Yalcin Aktar - cp's OEIS frontend

A202301 Next prime after the partial sum of the first n primes.

3, 7, 11, 19, 29, 43, 59, 79, 101, 131, 163, 199, 239, 283, 331, 383, 443, 503, 569, 641, 719, 797, 877, 967, 1061, 1163, 1277, 1373, 1481, 1597, 1721, 1861, 1993, 2129, 2281, 2437, 2591, 2749, 2917, 3089, 3271, 3449, 3643, 3833, 4049, 4229, 4441, 4663, 4889

Offset: 1

Yalcin Aktar, Jan 11 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A007504, A151800.

A007504 := proc(n)
        add( ithprime(k),k=1..n) ;
end proc:
A202301 := proc(n)
        nextprime(A007504(n)) ;
end proc:
seq(A202301(n),n=1..20) ; # R. J. Mathar, Jan 29 2012

s = 0; Table[s = s + Prime[n]; NextPrime[s], {n, 100}] (* T. D. Noe, Apr 10 2012 *)
NextPrime[#]&/@Accumulate[Prime[Range[50]]] (* Harvey P. Dale, Feb 01 2015 *)

a(n) = A151800(A007504(n)). - R. J. Mathar, Jan 29 2012 [corrected by Georg Fischer, Dec 19 2020]

A203817 Decimal expansion of gamma*Pi.

Original entry on oeis.org

1, 8, 1, 3, 3, 7, 6, 4, 9, 2, 3, 9, 1, 6, 0, 3, 4, 9, 9, 6, 1, 3, 1, 3, 4, 5, 3, 1, 2, 7, 1, 0, 4, 0, 0, 1, 9, 1, 0, 7, 6, 6, 4, 1, 8, 1, 3, 7, 2, 6, 0, 0, 8, 0, 8, 6, 3, 6, 1, 4, 8, 0, 7, 8, 1, 9, 7, 5, 0, 9, 2, 2, 9, 6, 3, 0, 0, 4, 2, 0, 3, 8, 6, 4, 5, 3, 4, 5, 5, 0, 3, 8, 7, 1, 5, 5, 0, 4, 4

Offset: 1

Yalcin Aktar, Jan 06 2012

			1.8133764923916034996131345312710400191076641813726...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 587.

Cf. A001620 (gamma), A000796 (Pi).

R:= RealField(100); EulerGamma(R)*Pi(R); // G. C. Greubel, Sep 06 2018

Maple
```
evalf(gamma*Pi) ;
```

RealDigits[EulerGamma*Pi, 10, 100][[1]] (* G. C. Greubel, Sep 06 2018 *)

PARI
```
Euler*Pi
```

Equals -2 * Integral_{x=0..oo} log(x)*sin(x)/x dx. - Amiram Eldar, Aug 18 2020

A203816 Decimal expansion of egammaPi, where gamma is Euler's constant.

Original entry on oeis.org

4, 9, 2, 9, 2, 6, 8, 3, 6, 7, 4, 2, 2, 8, 9, 7, 8, 9, 1, 5, 2, 6, 3, 0, 4, 7, 9, 8, 0, 3, 4, 2, 3, 1, 0, 2, 6, 2, 8, 6, 5, 2, 9, 9, 2, 3, 3, 2, 6, 6, 0, 7, 6, 5, 8, 0, 3, 2, 2, 6, 6, 6, 4, 7, 3, 9, 9, 9, 8, 6, 5, 6, 5, 6, 8, 1, 1, 5, 7, 3, 9, 6, 7, 3, 3, 5, 6, 2, 5, 5, 7, 9, 6, 1, 7, 9, 8, 6

Offset: 1

Yalcin Aktar, Jan 06 2012

			4.929268367422897891526304798034231...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001620 (Euler's constant), A019609 (e*Pi).

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(1)*Pi(R)*EulerGamma(R); // G. C. Greubel, Sep 03 2018

RealDigits[E*EulerGamma*Pi, 10, 105][[1]] (* Alonso del Arte, Jan 06 2012 *)

exp(1)*Euler*Pi \\ Charles R Greathouse IV, Jan 15 2012

A193418 Expansion of x(x^2+x-1)/(3x^6-4x^5+x^4-3x^2+4*x-1).

Original entry on oeis.org

1, 3, 8, 23, 69, 206, 616, 1846, 5537, 16609, 49824, 149469, 448405, 1345212, 4035632, 12106892, 36320673, 108962015, 326886040, 980658115, 2941974341, 8825923018, 26477769048, 79433307138, 238299921409, 714899764221, 2144699292656

Offset: 1

Yalcin Aktar, Jul 26 2011

nonn

Conjecture: log(A005960-a(n)) ~ (log(2)*(2*n-11)).

