A198245 -id:A198245 - cp's OEIS frontend

A365405 Erroneous version of A198245.

149, 241, 2124679, 16467631, 17613227, 327784727, 426369739, 1062232319

Offset: 1

dead

Included in accordance with OEIS policy of including published but erroneous sequences to serve as pointers to the correct versions.

A. R. Booker, S. Hathi, M. J. Mossinghoff and T. S. Trudgian, Wolstenholme and Vandiver primes, The Ramanujan Journal, 58 (2022), 913-941, arXiv:2101.11157. See Theorem 1, but beware errors.

A245204 The unique integer r with |r| < prime(n)/2 such that E_{prime(n)-3}(1/4) == r (mod prime(n)), where E_m(x) denotes the Euler polynomial of degree m.

Original entry on oeis.org

1, 2, 2, 4, 1, 1, 5, 1, -2, -6, 10, 14, 5, 7, 7, -28, -12, 13, 14, 26, -21, -31, -13, -10, -11, -7, -6, 5, 2, -21, 2, 33, -15, -24, 34, 71, -15, 24, 9, 37, 73, -18, -84, -65, 9, -90, -65, -47, 97, -64, -100, -8, 41, 81, -81, -71, -65, -70, 113, 10, -80, 119, 57, 78, 20, 124, 167, -71, -48

Offset: 2

Zhi-Wei Sun, Jul 13 2014

sign

Conjecture: a(n) = 0 infinitely often. In other words, there are infinitely many odd primes p such that E_{p-3}(1/4) == 0 (mod p) (equivalently, p divides A001586(p-3)).
This seems reasonable in view of the standard heuristic arguments. The first n with a(n) = 0 is 171 with prime(171) = 1019. The next such a number n is greater than 2600 and hence prime(n) > 23321.
Zhi-Wei Sun made many conjectures on congruences involving E_{p-3}(1/4), see the reference.

			a(3) = 2 since E_{prime(3)-3}(1/4) = E_2(1/4) = -3/16 == 2 (mod prime(3)=5).

Zhi-Wei Sun, Table of n, a(n) for n = 2..1300
Zhi-Wei Sun, Super congruences and Euler numbers, arXiv:1001.4453 [math.NT].
Zhi-Wei Sun, Super congruences and Euler numbers, Sci. China Math. 54(2011), 2509-2535.

Cf. A000040, A001586, A122045, A198245, A245089, A245206.

rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
a[n_]:=rMod[EulerE[Prime[n]-3,1/4],Prime[n]]
Table[a[n],{n,2,70}]

A196230 Euler primes: values of x^2 - x + k for x = 1..k-1, where k is one of Euler's "lucky" numbers 2, 3, 5, 11, 17, 41.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, 97, 101, 107, 113, 127, 131, 149, 151, 173, 197, 199, 223, 227, 251, 257, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601

Offset: 1

Jonathan Sondow, Oct 29 2011

See A198245 for another sequence of "Euler primes". - N. J. A. Sloane, May 29 2022
All terms are prime numbers.
k is an Euler "lucky" number iff 4k-1 is a Heegner number 1, 2, 3, 7, 11, 19, 43, 67, 163.
See A014556 (Euler's "lucky" numbers) and A003173 (Heegner numbers) for additional references and links.

			The prime 1601 is a member because 40^2-40+41 = 1601.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 225.

Cf. A003173, A014556, A198245.

H = {2, 3, 5, 11, 17, 41}; Union[Flatten[Table[ Array[ #^2 - # + H[[k]] &, H[[k]] - 1], {k, 1, 6}]]]

A245206 Odd primes p with E_{p-3}(1/4) == 0 (mod p), where E_n(x) denotes the Euler polynomial of degree n.

Original entry on oeis.org

1019

Offset: 1

Zhi-Wei Sun, Jul 13 2014

The conjecture in A245204 asserts that the current sequence contains infinitely many primes.

Our computation shows that the second term should be greater than prime(2600) = 23321.

			a(1) = 1019 since 1019 is a prime with E_{1019-3}(1/4) == 88*1019 (mod 1019^2).

Zhi-Wei Sun, Super congruences and Euler numbers, Sci. China Math., Vol. 54, No. 12 (2011), 2509-2535; arXiv preprint, arXiv:1001.4453 [math.NT], 2010-2011.
Index entries for one-term sequences.

Cf. A000040, A001586, A122045, A198245, A245089, A245204.

rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
n=0;Do[If[rMod[EulerE[Prime[k]-3,1/4],Prime[k]]==0,n=n+1;Print[n," ",Prime[k]]],{k,2,200}]

A352858 a(n) = abs(E_{p-3} (mod p)) for p = prime(n), where E_i is the i-th Euler number (A000364).

Original entry on oeis.org

1, 2, 1, 3, 8, 7, 1, 3, 9, 4, 4, 4, 14, 7, 12, 16, 25, 22, 25, 4, 23, 33, 42, 15, 46, 18, 23, 38, 58, 2, 6, 55, 0, 37, 74, 63, 10, 61, 21, 38, 92, 89, 70, 79, 69, 59, 85, 22, 27, 69, 0, 45, 58, 96, 106, 6, 50, 28, 91, 133, 46, 147, 133, 38, 29, 128, 167, 116

Offset: 3

Felix Fröhlich, Apr 06 2022

nonn

a(n) = 0 iff p is a term of A198245.

These are the absolute values of the "A-values" that can be used to define "near-misses" in a search for terms of A198245 (cf. Mestrovic, 2014).

Romeo Mestrovic, A search for primes p such that Euler number E_p-3 is divisible by p, Mathematics of Computation 83 (2014), 2967-2976.

Cf. A000364, A198245.

A-values: A258367 (near-Wieferich), A250406 (near-Wilson), A244801 and A241014 (near-Wall-Sun-Sun), A260209 and A260210 (near-Wolstenholme), A338558 (near-misses for A007659).

eulmod(n) = abs(centerlift(Mod(eulerfrac(n-3), n)))
a(n) = my(p=prime(n)); eulmod(p)

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A365405 Erroneous version of A198245.

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A245204 The unique integer r with |r| < prime(n)/2 such that E_{prime(n)-3}(1/4) == r (mod prime(n)), where E_m(x) denotes the Euler polynomial of degree m.

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A196230 Euler primes: values of x^2 - x + k for x = 1..k-1, where k is one of Euler's "lucky" numbers 2, 3, 5, 11, 17, 41.

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A245206 Odd primes p with E_{p-3}(1/4) == 0 (mod p), where E_n(x) denotes the Euler polynomial of degree n.

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A352858 a(n) = abs(E_{p-3} (mod p)) for p = prime(n), where E_i is the i-th Euler number (A000364).

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