A023150 -id:A023150 - cp's OEIS frontend

A092050 Numbers n such that prime(n) == -8 (mod n).

1, 63, 435, 100347, 100353, 100359, 637335, 129992911, 129993001, 129993007, 129993171, 8179002121, 8179002123, 8179002177, 382465573539

Offset: 1

Robert G. Wilson v, Feb 18 2004

nonn

No more terms < 2*10^12. - David Wasserman, Jun 09 2005

Cf. A023150, A045924, A092044, A092045, A092046, A092047, A092048, A092049, A092051, A092052.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == n - 8, Print[n]], {n, 1, 10^9}]

Corrected by Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 20 2004

More terms from David Wasserman, Jun 09 2005

A105446 Number of symbols in the Roman Fibonacci number representation of n.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Apr 09 2005

The Roman Fibonacci numbers are composed from the values of the Fibonacci Numbers (A000045) with the grammar of the Roman Numerals (A006968) and a few rules to disambiguate.
The alphabet: {1, 2, 3, 5, 8, A=13, B=21, C=34, D=55, E=89, F=144, ...}.
Rule one: of the infinite set of representations of integers by this grammar, always restrict to the subset of those with shortest length.
Rule two: if there are two or more in the subset of shortest representations, restrict to the subset with fewest subtractions [A31 preferred to 188, B31 preferred to 1AA, CA preferred to 8D, DB preferred to AE].
Rule three: if there are two or more representations per Rules one and two, restrict to the subset with the most duplications of characters [22 preferred to 31, 33 preferred to 51, 55 preferred to 82, 88 preferred to A3, BBB preferred to D53, CC preferred to BE]. We do not need a Rule four for a while...
Lemma: no Roman Fibonacci number requires three consecutive instances of the same symbol. Proof: 3*F(i) = F(i+2) + F(i-2).
Question: what is the asymptotic length of the Roman Fibonacci numbers?

			a(1) = 1 because 1 is a Fibonacci number, equal to its own representation as a Roman Fibonacci number.
a(4) = 2 because 4 is not a Fibonacci number, but can be represented as the sum or difference of two Fibonacci numbers, with its Roman Fibonacci number representation being "22" (not "31" per rule three).
a(17) = 3 because the Roman Fibonacci number representation of 17 has three symbols, namely "A22" (not "188" per rule two).
a(80) = 4 because the Roman Fibonacci number representation of 80 has four symbols, namely "DB22".

Cajori, F. A History of Mathematical Notations, 2 vols. Bound as One, Vol. 1: Notations in Elementary Mathematics. New York: Dover, pp. 30-37, 1993.
Menninger, K. Number Words and Number Symbols: A Cultural History of Numbers. New York: Dover, pp. 44-45 and 281, 1992.
Neugebauer, O. The Exact Sciences in Antiquity, 2nd ed. New York: Dover, pp. 4-5, 1969.

Eric Weisstein's World of Mathematics, Roman Numerals.
Eric Weisstein's World of Mathematics, Fibonacci Numbers.

A105447 = integers with A105446(n) = 2. A105448 = integers with A105446(n) = 3. A105449 = integers with A105446(n) = 4. A105450 = integers with A105446(n) = 5. A023150 = integers with A105446(n) = 6. A105452 = integers with A105446(n) = 7. A105453 = integers with A105446(n) = 8. A105454 = integers with A105446(n) = 9. A105455 = integers with A105446(n) = 10.
Cf. A000045, A006968.
Appears to be a duplicate of A058978.

a(n) = number of symbols in the Roman Fibonacci number representation of n, as defined in "Comments." a(n) = 1 iff n is an element of A000045. a(n) = 2 iff the shortest Roman Fibonacci number representation of n is as the sum or difference of two elements of A000045 and n is not an element of A000045.

A105451 Numbers k such that prime(k+1) == 8 (mod k).

Original entry on oeis.org

1, 5, 15, 73, 75, 100347, 637329, 27067271, 179993015, 1208198523, 55762149023, 55762149103, 382465573515

Offset: 1

Zak Seidov, May 02 2005

No additional terms up to 7 million. - Harvey P. Dale, Jan 23 2012

Integers k such that A004649(k+1) = 8. - Michel Marcus, Dec 30 2022

Cf. A004649, A105286, A105287, A105288, A105290, A105329.

Cf. A023150 (Numbers k such that prime(n) == 8 (mod k)).

bb={};Do[If[8==Mod[Prime[n+1], n], bb=Append[bb, n]], {n, 1, 1000000}];bb
Select[Range[700000],Mod[Prime[#+1],#]==8&] (* Harvey P. Dale, Jan 23 2012 *)

from sympy import nextprime
def A105451(max):
    terms = []
    p = 3
    for n in range(1, max+1):
        if (p - 8) % n == 0: terms.append(n)
        p = nextprime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

First two terms inserted by Eric M. Schmidt, Feb 05 2013
a(8)-a(10) from Michel Marcus, Dec 29 2022
a(11)-a(13) from Max Alekseyev, Aug 31 2024

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A092050 Numbers n such that prime(n) == -8 (mod n).

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A105446 Number of symbols in the Roman Fibonacci number representation of n.

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A105451 Numbers k such that prime(k+1) == 8 (mod k).

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