A014499 -id:A014499 - cp's OEIS frontend

A072583 Numbers k with the property that there is no match when comparing the numbers of 0's and 1's in the binary representations of k and the k-th prime.

2, 4, 9, 10, 11, 12, 14, 15, 17, 18, 27, 29, 33, 35, 36, 38, 39, 40, 43, 46, 48, 51, 52, 53, 54, 55, 56, 63, 66, 72, 73, 75, 76, 83, 85, 86, 90, 91, 92, 95, 96, 97, 100, 102, 104, 109, 111, 112, 113, 115, 117, 119, 120, 122, 123, 124, 126, 127, 129, 130, 131, 132, 133

Offset: 1

Reinhard Zumkeller, Jun 23 2002

In other words, A000120(k) <> A000120(A000040(k)) and A000120(k) <> A023416(A000040(k)) and A023416(k) <> A000120(A000040(k)) and A023416(k) <> A023416(A000040(k)).

A000120(k) <> A014499(k) and A000120(k) <> A035103(k) and A023416(k) <> A014499(k) and A023416(k) <> A035103(k).

			k = 40 = '101000', A000040(40) = 173 = '10101101'.

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

Cf. A000040, A000120, A014499, A023416, A035103, A049084, A071600, A072577, A072578, A072579, A072584.

With[{m = 150}, Select[Transpose[{Range[m], Prime[Range[m]]}], Intersection @@ DigitCount[#, 2] == {} &]][[;; , 1]] (* Amiram Eldar, Jul 28 2025 *)

a(n) = A049084(A072584(n)).

A176620 Primes p for which the factorization of p! over distinct terms of A050376 does not contain 2.

Original entry on oeis.org

7, 11, 13, 19, 31, 37, 41, 47, 59, 61, 67, 73, 79, 97, 103, 107, 109, 127, 131, 137, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443, 457, 463, 487

Offset: 1

Vladimir Shevelev, Apr 22 2010

nonn

Equivalent definition: primes p for which A007814(p!) is even. Apparently, the sequence is A027697 without the 2 (see A014499). [R. J. Mathar, Oct 26 2010]

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

Cf. A050376, A000040, A000142.

Select[Range[500], PrimeQ[#] && EvenQ @ IntegerExponent[#!, 2] &]  (* Amiram Eldar, Sep 13 2019 *)

Corrected (37 added, 41 added, 43 removed...) and extended by R. J. Mathar, Oct 26 2010

A177835 Primes p for which a smaller prime q exists with A000120(q) >= 2*A000120(p)-1.

Original entry on oeis.org

17, 37, 41, 67, 73, 97, 131, 137, 139, 149, 163, 193, 197, 257, 263, 269, 277, 281, 293, 337, 353, 389, 401, 449, 521, 523, 547, 577, 593, 641, 643, 673, 769, 773, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1091, 1093, 1097, 1109, 1123, 1129, 1153, 1163, 1171

Offset: 1

Vladimir Shevelev, May 14 2010

See A177836 for a comparison with A095075.

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

Cf. A000120, A014499, A095075, A061712, A177768, A177836.

read("transforms") ;A000120 := proc(n) wt(n) ; end proc:
isA177835 := proc(p) if isprime(p) then q := 2 ; while q < p do if A000120(q) >= 2*A000120(p)-1 then return true; end if; q := nextprime(q) ; end do: return false; else false; end if; end proc:
for i from 1 to 2000 do if isA177835(ithprime(i)) then printf("%d,",ithprime(i)) ; end if; end do: # R. J. Mathar, May 31 2010

With[{b = DigitCount[Prime[Range[200]], 2, 1]}, Rest@ Prime[Position[2*b - 1 - FoldList[Max, b], ?(# <= 0 &)] // Flatten]] (* _Amiram Eldar, Jul 25 2023 *)

keyword:base and more terms added by R. J. Mathar, May 31 2010

A177836 Terms of A095075 which are not in A177835.

Original entry on oeis.org

2, 541, 557, 563, 569, 587, 601, 613, 617, 647, 653, 659, 661, 677, 709, 787, 809, 929, 2141, 2203, 2221, 2251, 2281, 2333, 2347, 2357, 2381, 2389, 2393, 2417, 2467, 2473, 2617, 2659, 2699, 2707, 2713, 2729, 2837, 2851, 2857, 2897, 2953, 3221, 3347, 3461

Offset: 1

Vladimir Shevelev, May 14 2010

Note that the consecutive terms A095075(2)=17 up to A095075(27)=523 are all in A177835.

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

Cf. A095075, A177835, A000120, A014499, A061712, A177768.

