Jack Brennen - cp's OEIS frontend

A376807 Products of distinct prime Fibonacci numbers.

1, 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 89, 130, 178, 195, 233, 267, 390, 445, 466, 534, 699, 890, 1157, 1165, 1335, 1398, 1597, 2314, 2330, 2670, 3029, 3194, 3471, 3495, 4791, 5785, 6058, 6942, 6990, 7985, 9087, 9582, 11570, 15145, 15970, 17355, 18174

Offset: 1

Jack Brennen, Oct 04 2024

Each term is a product of a finite subsequence of A005478.

Cf. A005478, A160009.

import itertools, math, sympy
def fibprimegen(limit):  # Generate Fibonacci primes <= limit
  a,b = 1,2
  while b <= limit:
    if sympy.isprime(b):
      yield b
    a,b = b,a+b
LIMIT=1000000
fibprimes=list(fibprimegen(LIMIT))
fibprimeseqs=itertools.chain.from_iterable(
    itertools.combinations(fibprimes,n) for n in range(len(fibprimes)+1))
print(sorted(a for a in map(math.prod,fibprimeseqs) if a <= LIMIT))

A278586 Start with X = n^2. Repeatedly replace X with X - ceiling(X/n); a(n) is the number of steps to reach 0.

Original entry on oeis.org

1, 3, 5, 8, 11, 14, 17, 21, 24, 28, 32, 36, 40, 44, 49, 53, 57, 62, 66, 71, 75, 80, 84, 90, 94, 99, 103, 109, 113, 118, 123, 128, 133, 139, 143, 149, 154, 159, 164, 170, 175, 180, 185, 191, 196, 201, 207, 212, 217, 223, 229, 234, 240, 246, 251, 256, 262, 268, 273, 279, 284, 290, 296, 302, 308

Offset: 1

N. J. A. Sloane, Dec 02 2016, based on discussions about the Pythagoras article in the Sequence Fans Mailing List, Dec 01 2016. Jack Brennen provided the definition given here

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Matthijs Coster, Een Eigen Rij - Uitslag Prijsvraag, Pythagoras, Number 6, June 2016, pp. 20-21. The sequence was discovered by Pim Spelier.

Cf. A052488.

a:=[]; for n in [1..58] do k:=n^2; count:=0; while k gt 0 do count+:=1; k-:=Ceiling(k/n); end while; a[n]:=count; end for; a; // Jon E. Schoenfield, Dec 01 2016

A278586 := proc(n)
    local x,a;
    x := n^2 ;
    a := 0 ;
    while x <> 0 do
        x:= x-ceil(x/n) ;
        a := a+1 ;
    end do:
    a;
end proc: # R. J. Mathar, Dec 02 2016

f[n_] := Length@ NestWhileList[# - Ceiling[#/n] &, n^2, # > 1 &]; Array[f, 65] (* Robert G. Wilson v, Dec 01 2016 *)

A235599 Analog of A235598 if we start with 6.

Original entry on oeis.org

6, 8, 10, 24, 7, 25, 15, 9, 12, 5, 3, 4

Offset: 0

N. J. A. Sloane, Jan 17 2014, based on Jack Brennen's comments in A235598

See discussion in A235598.

A235598 Begin with a(0) = 3. Let a(n) for n > 0 be the smallest positive integer not yet in the sequence which forms part of a Pythagorean triple when paired with a(n-1).

Original entry on oeis.org

3, 4, 5, 12, 9, 15, 8, 6, 10, 24, 7, 25, 20, 16, 30, 18, 80, 39, 36, 27, 45, 28, 21, 29, 420, 65, 33, 44, 55, 48, 14, 50, 40, 32, 60, 11, 61, 1860, 341, 541, 146340, 15447, 20596, 25745, 32208, 2540, 1524, 635, 381, 508, 16125, 4515, 936, 75, 72, 54, 90, 56

Offset: 0

Jack Brennen, Dec 26 2013

Is the sequence infinite?  Can it "paint itself into a corner" at any point? Note that picking any starting point >= 5 seems to lead to a finite sequence ending in 5,3,4. For example, starting with 6 we get 6,8,10,24,7,25,15,9,12,5,3,4, stop (A235599).
By beginning with 3 or 4, we make sure that the 5,3,4 dead-end is never available.
If infinite, is it a permutation of the integers >= 3? This seems likely.  Proving it doesn't seem easy though.
Comment from Jim Nastos, Dec 30 2013: Your question about whether the sequence can 'paint itself into a corner' is essentially asking if the Pythagorean graph has a Hamiltonian path. As far as I know, the questions in the Cooper-Poirel paper (see link) are still unanswered. They ask whether the graph is k-colorable with a finite k, or whether it is even connected (sort of equivalent to your question of whether it is a permutation of the integers >=3).
Lars Blomberg has computed the sequence out to 3 million terms without finding a dead end.
Position of k>2: 0, 1, 2, 7, 10, 6, 4, 8, 35, 3, 67, 30, 5, 13, 89, 15, 143, 12, 22, 118, 385, 9, 11, ..., see A236243. - Robert G. Wilson v, Jan 17 2014