Cf. A005960.

a(n) = b(n+1,3)-b(n+1,4) with b(n,c) = sum(floor(3^m/2^c), m=1..n).
G.f.: x*(x^2+x-1) / (3*x^6-4*x^5+x^4-3*x^2+4*x-1).
a(n) = (9*3^n+4*n+1-(1+(-1)^n)*(1+4*i^n))/32, where i=sqrt(-1). - Bruno Berselli, Jul 30 2011

A193359 Decimal expansion of sum(1/floor(2^n/n),n=1..+oo).

Original entry on oeis.org

2, 1, 4, 2, 8, 6, 0, 8, 6, 3, 8, 5, 4, 9, 2, 9, 9, 5, 6, 7, 4, 9, 4, 0, 0, 0, 3, 2, 5, 0, 1, 9, 8, 8, 0, 8, 2, 0, 8, 2, 4, 3, 7, 4, 8, 3, 3, 4, 9, 9, 7, 0, 8, 5, 8, 9, 1, 9, 2, 2, 1, 8, 2, 0, 6, 3, 9, 9, 8, 2, 3, 8, 4, 8, 2, 6, 6, 0, 1, 5, 6, 5, 1, 8, 7, 1, 7, 4, 4, 6, 9, 1, 2, 0, 5, 6, 0, 0, 1, 1

Offset: 1

Yalcin Aktar, Jul 24 2011

			2.1428608638549299567494000...

suminf(n=1,1/(2^n\n)) \\ Charles R Greathouse IV, Jul 24 2011

A162845 Sum of digits of binomial(3n,n).

Original entry on oeis.org

1, 3, 6, 12, 18, 6, 24, 18, 27, 39, 18, 36, 36, 36, 42, 60, 63, 63, 78, 72, 72, 63, 72, 90, 72, 99, 90, 75, 117, 108, 90, 99, 117, 117, 99, 162, 126, 144, 153, 153, 153, 159, 150, 126, 153, 114, 144, 171, 171, 171, 162, 162, 198, 180, 186, 207, 180, 189, 180, 234, 207

Offset: 0

Yalcin Aktar, Jul 14 2009

			a(4)=sum of digits of binomial(3*4,4)=18 because binomial(3*4,4)=495.

a := proc (n) local nn: nn := convert(binomial(3*n, n), base, 10): add(nn[j], j = 1 .. nops(nn)) end proc: seq(a(n), n = 0 .. 70); # Emeric Deutsch, Jul 29 2009

Extended by Emeric Deutsch, Jul 29 2009

A161638 The largest number of steps in Euclid's algorithm applied to A157807(n) and A157813(n).

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 3, 2, 2, 1, 1, 2, 2, 3, 3, 2, 2, 4, 3, 1, 1, 2, 2, 3, 3, 2, 2, 3, 2, 1, 1, 2, 3, 3, 2, 3, 4, 4, 3, 2, 2, 4, 3, 1, 1, 2, 2, 2, 4, 2, 3, 5, 3, 3, 3, 2, 2, 4, 4, 3, 3, 1, 1, 2, 3, 2, 3, 4, 3, 2, 2, 3, 3, 4, 3, 2, 2, 1, 1, 2, 3, 2, 3, 3, 3, 2

Offset: 1

Yalcin Aktar, Jun 15 2009

The sequence of fractions is ordered as follows: 1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1,...

			a(8) = 3 because the algorithm applied to the pair (2,3) needs the steps 2 = 3 x 0 + 2 then 3 = 2 x 1 + 1 and 2 = 1 x 2 + 0.

Anonymous, Euclidean algorithm, Springer's Encyclopedia of Maths.
Eric W. Weisstein, Euclidean algorithm, MathWorld.
Wikipedia, Euclid's algorithm

from math import gcd
def euclid_steps(a, b):
  if b == 0:
    return 0
  else:
    return 1 + euclid_steps(b, a % b)
for s in range(2, 100, 2):
  for i in range(1, s):
    if gcd(i, s - i) != 1: continue
    print(euclid_steps(i, s - i))
  for i in range(s, 0, -1):
    if gcd(i, s + 1 - i) != 1: continue
    print(euclid_steps(i, s + 1 - i))
# Hiroaki Yamanouchi, Oct 06 2014

Partially edited by R. J. Mathar, Sep 23 2009

a(1) prepended and a(12)-a(87) added by Hiroaki Yamanouchi, Oct 06 2014

A130680 Numbers n such that n = (a_1 + a_2 + ... + a_p)*(a_1^3 + a_2^3 + ... + a_p^3), where n has the decimal expansion a_1a_2...a_p.

Original entry on oeis.org

1, 1215, 3700, 11680, 13608, 87949

Offset: 1

Yalcin Aktar, Jun 29 2007

This sequence is finite and all the terms are listed. Proof: Let a_1a_2...a_p be the decimal expansion of n. Then p <= log_10(n)+1. Furthermore we have a_i <= 9, therefore (a_1 + a_2 + ... + a_p) <= 9*(log_10(n)+1) and (a_1^3 + a_2^3 + ... + a_p^3) <= 9^3*(log_10(n)+1). On the other hand, for all n > 300000 we have 9^4*(log_10(n)+1)^2 < n. A computer search confirms that we indeed have found all terms.