With[{b = DigitCount[Prime[Range[600]], 2, 1]}, Join[{2}, Select[Prime[Position[ 2*b - 1 - FoldList[Max, b], ?(# > 0 &)] // Flatten], Differences[DigitCount[#, 2]][[1]] >= 0 &]]] (* _Amiram Eldar, Jul 27 2023 *)

A095075 \ A177835.

Keyword:base and more terms from R. J. Mathar, May 31 2010

A239694 Base 8 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 4, 6, 3, 5, 9, 8, 10, 9, 6, 8, 12, 11, 10, 12, 4, 8, 3, 9, 6, 5, 6, 10, 12, 9, 11, 8, 15, 5, 4, 6, 9, 11, 10, 9, 13, 12, 11, 13, 16, 4, 8, 10, 8, 13, 10, 12, 9, 15, 10, 13, 5, 11, 10, 12, 11, 8, 10, 13, 13, 17, 12, 16, 9, 8, 11, 13, 10, 16, 17, 16

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-8 dominance order on the natural numbers.

			The sixth prime is 13, 13 in base 8 is (1,5) so a(6)=1+5=6.

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, A053829, A239690, A239691, A239692, A239693.

[&+Intseq(NthPrime(n),8): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 8], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = sumdigits(prime(n), 8); \\ Michel Marcus, Mar 04 2023

[sum(i.digits(base=8)) for i in primes_first_n(200)]

a(n) = A053829(A000040(n)).

A373124 Sum of indices of primes between powers of 2.

Original entry on oeis.org

1, 2, 7, 11, 45, 105, 325, 989, 3268, 10125, 33017, 111435, 369576, 1277044, 4362878, 15233325, 53647473, 189461874, 676856245, 2422723580, 8743378141, 31684991912, 115347765988, 421763257890, 1548503690949, 5702720842940, 21074884894536, 78123777847065

Offset: 0

Gus Wiseman, May 31 2024

nonn

Sum of k such that 2^n+1 <= prime(k) <= 2^(n+1).

			Row-sums of the sequence of all positive integers as a triangle with row-lengths A036378:
   1
   2
   3  4
   5  6
   7  8  9 10 11
  12 13 14 15 16 17 18
  19 20 21 22 23 24 25 26 27 28 29 30 31
  32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

For indices of primes between powers of 2:
- sum A373124 (this sequence)
- length A036378
- min A372684 (except initial terms), delta A092131
- max A007053
For primes between powers of 2:
- sum A293697
- length A036378
- min A104080 or A014210
- max A014234, delta A013603
For squarefree numbers between powers of 2:
- sum A373123
- length A077643, run-lengths of A372475
- min A372683, delta A373125, indices A372540
- max A372889, delta A373126, indices A143658
Cf. A000040, A000120, A014499, A029837, A029931, A035100, A069010, A070939, A112925, A112926, A211997.

Table[Total[PrimePi/@Select[Range[2^(n-1)+1,2^n],PrimeQ]],{n,10}]

ip(n) = primepi(1<A007053
t(n) = n*(n+1)/2; \\ A000217
a(n) = t(ip(n+1)) - t(ip(n)); \\ Michel Marcus, May 31 2024

A129000 Start with an integer (in this case, 1). First, add 5 or 8 if the integer is odd or even, respectively. Then divide by 2.

Original entry on oeis.org

1, 3, 4, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7

Offset: 1

Adam F. Schwartz (adam_s(AT)mit.edu), May 01 2007

			a(7) = 6 because (7 + 5)/2 = 6.

Antti Karttunen, Table of n, a(n) for n = 1..1001
Tanya Khovanova, Arithmetic Progressions
Index entries for linear recurrences with constant coefficients, signature (0, 1).

Cf. A081742, A089610, A014499, A071673.

a={1};k=1;For[n=1,n<70,n++,If[EvenQ[k],k=k+8,k=k+5];k=k/2;AppendTo[a, k]]; a (* Stefan Steinerberger, May 26 2007 *)

a(n) = (a(n-1) + b)/d, if a(n) even = (a(n-1) + c)/d, if a(n) odd (starting with a(1)=1, b=5, c=8, d=2).

