A105454 -id:A105454 - cp's OEIS frontend

A152658 Beginnings of maximal chains of primes.

5, 13, 29, 37, 43, 61, 89, 109, 131, 139, 227, 251, 269, 277, 293, 359, 389, 401, 449, 461, 491, 547, 569, 607, 631, 743, 757, 773, 809, 857, 887, 947, 971, 991, 1069, 1109, 1151, 1163, 1187, 1237, 1289, 1301, 1319, 1373, 1427, 1453, 1481, 1499, 1549, 1601

Offset: 1

Klaus Brockhaus, Dec 10 2008

nonn

A sequence of consecutive primes prime(k), ..., prime(k+r), r >= 1, is called a chain of primes if i*prime(i) + (i+1)*prime(i+1) is prime (the linking prime for prime(i) and prime(i+1), cf. A119487) for i from k to k+r-1. A chain of primes prime(k), ..., prime(k+r) is maximal if it is not part of a longer chain, i.e. if neither (k-1)*prime(k-1) + k*prime(k) nor (k+r)*prime(k+r) + (k+r+1)*prime(k+r+1) is prime.

A chain of primes has two or more members; a prime is called secluded if it is not member of a chain of primes (cf. A152657).

			3*prime(3) + 4*prime(4) = 3*5 + 4*7 = 43 is prime and 4*prime(4) + 5*prime(5) = 4*7 + 5*11 = 83 is prime, so 5, 7, 11 is a chain of primes. 2*prime(2) + 3*prime(3) = 2*3 + 3*5 = 21 is not prime and 5*prime(5) + 6*prime(6) = 5*11 + 6*13 = 133 is not prime, hence 5, 7, 11 is maximal and prime(3) = 5 is the beginning of a maximal chain.

Klaus Brockhaus, Table of n, a(n) for n=1..10000

Cf. A152117 (n*(n-th prime) + (n+1)*((n+1)-th prime)), A152657 (secluded primes), A119487 (primes of the form i*(i-th prime) + (i+1)*((i+1)-th prime), linking primes).

Cf. A105454 - Zak Seidov, Feb 04 2016

[ p: n in [1..253] | (n eq 1 or not IsPrime((n-1)*PreviousPrime(p) +n*p) ) and IsPrime((n)*p+(n+1)*NextPrime(p)) where p is NthPrime(n) ];

A105455 Numbers k such that kprime(k)+(k+1)prime(k+1)+(k+2)*prime(k+2) is prime.

Original entry on oeis.org

1, 6, 12, 20, 22, 24, 28, 30, 34, 56, 60, 142, 144, 148, 168, 192, 196, 230, 252, 260, 276, 282, 304, 322, 334, 344, 346, 352, 366, 374, 380, 386, 394, 404, 418, 424, 432, 440, 444, 470, 478, 484, 572, 590, 610, 612, 630, 642, 662, 684, 754, 766, 784, 790, 840, 842, 874, 886

Offset: 1

Zak Seidov, May 02 2005

nonn

			k=1: 1*prime(1) + 2*prime(2) + 3*prime(3) = 1*2 + 2*3 + 3*5 = 23 prime,
k=6: 6*prime(6) + 7*prime(7) + 8*prime(8) = 6*13 + 7*17 + 8*19 = 349 prime. - _Zak Seidov_, Feb 18 2016

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

Cf. A033286, A105454, A152117, A119487.

[n: n in [1..1000] | IsPrime(n*NthPrime(n)+(n+1)*NthPrime(n+1)+(n+2)*NthPrime(n+2))]; // Vincenzo Librandi, Feb 06 2016

bb={};Do[If[PrimeQ[n Prime[n]+(n+1) Prime[n+1]+(n+2) Prime[n+2]], bb=Append[bb, n]], {n, 1, 400}];bb
Select[Range@ 900, PrimeQ[# Prime[#] + (# + 1) Prime[# + 1] + (# + 2) Prime[# + 2]] &] (* Michael De Vlieger, Feb 05 2016 *)

lista(nn) = {for(n=1, nn, if(ispseudoprime(n*prime(n)+(n+1)*prime(n+1)+(n+2)*prime(n+2)), print1(n, ", "))); } \\ Altug Alkan, Feb 05 2016

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 m
        m, p, q, r = m+1, q, r, nextprime(r)
print(list(islice(agen(), 58))) # Michael S. Branicky, May 17 2022

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.

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 *)

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A152658 Beginnings of maximal chains of primes.

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A105455 Numbers k such that k*prime(k)+(k+1)*prime(k+1)+(k+2)*prime(k+2) is prime.

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

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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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A105455 Numbers k such that kprime(k)+(k+1)prime(k+1)+(k+2)*prime(k+2) is prime.