Same length: A071814, zeros of A372441.
Same sum: A372427, zeros of A372428.
Same maxima: A372436, zeros of A372442.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A014499, A030101, A061712, A318283, A355536, A359401, A359402, A372429-A372433.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Min[bix[#]]==Max[prix[#]]&]

A001511(a(n)) = A061395(a(n)).

A372885 Prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.

Original entry on oeis.org

2, 3, 11, 23, 29, 41, 43, 61, 71, 79, 89, 101, 103, 113, 131, 137, 149, 151, 163, 181, 191, 197, 211, 239, 269, 271, 281, 293, 307, 331, 349, 353, 373, 383, 401, 433, 457, 491, 503, 509, 523, 541, 547, 593, 641, 683, 701, 709, 743, 751, 761, 773, 827, 863, 887

Offset: 1

Gus Wiseman, May 19 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

The indices of these primes are A372886.

			The binary indices of 89 are {1,4,5,7}, with sum 17, which is prime, so 89 is in the sequence.
The terms together with their binary expansions and binary indices begin:
    2:         10 ~ {2}
    3:         11 ~ {1,2}
   11:       1011 ~ {1,2,4}
   23:      10111 ~ {1,2,3,5}
   29:      11101 ~ {1,3,4,5}
   41:     101001 ~ {1,4,6}
   43:     101011 ~ {1,2,4,6}
   61:     111101 ~ {1,3,4,5,6}
   71:    1000111 ~ {1,2,3,7}
   79:    1001111 ~ {1,2,3,4,7}
   89:    1011001 ~ {1,4,5,7}
  101:    1100101 ~ {1,3,6,7}
  103:    1100111 ~ {1,2,3,6,7}
  113:    1110001 ~ {1,5,6,7}
  131:   10000011 ~ {1,2,8}
  137:   10001001 ~ {1,4,8}
  149:   10010101 ~ {1,3,5,8}
  151:   10010111 ~ {1,2,3,5,8}
  163:   10100011 ~ {1,2,6,8}
  181:   10110101 ~ {1,3,5,6,8}
  191:   10111111 ~ {1,2,3,4,5,6,8}
  197:   11000101 ~ {1,3,7,8}

Robert Israel, Table of n, a(n) for n = 1..10000

For prime instead of binary indices we have A006450, prime case of A316091.
Prime numbers p such that A029931(p) is also prime.
Prime case of A372689.
The indices of these primes are A372886.
A000040 lists the prime numbers, A014499 their binary indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A058698 counts partitions of prime numbers, strict A064688.
A372687 counts strict partitions of prime binary rank, counted by A372851.
A372688 counts partitions of prime binary rank, with Heinz numbers A277319.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000009, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372850, A372887.

filter:= proc(p)
  local L,i,t;
  L:= convert(p,base,2);
  isprime(add(i*L[i],i=1..nops(L)))
end proc:
select(filter, [seq(ithprime(i),i=1..200)]); # Robert Israel, Jun 19 2025

Select[Range[100],PrimeQ[#] && PrimeQ[Total[First/@Position[Reverse[IntegerDigits[#,2]],1]]]&]

A372886 Indices of prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.

Original entry on oeis.org

1, 2, 5, 9, 10, 13, 14, 18, 20, 22, 24, 26, 27, 30, 32, 33, 35, 36, 38, 42, 43, 45, 47, 52, 57, 58, 60, 62, 63, 67, 70, 71, 74, 76, 79, 84, 88, 94, 96, 97, 99, 100, 101, 108, 116, 124, 126, 127, 132, 133, 135, 137, 144, 150, 154, 156, 160, 161, 162, 164, 172

Offset: 1

Gus Wiseman, May 19 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

The prime numbers themselves are A372885(n).

			The binary indices of 89 = prime(24) are {1,4,5,7}, with sum 17, which is prime, so 24 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Numbers k such that A029931(prime(k)) is prime.
Indices of primes that belong to A372689.
The indexed prime numbers themselves are A372885.
A000040 lists the prime numbers, A014499 their binary indices
A006450 lists primes of prime index, prime case of A316091.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
A058698 counts partitions of prime numbers, strict A064688.
A372687 counts strict partitions of prime binary rank, counted by A372851.
A372688 counts partitions of prime binary rank, with Heinz numbers A277319.
Cf. A029837, A158704, A158705, A035100, A372429, A372471, A372850.

filter:= proc(p)
  local L,i,t;
  L:= convert(p,base,2);
  isprime(add(i*L[i],i=1..nops(L)))
end proc:
select(t -> filter(ithprime(t)), [$1..1000]); # Robert Israel, Jun 19 2025

Select[Range[100],PrimeQ[Total[First /@ Position[Reverse[IntegerDigits[Prime[#],2]],1]]]&]

A239619 Base 3 sum of digits of prime(n).

Original entry on oeis.org

2, 1, 3, 3, 3, 3, 5, 3, 5, 3, 3, 3, 5, 5, 5, 7, 5, 5, 5, 7, 5, 7, 3, 5, 5, 5, 5, 7, 3, 5, 5, 7, 5, 5, 7, 7, 7, 3, 5, 5, 7, 5, 5, 5, 7, 5, 7, 7, 7, 7, 9, 9, 9, 5, 5, 5, 7, 3, 5, 5, 5, 7, 5, 7, 7, 7, 5, 5, 7, 7, 5, 7, 7, 7, 5, 7, 7, 7, 9, 5, 7, 7, 9, 5, 7, 7, 9

Offset: 1

Tom Edgar, Mar 22 2014

			The fifth prime is 11, 11 in base 3 is (1,0,2) so a(5)=1+0+2=3.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A007605, A014499, A053735.

[&+Intseq(NthPrime(n),3): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 3], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = vecsum(digits(prime(n), 3)); \\ Michel Marcus, Mar 07 2020

[sum(i.digits(base=3)) for i in primes_first_n(200)]

a(n) = A053735(A000040(n)).

A239690 Base 4 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 2, 4, 5, 4, 2, 4, 5, 5, 7, 4, 5, 7, 8, 5, 8, 7, 4, 5, 4, 7, 5, 5, 4, 5, 7, 8, 7, 5, 10, 5, 5, 7, 5, 7, 7, 7, 8, 8, 8, 7, 11, 4, 5, 7, 7, 10, 8, 7, 8, 11, 7, 11, 2, 5, 5, 7, 4, 5, 7, 5, 7, 8, 7, 8, 7, 4, 8, 7, 5, 8, 10, 7, 10, 11, 5, 7, 5, 7, 8, 7, 11, 7

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-4 dominance order on the natural numbers.

			The sixth prime is 13, 13 in base 4 is (3,1) so a(6)=3+1=4.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.

Cf. A007605, A014499, A053737, A239619.

Cf. A000040, A007090.

a239690 = a053737 . a000040  -- Reinhard Zumkeller, Mar 20 2015

[&+Intseq(NthPrime(n),4): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 4], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

[sum(i.digits(base=4)) for i in primes_first_n(200)]

a(n) = A053737(A000040(n)).

cp's OEIS Frontend

A372689 Positive integers whose binary indices (positions of ones in reversed binary expansion) sum to a prime number.

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A071600 Numbers k such that k and prime(k) have the same number of 1's in their binary representation.

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A072084 In prime factorization of n replace all primes with their numbers of 1's in binary representation.

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A080085 Number of factors of 2 in the factorial of the n-th prime, counted with multiplicity.

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A177718 a(n) = |(number of 1's in binary representation of prime(n)) - (number of 0's in binary representation of prime(n))|.

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A372438 Least binary index equals greatest prime index.

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A372885 Prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.

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