A090418 -id:A090418 - cp's OEIS frontend

A008586 Multiples of 4.

0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 14 ).
A000466(n), a(n) and A053755(n) are Pythagorean triples. - Zak Seidov, Jan 16 2007
If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-3) is equal to the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
Number of n-permutations (n>=1) of 5 objects u, v, z, x, y with repetition allowed, containing n-1 u's. Example: if n=1 then n-1 = zero (0) u, a(1)=4 because we have v, z, x, y. If n=2 then n-1 = one (1) u, a(2)=8 because we have vu, zu, xu, yu, uv, uz, ux, uy. A038231 formatted as a triangular array: diagonal: 4, 8, 12, 16, 20, 24, 28, 32, ... - Zerinvary Lajos, Aug 06 2008
For n > 0: numbers having more even than odd divisors: A048272(a(n)) < 0. - Reinhard Zumkeller, Jan 21 2012
A214546(a(n)) < 0 for n > 0. - Reinhard Zumkeller, Jul 20 2012
A090418(a(n)) = 0 for n > 0. - Reinhard Zumkeller, Aug 06 2012
Terms are the differences of consecutive centered square numbers (A001844). - Mihir Mathur, Apr 02 2013
a(n)*Pi = nonnegative zeros of the cycloid generated by a circle of radius 2 rolling along the positive x-axis from zero. - Wesley Ivan Hurt, Jul 01 2013
Apart from the initial term, number of vertices of minimal path on an n-dimensional cubic lattice (n>1) of side length 2, until a self-avoiding walk gets stuck. A004767 + 1. - Matthew Lehman, Dec 23 2013
The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 2688. - Philippe A.J.G. Chevalier, Dec 29 2015
First differences of A001844. - Robert Price, May 13 2016
Numbers k such that Fibonacci(k) is a multiple of 3 (A033888). - Bruno Berselli, Oct 17 2017

T. D. Noe, Table of n, a(n) for n = 0..1000
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 316 [Broken link]
Milan Janjic, Two Enumerative Functions
Tanya Khovanova, Recursive Sequences
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014, 2015.
William A. Stein, The modular forms database
Eric Weisstein's World of Mathematics, Doubly Even Number
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000290, A000466, A001844, A004767, A008574, A030308, A033888, A035008, A038231, A048272, A053755, A090418, A214546.

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008585, A005843, A001477, A000217.

a008586 = (* 4)
a008586_list = [0, 4 ..]  -- Reinhard Zumkeller, May 13 2014

A008586:=n->4*n; seq(A008586(n), n=0..100); # Wesley Ivan Hurt, Feb 24 2014

Range[0, 500, 4] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)

a(n)=n<<2 \\ Charles R Greathouse IV, Oct 17 2011

a(n) = A008574(n), n>0. - R. J. Mathar, Oct 28 2008
a(n) = Sum_{k>=0} A030308(n,k)*2^(k+2). - Philippe Deléham, Oct 17 2011
a(n+1) = A000290(n+2) - A000290(n). - Philippe Deléham, Mar 31 2013
G.f.: 4*x/(1-x)^2. - David Wilding, Jun 21 2014
E.g.f.: 4*x*exp(x). - Stefano Spezia, May 18 2021

A090423 Primes that can be written in binary representation as concatenation of other primes.

Original entry on oeis.org

11, 23, 29, 31, 43, 47, 59, 61, 71, 79, 83, 109, 113, 127, 151, 157, 167, 173, 179, 181, 191, 223, 229, 233, 239, 241, 251, 271, 283, 317, 337, 347, 349, 353, 359, 367, 373, 379, 383, 431, 433, 439, 457, 463, 467, 479, 487, 491, 499, 503, 509, 541, 563, 599, 607

Offset: 1

Reinhard Zumkeller, Nov 30 2003

A090418(a(n)) > 1; subsequence of A090421.

			337 is 101010001 in binary,
10 is 2,
10 is 2,
10001 is 17, partition is 10_10_10001, so 337 is in the sequence.

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

Cf. A090422, A000040, A004676, A007088.

a090423 n = a090423_list !! (n-1)
a090423_list = filter ((> 1 ) . a090418 . fromInteger) a000040_list
-- Reinhard Zumkeller, Aug 06 2012

is_A090423(n)={isprime(n)&&for(i=2, #binary(n)-2, bittest(n, i-1)&&isprime(n%2^i)&&is_A090421(n>>i)&&return(1))} \\ M. F. Hasler, Apr 21 2015