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
J. Cooper and C. Poirel, Note on the Pythagorean Triple System

Cf. A020882, A020883, A020884, A235599, A236243, A236244.

f[s_List] := Block[{n = s[[-1]]}, sol = Solve[ x^2 + y^2 == z^2 && x > 0 && y > 0 && z > 0 && (x == n || z == n), {x, y, z}, Integers]; Append[s, Min[ Complement[ Union[ Extract[sol, Position[ sol, Integer]]], s]]]]; lst = Nest[f, {3}, 250] (* _Robert G. Wilson v, Jan 17 2014 *)

A228170 The least semiprime (A001358) such that between it and the next n semiprimes, but not the next n+1 semiprimes, there are no primes.

Original entry on oeis.org

9, 33, 91, 141, 115, 213, 1382, 1639, 1133, 2558, 2973, 1329, 15685, 16143, 9974, 35678, 34063, 43333, 19613, 107381, 162145, 44294, 404599, 461722, 838259, 155923, 535403, 492117, 396737, 2181739, 370262, 1468279, 6034249, 3933601, 1671783, 25180174, 1357203

Offset: 1

Jack Brennen, Jonathan Vos Post, Zak Seidov, and Robert G. Wilson v, Aug 14 2013

nonn

If prime_omega(n) as defined as A001222 and a set of values becomes a string, then the 'just' means that its string is not a substring of some larger string. See the example below.
Yet another way to think of this is that between any two consecutive primes there are 'just' n semiprimes with the first one being cited above.
a(91) > 1.8*10^12. - Giovanni Resta, Aug 15 2013

			a(1) = 9 because between 9 and 10 there are no primes;
a(2) = 33 because between 33 and 35 (the second semiprime past 33) there are no primes;
a(3) = 91 because between 91 and 95 (the third semiprime past 91 with 93 & 94 also semiprimes) there are no primes;
a(4) = 141 because between 141 and 146 (the fourth semiprime past 141 with 142, 143 & 145 also being semiprimes) there are no primes;
the reason a(4) is not 115 is because although there are no primes between 115 and 121, the string "2, 3, 3, 2, 2, 5, 2, 2" is a substring of the string generated by 115 through 123. See the next line.
a(5) = 115 because between 115 and 123 (the fifth semiprime past 115 with 118, 119, 121, and 122 also being semiprimes) there are no primes;

Giovanni Resta, Table of n, a(n) for n = 1..90

Cf. A001358, A133478, A112888, A228171.

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; t = Table[0, {100}]; p=3; While[p < 3100000000, q = NextPrime[p]; a = Count[ PrimeOmega[ Range[p, q]], 2]; If[ t[[a]] == 0, t[[a]] = p; Print[{p, a}]]; p = q]; NextSemiPrime@# & /@ t

a(n) is the next semiprime after A228171(n+1).

A228171 Least prime such that between it and the next prime there are exactly n semiprimes.

Original entry on oeis.org

2, 3, 7, 31, 89, 139, 113, 211, 1381, 1637, 1129, 2557, 2971, 1327, 15683, 16141, 9973, 35677, 34061, 43331, 19609, 107377, 162143, 44293, 404597, 461717, 838249, 155921, 535399, 492113, 396733, 2181737, 370261, 1468277, 6034247, 3933599, 1671781, 25180171

Offset: 0

Jack Brennen, Jonathan Vos Post, Zak Seidov, and Robert G. Wilson v, Aug 14 2013

nonn

a(62) = 1294268491, a(64) = 2300942549.

			a(2) = 7 since between 7 and the next prime, 11, there are 2 semiprimes (9, 10).

Giovanni Resta, Table of n, a(n) for n = 0..91

Cf. A133478, A228170.

t = Table[0, {100}]; p=3; While[p < 3100000000, q = NextPrime[p]; a = Count[ PrimeOmega[ Range[p, q]], 2]; If[ t[[a]] == 0, t[[a]] = p; Print[{p, a}]]; p = q]; t

a(n) is the prime precessing A228170(n-1).

A216405 Numbers which start a run of nine consecutive zero-digit-free decimal integers, each of which is divisible by the sum of its digits.

Original entry on oeis.org

1, 142813628717821, 253323932621811, 1234954171531131, 1713763544613181, 3713154346661821, 5953112416611411, 8711631351783421, 11853531183574141, 12191214257422251, 17137635446131261, 19941476493818971, 21342541323383331, 25628491758925521, 28665872459864731

Offset: 1

Jack Brennen, Oct 16 2012

Each term of the sequence ends with the digit 1.

No run of ten consecutive zero-digit-free decimal integers is possible.

			The numbers from a(2)=142813628717821 to 142813628717829 are each divisible by their digit sums, which are 61 to 69 respectively.

