A080567 -id:A080567 - cp's OEIS frontend

A080545 Characteristic function of {1} union {odd primes}: 1 if n is 1 or an odd prime, else 0.

1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

N. J. A. Sloane, Mar 21 2003

			a(2) = 0 because 2 is prime but even.
a(3) = 1 because 3 is prime and odd. Likewise for a(5) and a(7).
a(4) = 0 because 4 is neither prime nor odd. Likewise for a(6) and a(8).
a(9) = 0 because 9 is odd but composite.

Antti Karttunen, Table of n, a(n) for n = 1..1001
Index entries for characteristic functions

Cf. A010051, A080355, A080567, A171387.

Differs from A080339 only at a(2).

Table[Boole[PrimeOmega[n] < 2 && OddQ[n]], {n, 100}] (* Alonso del Arte, Nov 19 2013 *)

a(n) = (delta(Omega(n), 0) + delta(Omega(n), 1)) * d_0(n), where delta is the Kronecker delta function, Omega is the number of prime factors function (counted with multiplicity), and d_0(n) is the least significant digit of n in binary. - Alonso del Arte, Nov 19 2013

Added missing a(2) - Walter Roscello, Nov 19 2013

A113824 a(1)=1; a(n+1) = the least prime greater than 2*a(n) which is a(n) plus a power of two.

Original entry on oeis.org

1, 3, 7, 23, 151, 65687, 9007199254806679, 73795983494093013143, 205688069665150755269371147819668813122842057000180977011589271

Offset: 1

Artur Jasinski, Jan 23 2006

nonn

Next term 205688069665150755269371147819668813122842057000180977011589271 + 2^1752 is too large to include here.

Those powers of two are A073924.

			151 is there because 23 + 2^7 = 151 is prime.

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972.

Edited by Don Reble, Jan 25 2006

A113835 a(n) = a(n-1) + 2^(A007494(n-1)).

Original entry on oeis.org

1, 5, 13, 45, 109, 365, 877, 2925, 7021, 23405, 56173, 187245, 449389, 1497965, 3595117, 11983725, 28760941, 95869805, 230087533, 766958445, 1840700269, 6135667565, 14725602157, 49085340525, 117804817261, 392682724205

Offset: 1

Artur Jasinski, Jan 27 2006

nonn

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971.

Partial sums of A094015.

Empirical g.f.: x*(4*x+1) / ((x-1)*(8*x^2-1)). - Colin Barker, Sep 01 2013

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009

A113878 a(1)=0; a(n+1) is the least number > a(n) such that Sum_{k=1..n+1} 2^a(k) is not composite.

Original entry on oeis.org

0, 1, 2, 4, 7, 16, 53, 66, 207, 1752, 5041, 6310

Offset: 1

Artur Jasinski, Jan 27 2006

Base-2 logarithms of A073924.

a(13) > 50000. - Don Reble

Cf. A073924, A080355, A080567, A113824.

a[1] = 0; a[n_] := a[n] = Block[{k = a[n - 1] + 1, s = Plus @@ (2^Array[a, n - 1])}, While[ !PrimeQ[s + 2^k], k++ ]; k]; Array[a, 12] (* Robert G. Wilson v *)

from sympy import isprime
def afind(limit):
    print("0, 1", end=", ")
    s, pow2 = 2**0 + 2**1, 2**2
    for m in range(2, limit+1):
        if isprime(s+pow2): print(m, end=", "); s += pow2
        pow2 *= 2
afind(2000) # Michael S. Branicky, Jul 11 2021

Edited by Don Reble, Feb 17 2006

A113829 a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence of numbers that are congruent to {0,3,4,5,7,8} mod 12.

Original entry on oeis.org

1, 9, 25, 57, 185, 441, 4537, 37305, 102841, 233913, 758201, 1806777, 18583993, 152801721, 421237177, 958108089, 3105591737, 7400559033, 76120035769, 625875849657, 1725387477433, 3924410732985, 12720503755193, 30312689799609

Offset: 1

Artur Jasinski, Jan 27 2006

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,4096,-4096).

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971, A154571.

LinearRecurrence[{1,0,0,0,0,4096,-4096},{1,9,25,57,185,441,4537},30] (* Harvey P. Dale, Aug 04 2018 *)

Vec((-4096*x^6+4096*x^5+256*x^4+128*x^3+32*x^2+16*x+9)/(4096*x^7 - 4096*x^6-x+1)+O(x^99)) \\ Charles R Greathouse IV, Apr 05 2012

G.f.: (9+16*x+32*x^2+128*x^3+256*x^4+4096*x^5-4096*x^6)/(1-x-4096*x^6+4096*x^7). - Charles R Greathouse IV, Apr 05 2012

Better definition, corrected offset and edited by Omar E. Pol, Jan 08 2009

A113841 a(n) = a(n-1) + 2^A047240(n) for n>1, a(1)=1.

Original entry on oeis.org

1, 3, 7, 71, 199, 455, 4551, 12743, 29127, 291271, 815559, 1864135, 18641351, 52195783, 119304647, 1193046471, 3340530119, 7635497415, 76354974151, 213793927623, 488671834567, 4886718345671, 13682811367879, 31274997412295

Offset: 1

Artur Jasinski, Jan 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 64, -64).

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971.