# Primes = [2,...,607]
from sympy import sieve
primes = list(sieve.primerange(1, 608))
def tryPartioning(binString):   # First digit is not 0
    l = len(binString)
    for t in range(2, l-1):
        substr1 = binString[:t]
        if (int('0b'+substr1,2) in primes) or (t>=4 and tryPartioning(substr1)):
            substr2 = binString[t:]
            if substr2[0]!='0':
                if (int('0b'+substr2,2) in primes) or (l-t>=4 and tryPartioning(substr2)):
                    return 1
    return 0
for p in primes:
    if tryPartioning(bin(p)[2:]):
        print(p, end=',')

from sympy import isprime, primerange
def ok(p):
  b = bin(p)[2:]
  for i in range(2, len(b)-1):
    if isprime(int(b[:i], 2)) and b[i] != '0':
      if isprime(int(b[i:], 2)) or ok(int(b[i:], 2)): return True
  return False
def aupto(lim): return [p for p in primerange(2, lim+1) if ok(p)]
print(aupto(607)) # Michael S. Branicky, May 16 2021

Corrected by Alex Ratushnyak, Aug 03 2012

A090421 Numbers that can be written in binary representation as concatenation of primes.

Original entry on oeis.org

2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 29, 30, 31, 37, 41, 42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 63, 67, 70, 71, 73, 78, 79, 81, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 101, 103, 107, 109, 111, 113, 115, 117, 118, 119, 122, 123

Offset: 1

Reinhard Zumkeller, Nov 30 2003

A090418(a(n)) > 0; complement of A090419.

Terms in the sequence cannot end in 00, 0010, 001010, etc., and thus a(n) > 3n/2 + O(log n). - Charles R Greathouse IV, Apr 21 2015

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

Cf. A004676, A007088, A090422.

Cf. A000040, A090420, and A090423 are subsequences.

a090421 n = a090421_list !! (n-1)
a090421_list = filter ((> 0) . a090418) [1..]
-- Reinhard Zumkeller, Aug 06 2012

is_A090421(n)={isprime(n) || if(bittest(n,0),for(k=2,#binary(n)-2,bittest(n,k-1) && isprime(n%2^k) && is_A090421(n>>k) && return(1)), bittest(n,1) && is_A090421(n>>2))} \\ Charles R Greathouse IV and M. F. Hasler, Apr 21 2015

Based on corrections in A090418, data recomputed by Reinhard Zumkeller, Aug 06 2012

A090422 Primes that cannot be written in binary representation as concatenation of other primes.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 37, 41, 53, 67, 73, 89, 97, 101, 103, 107, 131, 137, 139, 149, 163, 193, 197, 199, 211, 227, 257, 263, 269, 277, 281, 293, 307, 311, 313, 331, 389, 397, 401, 409, 419, 421, 443, 449, 461, 521, 523, 547, 557, 569, 571, 577, 587, 593

Offset: 1

Reinhard Zumkeller, Nov 30 2003

A090418(a(n)) = 1; subsequence of A090421.

This sequence is indeed infinite, as we need infinitely many terms to cover the primes with arbitrarily large runs of 0's in their base-2 representation. - Jeffrey Shallit, Mar 07 2021

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

Cf. A090423, A000040, A004676, A007088.

A342244 handles the case where the primes are allowed to have leading zeros.

a090422 n = a090422_list !! (n-1)
a090422_list = filter ((== 1) . a090418 . fromInteger) a000040_list
-- Reinhard Zumkeller, Aug 07 2012

from sympy import isprime, primerange
def ok(p):
  b = bin(p)[2:]
  for i in range(2, len(b)-1):
    if isprime(int(b[:i], 2)) and b[i] != '0':
      if isprime(int(b[i:], 2)) or not ok(int(b[i:], 2)): return False
  return True
def aupto(lim): return [p for p in primerange(2, lim+1) if ok(p)]
print(aupto(593)) # Michael S. Branicky, Mar 07 2021

Based on corrections in A090418, data recomputed by Reinhard Zumkeller, Aug 07 2012

A090419 Numbers that cannot be written in binary representation as concatenation of primes.

Original entry on oeis.org

1, 4, 6, 8, 9, 12, 16, 18, 20, 24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 40, 44, 48, 49, 50, 51, 52, 56, 57, 60, 64, 65, 66, 68, 69, 72, 74, 75, 76, 77, 80, 82, 84, 88, 92, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 120, 121, 124

Offset: 1

Reinhard Zumkeller, Nov 30 2003

A090418(a(n)) = 0; complement of A090421.

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

Cf. A004676, A007088.

a090419 n = a090419_list !! (n-1)
a090419_list = filter ((== 0) . a090418) [1..]
-- Reinhard Zumkeller, Aug 06 2012

Based on corrections in A090418, data recomputed by Reinhard Zumkeller, Aug 06 2012

A090420 Numbers having a unique representation as concatenation of primes in binary.

Original entry on oeis.org

2, 3, 5, 7, 10, 13, 14, 15, 17, 19, 21, 22, 30, 37, 41, 42, 53, 54, 55, 58, 62, 67, 70, 73, 78, 81, 85, 86, 89, 90, 97, 101, 103, 107, 111, 115, 117, 122, 131, 137, 139, 141, 143, 149, 150, 159, 163, 165, 166, 169, 170, 177, 193, 197, 199, 211, 214, 215, 218

Offset: 1

Reinhard Zumkeller, Nov 30 2003

A090418(a(n)) = 1; subsequence of A090421.

