A105455 -id:A105455 - cp's OEIS frontend

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

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.

A105447 Positive integers whose representation as Roman Fibonacci numbers has exactly two symbols.

Original entry on oeis.org

4, 6, 7, 9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 29, 31, 32, 33, 35, 36, 37, 39, 42, 47, 50, 52, 53, 54, 56, 57, 58, 60, 63, 68, 76, 81, 84, 86, 87, 88, 90, 91, 92, 94, 97, 102, 110, 123, 131, 139, 141, 142, 143, 145, 146, 147, 149, 152, 157

Offset: 1

Jonathan Vos Post, Apr 09 2005

			In Roman Fibonacci number representation:
4 = "22", 6 = "33", 7 = "52", 9 = "81", 10 = "55", 11 = "83", 12 = "1A",
14 = "A1", 15 = "A2", 16 = "88", 18 = "3B", 19 = "2B", 20 = "1B", ...
143 = "1F", 145 = "F1", 146 = "F2", 147 = "F3", 149 = "F5", 152 = "F8".

Cf. A000045, A006968, A105446, A105448-A105455.

a(n) is an element of A105447 iff A105446(n) = 2. a(n) is the sum or difference of two Fibonacci numbers A000045 and a(n) is not itself a Fibonacci number A000045.

A105448 Positive integers whose representation as Roman Fibonacci numbers has exactly three symbols.

Original entry on oeis.org

17, 25, 27, 28, 30, 38, 40, 41, 43, 44, 45, 46, 48, 49, 51, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 82, 83, 85, 93, 95, 96, 98, 99, 100, 101, 103, 104, 105, 107, 108, 109, 111, 112, 113, 115, 118, 124, 125, 126, 128, 132, 133, 134, 136, 140

Offset: 1

Jonathan Vos Post, Apr 09 2005

A105446 gives the number of symbols in Roman Fibonacci numbers. A105447 gives positive integers whose representation as Roman Fibonacci numbers has exactly two symbols. A105449 gives positive integers whose representation as Roman Fibonacci numbers has exactly four symbols. A105450 gives positive integers whose representation as Roman Fibonacci numbers has exactly five symbols.

			In Roman Fibonacci number representation:
17 = "A22", 25 = "B22", 27 = "AA1", 28 = "AA2", 30 = "B81", 38 = "C22", 40 = "C33", 41 = "C52", 43 = "BB1", 44 = "BB2", 45 = "BB3", 46 = "1CA", 48 = "CA1", ..., 155 = "2FA", 156 = "1FA", 158 = "FA1", 159 = "FA2", 160 = "FA3".

Cf. A000045, A006968, A105446, A105447, A105449-A105455.

a(n) is an element of A105447 iff A105446(n) = 3.

A105449 Positive integers whose representation as Roman Fibonacci numbers has exactly four symbols.

Original entry on oeis.org

80, 106, 114, 116, 117, 119, 120, 121, 122, 127, 129, 130, 135, 137, 138

Offset: 1

Jonathan Vos Post, Apr 09 2005

A105446 gives the number of symbols in Roman Fibonacci numbers. A105447 gives positive integers whose representation as Roman Fibonacci numbers has exactly two symbols. A105448 gives positive integers whose representation as Roman Fibonacci numbers has exactly three symbols. A105450 gives positive integers whose representation as Roman Fibonacci numbers has exactly five symbols.

			In Roman Fibonacci number representation:
80 = "DB22", 106 = "EA22", 114 = "DD22", 116 = "DD33", 117 = "DD52", 119 = "DD81", 120 = "DD55", 121 = "DD2A", 122 = "DD1A", 127 = "BF22", 129 = "BF33", 130 = "BF52", 135 = "AF22", 137 = "AF33", 138 = "AF52".

Cf. A000045, A006968, A105446-A105448, A105450-A105455.

a(n) is an element of A105449 iff A105446(n) = 4.

A268465 Primes of the form mprime(m) + (m + 1)prime(m + 1) + (m + 2)*prime(m + 2).

Original entry on oeis.org

23, 349, 1579, 4691, 5783, 7187, 9547, 11519, 15377, 45779, 52289, 353359, 361787, 384277, 510227, 678413, 710599, 1007861, 1218709, 1301617, 1484449, 1567567, 1839469, 2073989, 2264959, 2409163, 2438377, 2520779, 2735281, 2882653, 2998867, 3100271, 3211751

Offset: 1

Zak Seidov, Feb 05 2016

nonn

Primes arising in A105455.

Primes of the form A033286(m)+A033286(m+1)+A033286(m+2).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A033286.

from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    m, p, q, r = 1, 2, 3, 5
    while True:
        t = m*p + (m+1)*q + (m+2)*r
        if isprime(t): yield t
        m, p, q, r = m+1, q, r, nextprime(r)
print(list(islice(agen(), 33))) # Michael S. Branicky, May 17 2022

Typo in a(28) fixed by Seth A. Troisi, May 17 2022

a(29) and beyond from Michael S. Branicky, May 17 2022

A242936 Numbers n such that nprime(n) + (n+1)(prime(n+1)) is semiprime.