Subsequence of A217973 and of A017281.

\\ Algorithm from Jack Brennen
list(lim)=my(v=List([1]),m); forstep(d=11, (40320*lim)^(1/9), 10, m=lcm(vector(9,k,d+k-1)); forstep(x=m+d, lim, m, if(sumdigits(x)==d && vecsort(digits(x))[1], listput(v,x)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Oct 16 2012

A136111 Positive integers n such that for every k in [n,n+1,n+2,n+3,n+4] we have sigma(k) coprime to k.

Original entry on oeis.org

1, 575, 202605639573839041, 478502736827135487987972323577847681

Offset: 1

Jack Brennen, Mar 17 2008

nonn

Note that it is impossible to have three consecutive positive even integers which have odd sigma() values.
In order to have an odd sigma() value, the integer must be a square or twice a square; it's not too hard to see that three consecutive positive even integers can't each be a square or twice a square.
So any solution must have n odd and among n+1 and n+3, one of them must be a square and the other must be twice a square.
Any other terms exceed 10^128.

A133478 a(n) = smallest semiprime s such that s + n is the next semiprime and there is no prime between s and s + n.

Original entry on oeis.org

9, 49, 62, 403, 341, 843, 6722, 3473, 2869, 14059, 18467, 26603, 166126, 41779, 74491, 192061, 463161, 226489, 344119, 517421, 943606, 2171131, 4577519, 584213, 2560177, 4356633, 8367139, 11174753, 13191293, 7319797, 27841051, 10644122, 13683034, 29492893, 47735342, 26837363, 63305661, 90455097, 189662206, 120175651

Offset: 1

Jack Brennen and Zak Seidov, Dec 23 2007

nonn

			a(1)=9 because there is no prime between 9 and 10.
a(2)=49 because there is no prime between 49 and 51.
a(3)=62 because there is no prime between 62 and 65.

Robert G. Wilson v, Table of n, a(n) for n = 1..60.

nn = 20; sp1 = 4; t = Table[0, {nn}]; found = 0; fQ[s_] := Union@ PrimeQ[s] == {False}; NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[ sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; While[found < nn, sp2 = NextSemiPrime[sp1]; d = sp2 - sp1; If[d <= nn && t[[d]] == 0 && fQ[Range[sp1, sp2]], t[[d]] = sp1; found++; Print[{d, sp1}]]; sp1 = sp2]; t (* Robert G. Wilson v, Jun 13 2013 *)

a(37)-a(40) from Jacques Tramu, Dec 23 2007

A114040 a(0) = 1, a(1) = 9, a(n) = 6*a(n-1) - a(n-2) - 4.

Original entry on oeis.org

1, 9, 49, 281, 1633, 9513, 55441, 323129, 1883329, 10976841, 63977713, 372889433, 2173358881, 12667263849, 73830224209, 430314081401, 2508054264193, 14618011503753, 85200014758321, 496582077046169, 2894292447518689, 16869172608065961, 98320743200877073

Offset: 0

N. J. A. Sloane, based on email from Jack Brennen, Feb 01 2006

nonn

The most straightforward test for "triangularity" is istriangle(n) <===> issquare(8*n+1). If this sequence could be proved to be free of squares beyond the first three terms, that would lead directly to a proof that 0, 1 and 6 are the only triangular numbers whose squares are triangular numbers.

Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Equals 8*A001109(n)+1. It is also A081554(n)+1.

LinearRecurrence[{7,-7,1},{1,9,49},30] (* Harvey P. Dale, Aug 18 2018 *)

G.f.: (1+2x-7x^2)/((1-x)(1-6x+x^2)). [R. J. Mathar, Sep 09 2008]

cp's OEIS Frontend

User: Jack Brennen

A376807 Products of distinct prime Fibonacci numbers.

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A278586 Start with X = n^2. Repeatedly replace X with X - ceiling(X/n); a(n) is the number of steps to reach 0.

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A235599 Analog of A235598 if we start with 6.

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A235598 Begin with a(0) = 3. Let a(n) for n > 0 be the smallest positive integer not yet in the sequence which forms part of a Pythagorean triple when paired with a(n-1).

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Mathematica

A228170 The least semiprime (A001358) such that between it and the next n semiprimes, but not the next n+1 semiprimes, there are no primes.

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A228171 Least prime such that between it and the next prime there are exactly n semiprimes.

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Mathematica

Formula

A216405 Numbers which start a run of nine consecutive zero-digit-free decimal integers, each of which is divisible by the sum of its digits.

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PARI

A136111 Positive integers n such that for every k in [n,n+1,n+2,n+3,n+4] we have sigma(k) coprime to k.

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A133478 a(n) = smallest semiprime s such that s + n is the next semiprime and there is no prime between s and s + n.

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Extensions

A114040 a(0) = 1, a(1) = 9, a(n) = 6*a(n-1) - a(n-2) - 4.