More terms from Stefan Steinerberger, May 26 2007

A166007 Number of ones in the binary representation of the middle member q of the prime triple (p,q,r) with p

Original entry on oeis.org

3, 3, 3, 2, 3, 3, 4, 4, 4, 5, 5, 5, 3, 4, 5, 5, 4, 6, 5, 6, 6, 7, 5, 4, 7, 7, 6, 7, 6, 7, 4, 4, 9, 5, 6, 6, 6, 7, 7, 8, 6, 5, 5, 5, 9, 8, 6, 7, 8, 9, 4, 5, 6, 8, 7, 6, 6, 9, 4, 7, 7, 8, 7, 7, 6, 7, 7, 7, 7, 7, 9, 8, 3, 6, 6, 7, 7, 7, 7, 6, 7, 8, 6, 6, 5, 8

Offset: 1

Steven Lubars (lubars(AT)gmail.com), Oct 03 2009

			For n = 3, (p, q, r) = (11, 13, 17), q = 13
Decimal 13 = Binary 1101
a(3) = Number of ones in 1101 = 3

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A098414, A014499

DigitCount[#,2,1]&/@Transpose[Select[Partition[Prime[Range[1000]],3,1], Last[#]-First[#]==6&]][[2]] (* Harvey P. Dale, Dec 03 2014 *)

More terms from Harvey P. Dale, Dec 03 2014

A166008 Number of ones in the binary representation of the average of twin prime pairs.

Original entry on oeis.org

1, 2, 2, 2, 4, 3, 4, 2, 4, 4, 3, 4, 4, 2, 4, 4, 4, 4, 4, 4, 5, 4, 4, 6, 3, 5, 4, 5, 3, 4, 5, 6, 6, 6, 6, 8, 2, 4, 4, 3, 2, 6, 8, 4, 5, 4, 5, 6, 6, 5, 4, 5, 4, 5, 7, 8, 5, 7, 6, 8, 8, 8, 3, 3, 2, 4, 6, 7, 6, 4, 4, 6, 8, 3, 5, 3, 5, 6, 7, 7, 7, 7, 4, 4, 6, 7, 6

Offset: 1

Steven Lubars (lubars(AT)gmail.com), Oct 03 2009

			Third twin prime pair = (11,13) with average 12 = 1100_2, with 2 ones, so a(3)=2.

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

Cf. A000120, A014574, A014499.

[&+Intseq(p+1,2):p in PrimesUpTo(3000)|IsPrime(p+2)]; // Marius A. Burtea, Dec 19 2019

seq={1}; Do[If[And @@ PrimeQ[6n + {-1, 1}], AppendTo[seq, DigitCount[6n, 2, 1]]], {n, 1, 600}]; seq (* Amiram Eldar, Dec 19 2019 *)
DigitCount[#,2,1]&/@(Mean/@Select[Partition[Prime[Range[1000]],2,1],#[[2]]- #[[1]] == 2&]) (* Harvey P. Dale, Dec 12 2021 *)

a(n) = A000120(A014574(n)). - Michel Marcus, Dec 19 2019

More terms from Amiram Eldar, Dec 19 2019

A177959 n-th prime minus number of 0's in binary representation of n-th prime.

Original entry on oeis.org

1, 3, 4, 7, 10, 12, 14, 17, 22, 28, 31, 34, 38, 41, 46, 51, 58, 60, 63, 68, 69, 77, 80, 86, 93, 98, 101, 105, 107, 110, 127, 126, 132, 135, 145, 148, 154, 159, 164, 170, 176, 178, 190, 188, 193, 196, 208, 222, 224, 226, 230, 238, 238, 250, 250, 258, 264, 267, 272, 276

Offset: 1

Juri-Stepan Gerasimov, May 16 2010

nonn

Cf. A014499, A035193, A177687.

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:
A035103 := proc(n) A023416(ithprime(n)) ; end proc:
A177959 := proc(n) ithprime(n)-A035103(n) ; end proc:
seq(A177959(n),n=1..120) ; # R. J. Mathar, May 30 2010

a(n) = A000040(n) - A035103(n).

Corrected (39 removed, 124 replaced by 224, 126 replaced by 226) by R. J. Mathar, May 30 2010

cp's OEIS Frontend

A072583 Numbers k with the property that there is no match when comparing the numbers of 0's and 1's in the binary representations of k and the k-th prime.

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A176620 Primes p for which the factorization of p! over distinct terms of A050376 does not contain 2.

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A177835 Primes p for which a smaller prime q exists with A000120(q) >= 2*A000120(p)-1.

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Maple

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A177836 Terms of A095075 which are not in A177835.

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A239694 Base 8 sum of digits of prime(n).

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Magma

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Sage

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A373124 Sum of indices of primes between powers of 2.

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A129000 Start with an integer (in this case, 1). First, add 5 or 8 if the integer is odd or even, respectively. Then divide by 2.

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A166007 Number of ones in the binary representation of the middle member q of the prime triple (p,q,r) with p