CoefficientList[Series[(1 + 2 x + 4 x^2) / ((-1 + x) (-1 + 4 x) (1 + 4 x + 16 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, May 19 2013 *)
LinearRecurrence[{1,0,64,-64},{1,3,7,71},30] (* Harvey P. Dale, Nov 18 2013 *)

G.f.: x*(1+2*x+4*x^2)/((-1+x)*(-1+4*x)*(1+4*x+16*x^2)). - Vaclav Kotesovec, Nov 28 2012

a(1)=1, a(2)=3, a(3)=7, a(4)=71, a(n)=a(n-1)+64*a(n-3)-64*a(n-4). - Harvey P. Dale, Nov 18 2013

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009

A113860 Start with the binary representation of the Catalan constant (A104338, A006752) = 0.91596559... = sum_{i=1..infinity} b(i)/2^i, where b(i)=1,1,1,0,1,0,1,0,0,1,1,1,1.... Then a(n-1)=sum_{i=1..k: sum_{ j = 1..k} b(j)=n} b(i) * 2^(i-1). In words: scan the binary digits of the number, halt at each nonzero binary digit, add a power of 2 corresponding to the place of this digit after the comma, assign current partial sum to a(n), increment n.

Original entry on oeis.org

1, 3, 7, 23, 87, 599, 1623, 3671, 7767, 15959, 81495, 343639, 867927, 1916503, 18693719, 152911447, 421346903, 958217815, 2031959639, 4179443287, 12769377879, 1112281005655, 9908374027863, 27500560072279, 97869304249943

Offset: 0

Artur Jasinski, Jan 25 2006

An instance of a Jasinski Integer Sequence using the convention JIS[number,counting system] as defined for example in A080355. This is JIS [Catalan constant,binary]=JIS[0.9159655941772190150546..,2].

Cf. A080355, A080567, A099969, A099970, A099971, A099972.

Naming a sequence after oneself is deprecated. - N. J. A. Sloane.

Corrected and extended by R. J. Mathar, Aug 31 2007

A113867 a(n) = a(n-1) + 2^(A047258(n)) for n>1, a(1)=1.

Original entry on oeis.org

1, 17, 49, 113, 1137, 3185, 7281, 72817, 203889, 466033, 4660337, 13048945, 29826161, 298261617, 835132529, 1908874353, 19088743537, 53448481905, 122167958641, 1221679586417, 3420702841969, 7818749353073, 78187493530737

Offset: 1

Artur Jasinski, Jan 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971.

CoefficientList[Series[(1 + 16 x + 32 x^2) / ((-1 + x) (- 1 + 4 x) (1 + 4 x + 16 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, May 20 2013 *)

G.f.: x*(1+16*x+32*x^2)/((-1+x)*(-1+4*x)*(1+4*x+16*x^2)). - Vaclav Kotesovec, Nov 28 2012

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009

A113870 a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence formed by k(1)=0 together with the numbers A042963.

Original entry on oeis.org

1, 3, 7, 39, 103, 615, 1639, 9831, 26215, 157287, 419431, 2516583, 6710887, 40265319, 107374183, 644245095, 1717986919, 10307921511, 27487790695, 164926744167, 439804651111, 2638827906663, 7036874417767, 42221246506599

Offset: 1

Artur Jasinski, Jan 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (1, 16, -16).

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971.

a(n)=([0,1,0; 0,0,1; -16,16,1]^(n-1)*[1;3;7])[1,1] \\ Charles R Greathouse IV, Jul 08 2024

G.f.: (3+x-40*x^2)/(4*(-1+x)*(-1+4*x)*(1+4*x)). - Vaclav Kotesovec, Nov 28 2012

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009

A113876 a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence formed by k(1)=0 together with the numbers A042964.

Original entry on oeis.org

1, 5, 13, 77, 205, 1229, 3277, 19661, 52429, 314573, 838861, 5033165, 13421773, 80530637, 214748365, 1288490189, 3435973837, 20615843021, 54975581389, 329853488333, 879609302221, 5277655813325, 14073748835533, 84442493013197, 225179981368525, 1351079888211149

Offset: 1

Artur Jasinski, Jan 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (1,16,-16).

Cf. A042964, A099974, A112627, A080355, A080567, A099969, A099970, A099971.

a(n)=(4+(-4)^n+5*4^n)/20 \\ Charles R Greathouse IV, Jul 08 2024

G.f.: (1+x-8*x^2)/(2*(-1+x)*(-1+4*x)*(1+4*x)). - Vaclav Kotesovec, Nov 28 2012

a(n) = (4 + (-4)^n + 5*4^n)/20. - Gerry Martens, May 26 2024

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009

cp's OEIS Frontend

A080545 Characteristic function of {1} union {odd primes}: 1 if n is 1 or an odd prime, else 0.

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A113824 a(1)=1; a(n+1) = the least prime greater than 2*a(n) which is a(n) plus a power of two.

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A113835 a(n) = a(n-1) + 2^(A007494(n-1)).

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A113878 a(1)=0; a(n+1) is the least number > a(n) such that Sum_{k=1..n+1} 2^a(k) is not composite.

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A113829 a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence of numbers that are congruent to {0,3,4,5,7,8} mod 12.

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A113841 a(n) = a(n-1) + 2^A047240(n) for n>1, a(1)=1.

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A113867 a(n) = a(n-1) + 2^(A047258(n)) for n>1, a(1)=1.

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A113870 a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence formed by k(1)=0 together with the numbers A042963.

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