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

Cf. A004676, A007088.

Cf. A090423 (subsequence).

a090420 n = a090420_list !! (n-1)
a090420_list = filter ((== 1) . a090418) [1..]
-- Reinhard Zumkeller, Aug 07 2012

Based on corrections in A090418, data recomputed by Reinhard Zumkeller, Aug 07 2012

A257318 Numbers n whose binary expansion can be written as the concatenation of the binary expansion of prime numbers in at least two different ways (not allowing leading zeros).

Original entry on oeis.org

11, 23, 29, 31, 43, 45, 46, 47, 59, 61, 63, 71, 79, 83, 87, 91, 93, 94, 95, 109, 113, 118, 119, 123, 125, 126, 127, 151, 157, 167, 171, 173, 174, 175, 179, 181, 182, 183, 186, 187, 189, 190, 191, 219, 223, 229, 233, 235, 237, 238, 239, 241, 245, 246, 247, 251, 253, 254, 255, 271, 283, 286, 287

Offset: 1

Jeffrey Shallit, Apr 20 2015

Numbers such that A090418(n)>1. A090423 is a subsequence. - M. F. Hasler, Apr 21 2015

			The first term is 11, as 11 in base 2 is 1011, which can be written either as (1011) or (10)(11).

M. F. Hasler, Table of n, a(n) for n = 1..1000

Cf. A090421.

is_A257318(n)={A090418(n)>1} \\ M. F. Hasler, Apr 21 2015

A090424 Smallest number that can be written in binary representation as concatenation of other primes in exactly n ways.

Original entry on oeis.org

2, 11, 23, 47, 175, 95, 189, 375, 191, 381, 763, 1015, 751, 383, 765, 1023, 1407, 2045, 767, 8123, 1919, 5999, 1533, 5623, 4063, 3067, 3007, 3039, 1535, 6013, 6077, 8183, 7935, 11247, 3069, 12023, 12143, 6139, 6015, 6111, 6127, 3071, 6079, 6135, 7679, 32507

Offset: 1

Reinhard Zumkeller, Nov 30 2003

A090418(a(n)) = n and A090418(m) <> n for m < a(n).

			n=6: a(6)=95 -> '1011111': '10"11111'==2"31, '10"11"111'==2"3"7, '10"111"11'==2"7"3, '101"11"11'==5"3"3, '1011"111'==11"7 and '10111"11'==23"3, therefore A090418(95)=6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..100

Cf. A004676, A007088.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a090424 = (+ 1) . fromJust . (`elemIndex` a090418_list)
-- Reinhard Zumkeller, Aug 07 2012

Based on corrections of A090418, data recomputed by Reinhard Zumkeller, Aug 07 2012

A256872 Numbers whose binary expansion is the concatenation of the binary expansion of two prime numbers in at least two ways.

Original entry on oeis.org

23, 31, 45, 47, 61, 93, 95, 119, 125, 127, 175, 187, 189, 191, 239, 247, 253, 255, 335, 357, 359, 363, 369, 379, 381, 383, 431, 439, 455, 477, 485, 491, 493, 495, 507, 509, 511, 573, 575, 631, 637, 639, 669, 671

Offset: 1

M. F. Hasler, Apr 21 2015

A simplified variant (and subsequence) of A257318 (and A090421) where the concatenation of any number of primes is considered.
The subsequence of numbers which are concatenation of 2 primes in at least 3 ways is (93, 95, 189, 191, 239, 253, 335, 381, 383, 669, ...).
All terms are odd. Indeed, if an even number n > 2 is concatenation of two primes (in binary), then it is of the form 'n' = 'floor(n/4)''2' (where 'x' is x in binary), and there is no other possible decomposition.

			23 = 10111[2] = (10[2])(111[2]) = (101[2])(11[2]) which is (2)(7) resp. (5)(3).

Cf. A090418, A090421, A090423, A257318.

is(n,c=2)={for(i=2,#binary(n)-2,bittest(n,i-1)&&isprime(n>>i)&&isprime(n%2^i)&&!c--&&return(1))}

A090418(a(n)) >= 2. (Necessary but not sufficient condition. This actually characterizes elements of A257318. For example, all terms of A090423 satisfy this but many of them are not terms of this sequence.)

cp's OEIS Frontend

A008586 Multiples of 4.

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A090423 Primes that can be written in binary representation as concatenation of other primes.

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A090421 Numbers that can be written in binary representation as concatenation of primes.

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A090422 Primes that cannot be written in binary representation as concatenation of other primes.

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A090419 Numbers that cannot be written in binary representation as concatenation of primes.

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A090420 Numbers having a unique representation as concatenation of primes in binary.

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A257318 Numbers n whose binary expansion can be written as the concatenation of the binary expansion of prime numbers in at least two different ways (not allowing leading zeros).

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