Stefan Krämer - cp's OEIS frontend

User: Stefan Krämer

Stefan Krämer has authored 4 sequences.

A130673 Smallest m of r=1,2,3,... where the generalized Euler constants (of D. H. Lehmer) E(r,m) change their sign: E(r,m) > 0 and E(r+1,m) < 0.

2, 3, 6, 9, 13, 17, 21, 25, 29, 34, 39, 43, 48, 53, 58, 63, 68

Offset: 1

Stefan Krämer, Jun 28 2007

Maple produces the following table:
...m................|..r
...2,...............|..1
...3,.4,.5..........|..2
...6,.7,.8..........|..3
...9,10,11,12.......|..4
..13,14,15,16.......|..5
..17,18,19,20.......|..6
..21,22,23,24.......|..7
..25,26,27,28.......|..8
..29,30,31,32,33....|..9
..34,35,36,37,38....|.10
..39,40,41,42.......|.11
..43,44,45,46,47....|.12
..48,49,50,51,52....|.13
..53,54,55,56,57....|.14
..58,59,60,61,62....|.15
..63,64,65,66,67....|.16
..68,69,70,71,72,73.|.17

Stefan Kraemer, Eulers constant and related numbers, preprint, 2005.

Stefan Kraemer, Euler's Constant 0.577... Its Mathematics and History
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975) 125-142.

E(r,m) = lim_{n->oo} (H_{r,m}(n) - log n / m); E(r,m) = -1/m * (log m + Psi(r/m))

A120337 Euler-irregular primes p dividing E(2k) for some 2k < p-1.

Original entry on oeis.org

19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587, 619, 677, 691, 709, 739, 751, 761, 769, 773, 811, 821, 877, 887, 907, 929, 941, 967, 971, 983

Offset: 1

Stefan Krämer, Jun 22 2006

nonn

Conjecture (Ernvall and Metsänkylä, 1978): The asymptotic density of this sequence within the primes is 1 - 1/sqrt(e) = 0.393469... (A290506), the same as the corresponding conjectured density of the irregular primes (A000928). - Amiram Eldar, Dec 06 2022

			a(1) = 19 because 19 divides E(10) = -19*2659 and 10 + 1 < 19.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Reijo Ernvall, On the distribution mod 8 of the E-irregular primes, Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, Vol. 1, 1975, pp. 195-198.
Reijo Ernvall and Tauno Metsänkylä, Cyclotomic invariants and E-irregular primes, Mathematics of Computation, Vol. 32, No. 142 (1978), pp. 617-629; Corrigenda, ibid., Vol. 33, No. 145 (1979), p. 433.
Su Hu and Min-Soo Kim, A note on the irregular primes with respect to Euler polynomials, arXiv:1510.01558 [math.NT], 2015.
Su Hu, Min-Soo Kim, Pieter Moree and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, arXiv:1809.08431 [math.NT], 2018.
Romeo Mestrovic, A search for primes p such that Euler number E_{p-3} is divisible by p, arXiv preprint arXiv:1212.3602 [math.NT], 2012. - From _N. J. A. Sloane_, Jan 25 2013
Prime Pages, Euler Irregular
Samuel S. Wagstaff, Prime divisors of the Bernoulli and Euler numbers, Number theory for the millennium, III, 2002, pp. 357-374, 2002. MR 1956285.

Cf. A000928, A092218, A120115, A122045, A290506.

A120337_list := proc(bound)
local ae, F, p, m, maxp; F := NULL;
for m from 2 by 2 to bound do
  p := nextprime(m+1);
  ae := abs(euler(m));
  maxp := min(ae, bound);
  while p <= maxp do
      if ae mod p = 0
      then F := F,p fi;
      p := nextprime(p);
   od;
od;
sort([F]) end: # Peter Luschny, Apr 25 2011

fQ[p_] := Block[{k = 1}, While[ 2k +1 < p && Mod[ EulerE[ 2k], p] != 0, k++]; p > 2k +1]; Select[ Prime@ Range@ 168, fQ@# &] (* Robert G. Wilson v, Dec 10 2014 *)

The (trivial) divisors of E(2n) are given by the theorem of Sylvester (1861): Let p prime with p=1 (mod 4), p-1|2n, p^k|2n then p^{k+1} | E(2n).

Terms 251 through 983 from Peter Luschny, Apr 25 2011

A109997 Primes where the arithmetic mean of the fractional parts of p/1,p/2,..., p/p is less than 1-gamma=0.422784...

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 61, 67, 73, 79, 97, 101, 103, 109, 113, 127, 137, 151, 157, 163, 173, 181, 193, 197, 199, 211, 229, 233, 241, 257, 271, 277, 281, 283, 307, 313, 331, 337, 353, 367, 373, 379, 397, 401, 409, 421, 433, 443, 457, 463

Offset: 1

Stefan Krämer, Sep 01 2005

nonn

S. R. Finch. Mathematical Constants. Cambridge University Press, 2003 ISBN 0-521-81802-2 p. 29
Stefan Kraemer. Eulers constant and related numbers, preprint, 2005.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A104885, A109996.

With[{c=1-EulerGamma},Select[Prime[Range[100]],Mean[FractionalPart/@(#/ Range[#])]Harvey P. Dale, Sep 19 2020 *)

A109996 Primes p such that the arithmetic mean of the fractional parts of p/1, p/2, ..., p/p is larger than 1 - gamma = 0.422784...

Original entry on oeis.org

23, 47, 53, 59, 71, 83, 89, 107, 131, 139, 149, 167, 179, 191, 223, 227, 239, 251, 263, 269, 293, 311, 317, 347, 349, 359, 383, 389, 419, 431, 439, 449, 461, 467, 479, 491, 503, 509, 557, 569, 571, 587, 593, 599, 607, 619, 643, 647, 659, 683, 701, 719, 727

Offset: 1

Stefan Krämer, Sep 01 2005

nonn

S. R. Finch. Mathematical Constants. Cambridge University Press, 2003 ISBN 0-521-81802-2 p. 29.
Stefan Kraemer. Eulers constant and related numbers, preprint, 2005.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Stefan Kraemer, Euler's Constant 0.577... Its Mathematics and History

Cf. A104885, A109997.

Cf. A153810 (1-gamma).

H:= proc(n) H(n):= 1/n+`if`(n=1, 0, H(n-1)) end:
a:= proc(n) option remember; local c, p; Digits := 1000;
      c:= evalf(1-gamma);
      p:=`if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
         if H(p)-add(iquo(p, i), i=1..p)/p>c
         then return p fi
      od
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Jun 14 2013

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[Mean[FractionalPart /@ (p/Range[p])] > 1-EulerGamma, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 28 2021 *)

lista(nn) = {forprime(p=2, nn, if (sum (i=1, p, p/i - floor(p/i))/p > 1- Euler, print1(p, ", ")););} \\ Michel Marcus, Jun 14 2013

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User: Stefan Krämer

A130673 Smallest m of r=1,2,3,... where the generalized Euler constants (of D. H. Lehmer) E(r,m) change their sign: E(r,m) > 0 and E(r+1,m) < 0.

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A120337 Euler-irregular primes p dividing E(2k) for some 2k < p-1.

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A109997 Primes where the arithmetic mean of the fractional parts of p/1,p/2,..., p/p is less than 1-gamma=0.422784...

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A109996 Primes p such that the arithmetic mean of the fractional parts of p/1, p/2, ..., p/p is larger than 1 - gamma = 0.422784...

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Author

Keywords

References

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Crossrefs

Programs

Maple

Mathematica

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