			87949 = (8+7+9+4+9)*(8^3+7^3+9^3+4^3+9^3).

Cf. A115518.

For[n = 1, n < 1000000, n++, b = IntegerDigits[n]; If[Sum[b[[i]], {i, 1, Length[b]}] * Sum[b[[i]]^3, {i, 1, Length[b]}] == n, Print[n]]]
ffQ[n_]:=Module[{c=IntegerDigits[n]},Total[c]Total[c^3]==n]; Select[ Range[ 90000],ffQ] (* Harvey P. Dale, Oct 18 2013 *)

Edited by Stefan Steinerberger, Jul 13 2007

A130688 Numbers n with following property: suppose n^6 = d1 d2 d3 ...dk in decimal; then d1! + d2! + ... + dk! is a square.

Original entry on oeis.org

1, 6, 747, 2802, 10000, 10256, 11876, 13875, 14623, 14710, 17117, 18090, 23919, 26569, 34282, 35402, 40515, 41202, 41850, 42195, 44684, 48396, 54698, 58509, 59293, 59644, 59900, 65502, 67795, 74004, 75320, 79593, 82677, 82713, 83402

Offset: 1

Yalcin Aktar, Jun 30 2007

Numbers n such that n^6 is in A130687.

			a(2) = 6, because 6^6 = 46656, and (4!+6!+6!+5!+6!)^(1/2) = 48 is an integer.

Cf. A130687, A089185, A126076, A126077, A126078.

A061602 := proc(n) local digs ; digs := convert(n,base,10) ; add(factorial(op(i,digs)),i=1..nops(digs)) ; end: isA130687 := proc(n) RETURN(issqr(A061602(n))) ; end: isA130688 := proc(n) RETURN(isA130687(n^6)) ; end: for n from 1 to 130000 do if isA130688(n) then printf("%d, ",n) ; fi : od:

for(n=1,10^5,m=n^6;s=0;while(m,s+=(m%10)!;m\=10);if(issquare(s),print1(n",")))

Edited by R. J. Mathar and Martin Fuller, Jul 13 2007

A130687 Numbers n such that a_1! + a_2! + ... + a_m! is a square number, where a_1a_2...a_m is the decimal expansion of n.

Original entry on oeis.org

1, 14, 15, 17, 22, 40, 41, 45, 50, 51, 54, 70, 71, 102, 112, 120, 121, 123, 132, 144, 156, 165, 200, 201, 203, 210, 211, 213, 230, 231, 302, 312, 320, 321, 334, 343, 404, 414, 433, 440, 441, 457, 475, 506, 516, 547, 560, 561, 574, 605, 615

Offset: 1

Yalcin Aktar, Jun 30 2007

			1! + 4! = 4! + 1! = 5^2, hence 14 and 41 are in the sequence.

A061602 := proc(n) local digs ; digs := convert(n,base,10) ; add(factorial(op(i,digs)),i=1..nops(digs)) ; end: isA130687 := proc(n) issqr(A061602(n)) ; end: for n from 1 to 3000 do if isA130687(n) then printf("%d, ",n) ; fi ; od ; # R. J. Mathar, Jul 12 2007

Select[Range[755], IntegerQ[Sqrt[DigitCount[ # ][[10]]+Sum[DigitCount[ # ][[i]]*i!, {i, 1, 9}]]] &]

A010052(A061602(a(n)))=1. - R. J. Mathar, Jul 12 2007

Edited by Stefan Steinerberger and R. J. Mathar, Jul 12 2007

cp's OEIS Frontend

User: Yalcin Aktar

A202301 Next prime after the partial sum of the first n primes.

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A203817 Decimal expansion of gamma*Pi.

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Magma

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A203816 Decimal expansion of e*gamma*Pi, where gamma is Euler's constant.

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Magma

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A193418 Expansion of x*(x^2+x-1)/(3*x^6-4*x^5+x^4-3*x^2+4*x-1).

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Formula

A193359 Decimal expansion of sum(1/floor(2^n/n),n=1..+oo).

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PARI

A162845 Sum of digits of binomial(3n,n).

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Maple

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A161638 The largest number of steps in Euclid's algorithm applied to A157807(n) and A157813(n).

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Python

Extensions

A130680 Numbers n such that n = (a_1 + a_2 + ... + a_p)*(a_1^3 + a_2^3 + ... + a_p^3), where n has the decimal expansion a_1a_2...a_p.

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Mathematica

Extensions

A130688 Numbers n with following property: suppose n^6 = d1 d2 d3 ...dk in decimal; then d1! + d2! + ... + dk! is a square.

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A203816 Decimal expansion of egammaPi, where gamma is Euler's constant.

A193418 Expansion of x(x^2+x-1)/(3x^6-4x^5+x^4-3x^2+4*x-1).