A080355 -id:A080355 - cp's OEIS frontend

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

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

A135482 a(n) = (1/4)*Sum_{j=1..n} 2^prime(j).

Original entry on oeis.org

0, 1, 3, 11, 43, 555, 2603, 35371, 166443, 2263595, 136481323, 673352235, 35033090603, 584788904491, 2783812160043, 37968184248875, 2289767997934123, 146404956073789995, 722865708377213483, 37616353855796316715, 627912164214501968427, 2989095405649324575275

Offset: 0

Ctibor O. Zizka, Feb 07 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..450 (a(0) = 0 added by _M. F. Hasler_).

Cf. A000040, A000079, A076793, A080355.

Partial sums of A135620.

[&+[2^(NthPrime(k)-2): k in [1..n]]: n in [1..25]]; // Bruno Berselli, Sep 24 2015

A135482:= n-> add(2^ithprime(i)/4, i=1..n): seq(A135482(n), n=0..20); # Wesley Ivan Hurt, Feb 02 2014

Accumulate[Table[Floor[2^i/4],{i,Prime[Range[20]]}]] (* Harvey P. Dale, Dec 05 2013 *)

a(n) = sum(k=1, n, 2^prime(k))/4; \\ Michel Marcus, Oct 15 2016

a(n) = A076793(n)/4. - M. F. Hasler, Oct 30 2018

More terms from Harvey P. Dale, Dec 05 2013

a(0) = 0 prepended by M. F. Hasler, Oct 30 2018

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

A113914 (1,2,3) Jasinski-like positive power sequence.

Original entry on oeis.org

1, 5, 13, 29, 61, 131, 271, 569, 1381, 2789, 5581, 11171, 22369, 44741, 89491, 185543, 373273, 766229, 1532701, 3065411, 6130849, 12261701, 24700549, 49401101, 98802211, 202387391, 409557751, 819116231, 1638232471, 3276464969

Offset: 1

Jonathan Vos Post, Jan 29 2006

In general, the (b,c,d) Jasinski-like positive power sequence is defined as follows: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for positive integer k. The (b,c,d) Jasinski-like nonnegative power sequence is defined: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for integer k. In this notation, A113824 is the (1,2,2) Jasinski-like nonnegative power sequence. The first differences of such sequences are powers of d, with no closed-form known upper bound.

			a(1) = 1 by definition.
a(2) = 2*1 + 3^1 = 5.
a(3) = 2*5 + 3^1 = 13.
a(4) = 2*13 + 3^1 = 29.
a(5) = 2*29 + 3^1 = 61.
a(6) = 2*61 + 3^2 = 271.
a(7) = 2*271 + 3^2 = 569.
a(32) = 2*6553461379 + 3^49 = 239299329230630636512841. Here 49 is a record value for the exponent.

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972, A113824.

a(1) = 1, a(n+1) = the least prime p such that p = 2*a(n) + 3^k for integer k>0.

A333392 a(0) = 1; thereafter a(n) = 2^(prime(n)-1) + Sum_{k=1..n} 2^(prime(n)-prime(k)).

Original entry on oeis.org

1, 3, 7, 29, 117, 1873, 7493, 119889, 479557, 7672913, 491066433, 1964265733, 125713006913, 2011408110609, 8045632442437, 128730119078993, 8238727621055553, 527278567747555393, 2109114270990221573, 134983313343374180673, 2159733013493986890769, 8638932053975947563077

Offset: 0

Ilya Gutkovskiy, Mar 18 2020

nonn

			a(7) = 119889 (in base 10) = 11101010001010001 (in base 2).
                             ||| | |   | |   |
                             123 5 7  1113  17

Cf. A000040, A008578, A010051, A034785, A051006, A072762, A076793, A080339, A080355, A121240, A139104, A333393.

a[0] = 1; a[n_] := 2^(Prime[n] - 1) + Sum[2^(Prime[n] - Prime[k]), {k, 1, n}]; Table[a[n], {n, 0, 21}]

a(n) = if (n==0, 1, 2^(prime(n)-1) + sum(k=1, n, 2^(prime(n)-prime(k)))); \\ Michel Marcus, Mar 18 2020

a(n) = floor(c * 2^prime(n)) for n > 0, where c = 0.91468250985... = 1/2 + A051006.

A113927 a(1)=1, and recursively a(n+1) is the smallest prime p of the form p = 2*a(n) + 5^k for some k>0.

Original entry on oeis.org

1, 7, 19, 43, 211, 547, 4219, 8443, 17011, 34147, 71419, 142963, 1220989051, 3662681227, 19080811690579, 38161623381163, 76324467465451, 152648936884027, 305299094471179, 4656613483675581520483

Offset: 1

Jonathan Vos Post, Jan 30 2006

Note that last digits cycle 7, 9, 3, 1; 7, 9, 3, 1. Note that the exponent k of 5^k is always odd. This follows from taking this sequence mod 6.
Since the first prime value a(2) = 7 == 1 mod 6, all values a(n) thereafter are primes of the form 6*d+1. Hence a(n+1) = [2*(6*d+1) + 5^2] mod 6 == 12*d + 2 + 1 == 3 mod 6 and would be divisible by 3; a(n+1) = [2*(6*d+1) + 5^4] mod 6 == 12*d + 2 + 1 == 3 mod 6 and would be divisible by 3; and so for all even exponents.
In general, the (b,c,d) Jasinski-like positive power sequence is defined as follows: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for positive integer k. The (b,c,d) Jasinski-like nonnegative power sequence is defined: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for integer k. In this notation, A113824 is the (1,2,2) Jasinski-like nonnegative power sequence. A113914 is the (1,2,3) Jasinski-like positive power sequence, and this here the (1,2,5) Jasinski-like power sequence.

			a(1) = 1 by definition.
a(2) = 2*1 + 5^1 = 7.
a(3) = 2*7 + 5^1 = 19.
a(4) = 2*19 + 5^1 = 43.
a(5) = 2*43 + 5^3 = 211.
a(6) = 2*211 + 5^3 = 547.
a(7) = 2*547 + 5^5 = 4219.
a(13) = 2*142963 + 5^13 = 1220989051.
a(20) = 2*305299094471179 + 5^31 = 4656613483675581520483, where 31 is a record exponent.
a(22) = 2*9313226967351163119091 + 5^45 = 28421709449030461369547296941307 and 45 is the new record exponent.

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972, A113824, A113914.

A151891 a(n) = Pi(A113824(n+1)).

Original entry on oeis.org

2, 4, 9, 36, 6561, 252252704150178, 1650016588712720468

Offset: 1

Artur Jasinski, Apr 04 2008

Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972, A113824.

a(6)-a(7) using Kim Walisch's primecount, from Amiram Eldar, Mar 13 2020

cp's OEIS Frontend

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

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A135482 a(n) = (1/4)*Sum_{j=1..n} 2^prime(j).

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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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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.

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A113914 (1,2,3) Jasinski-like positive power sequence.

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A333392 a(0) = 1; thereafter a(n) = 2^(prime(n)-1) + Sum_{k=1..n} 2^(prime(n)-prime(k)).

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A113927 a(1)=1, and recursively a(n+1) is the smallest prime p of the form p = 2*a(n) + 5^k for some k>0.

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