Original entry on oeis.org

2, 5, 9, 11, 13, 16, 17, 22, 23, 27, 28, 30, 31, 33, 37, 38, 41, 42, 44, 45, 47, 53, 56, 58, 61, 64, 65, 67, 68, 70, 73, 74, 75, 76, 80, 81, 84, 85, 88, 90, 92, 93, 96, 99, 102, 105, 106, 107, 108, 109, 110, 112, 114, 116, 117, 119, 122, 123, 124, 125, 126, 129

Offset: 1

K. D. Bajpai, May 27 2014

nonn

			a(2) = 5: 5*prime(5) + (5+1)*prime(5+1) = 133 = 7 * 19 which is semiprime.
a(3) = 9: 9*prime(9) + (9+1)*prime(9+1) = 497 = 7 * 71 which is semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A105455.

with(numtheory): A242936:= proc() if bigomega(n*ithprime(n) + (n+1)*ithprime(n+1) = 2 then return (n) : fi; end:  seq(A242936 (), n=1..500);

c = 0; Do[If[PrimeOmega[n*Prime[n] + (n+1)*Prime[n+1]]==2, c++; Print[c,"  ",n]], {n,1,35000}];

A242937 Numbers n such that nprime(n) + (n+1)(prime(n+1)) + (n+2)*(prime(n+2)) is semiprime.

Original entry on oeis.org

2, 4, 7, 8, 9, 11, 14, 16, 18, 19, 26, 32, 36, 38, 42, 44, 48, 50, 53, 71, 74, 78, 79, 82, 86, 94, 95, 98, 104, 108, 111, 114, 118, 122, 124, 126, 136, 141, 146, 154, 162, 166, 172, 174, 178, 180, 184, 186, 189, 190, 197, 200, 203, 206, 216, 219, 221, 223, 224

Offset: 1

K. D. Bajpai, May 27 2014

nonn

			a(2) = 4: 4*prime(4) + (4+1)*prime(4+1) + (4+2)*prime(4+2) = 161 = 7 * 23 is semiprime.
a(5) = 9: 9*prime(9) + (9+1)*prime(9+1) + (9+2)*prime(9+2) = 838 = 2 * 419 is semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A105455.

with(numtheory):A242937:= proc() if bigomega(ithprime(n)*n + ithprime(n+1)*(n+1) + ithprime(n+2)*(n+2)) = 2 then return (n) : fi; end:  seq(A242937 (), n=1..500);

c = 0; Do[If[PrimeOmega[n*Prime[n] + (n+1)*Prime[n+1] + (n+2)*Prime[n+2]]==2, c++; Print[c,"  ",n]], {n,1,85000}];

A268467 Smallest prime that is the (sum, k*prime(k),k=m,..n+m-1) for some m, or 0 if no such m exists.

Original entry on oeis.org

2, 43, 23, 0, 1109, 1187, 929, 0, 4973, 1291, 11197, 0, 26099, 15583, 4423, 0, 42139, 10729, 21283, 0, 36899, 27179, 21563, 0, 24359, 33863, 27361, 0, 223423, 51239, 293467, 42043, 67699, 56503, 118361, 0, 80449, 94693, 136739, 0, 127837, 136991, 387913, 0, 304259, 192013, 321721, 0, 339517, 357683

Offset: 1

Zak Seidov, Feb 05 2016

nonn

Smallest prime that is the sum of n consecutive terms of A033286.
Apparently a(n) exists for any odd n.
Values of m = {1, 3, 1, 0, 7, 6, 4, 0, 9, 2, 12, 0, 17, 11, 2, 0, 17, 4, 8, 0, 11, 7, 4, 0, 3, 5, 2, 0, 27, 5, 30, 1, 5, 2, 10, 0, 3, 4, 8, 0, 5, 5, 22, 0, 15, 6, 14, 0, 13, 13, ...}. - Michael De Vlieger, Feb 05 2016

			n=1: m=1 and 1*prime(1) = 1*2 = 2 = a(1),
n=2: m=3 and 3*prime(3)+4*prime(4) = 3*5+4*7 = 43 = a(2),
n=3: m=1 and 1*prime(1)+2*prime(2)+3*prime(3) = 1*2+2*3+3*15 = 23 = a(3),
n=4: no solution => a(4) = 0,
n=5: m=7 and 7*prime(7)+..11*prime(11) = 119+152+207+290+341 = 1109 = a(5).

Cf. A000040, A033286, A105455, A268465, A105454, A152117, A119487.

Table[If[# == 0, 0, Sum[k Prime@ k, {k, #, n + # - 1}]] &@(SelectFirst[Range[10^3], PrimeQ@ Sum[k Prime@ k, {k, #, n + # - 1}] &] /. x_ /; MissingQ@ x -> 0), {n, 50}] (* Michael De Vlieger, Feb 05 2016, Version 10.2 *)

cp's OEIS Frontend

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

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A105447 Positive integers whose representation as Roman Fibonacci numbers has exactly two symbols.

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A105448 Positive integers whose representation as Roman Fibonacci numbers has exactly three symbols.

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A105449 Positive integers whose representation as Roman Fibonacci numbers has exactly four symbols.

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A268465 Primes of the form m*prime(m) + (m + 1)*prime(m + 1) + (m + 2)*prime(m + 2).

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A242936 Numbers n such that n*prime(n) + (n+1)*(prime(n+1)) is semiprime.

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A242937 Numbers n such that n*prime(n) + (n+1)*(prime(n+1)) + (n+2)*(prime(n+2)) is semiprime.

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A268467 Smallest prime that is the (sum, k*prime(k),k=m,..n+m-1) for some m, or 0 if no such m exists.

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Mathematica

A268465 Primes of the form mprime(m) + (m + 1)prime(m + 1) + (m + 2)*prime(m + 2).

A242936 Numbers n such that nprime(n) + (n+1)(prime(n+1)) is semiprime.

A242937 Numbers n such that nprime(n) + (n+1)(prime(n+1)) + (n+2)*(prime(n+2)) is semiprime.