Daniel Tisdale - cp's OEIS frontend

A209934 a(n) is the first value to occur consecutively in the sequence b_n defined by p_2k(b_n(k)) = p_k(n)^2, k=1,2,3,..., where p_k(n) is the n-th k-almost prime.

1, 3, 8, 12, 23, 26, 32, 66, 68, 78, 83, 106, 116, 169, 181, 201, 210, 216, 234, 273, 282, 296, 427, 436, 501, 504, 513, 538, 547, 583, 655, 688, 711, 738, 751, 851, 866, 947, 1065, 1088, 1155, 1274, 1277, 1285, 1350, 1369, 1389, 1456, 1594, 1615, 1702, 1734

Offset: 1

Daniel Tisdale, Mar 15 2012

nonn

A k-almost prime has exactly k prime factors, repetitions included.
Conjecture: Each sequence b_n repeats indefinitely. (Example: for n=3, b_n = 9, 8, 8, 8, 8, 8, ....  It looks like b_3(k) is 8 for all k > 1.)
The conjecture follows from the formula that uses A078843 below (and the strict monotonicity of A078843). However the first repeated value is not for every n the value that repeats indefinitely. For example a(8) = b_8(2) = b_8(3) = 66, but b_8(k) = 64 for k >= 4. - Peter Munn, Aug 05 2019

			for k = 1, 2, 3, 4, 5, 6, ...:
p_k(3) = 5, 9, 18, 36, 72, 144, ... (the 3rd k-almost prime);
p_k(3)^2 = 25, 81, 324, 1296, 5184, 20736, ...;
b_3(k) = 9, 8, 8, 8, 8, 8, ... (index in the 2k-almost primes);
so since b_3(3) = b_3(2) = 8, a(3) = 8.

Cf. A058933, A078840, A078843, A091538.

get_p(m,k) = {local(i,n);i=0;n=1;while(iA209934(n) = {local(m,k,k_old);m=3;k_old=get_k(2,get_p(1,n)^2);k=get_k(4,get_p(2,n)^2);while(kMichael B. Porter, Mar 20 2012

From Peter Munn, Aug 05 2019: (Start)
b_n(k) = A058933(A078840(k,n)^2).
a(n) = b_n(min {k : b_n(k) = b_n(k+1)}).
If n < A078843(k+1) and b_n(k) < A078843(2k+1) then b_n(i) = b_n(k) for i >= k.
(End)

Edited, correcting the subscripting, by Peter Munn, Aug 04 2019

A201250 Integers k such that Sum_{i=1..k-1} (-1)^(i+1)primepi((k-i+1)^2) = Sum_{i=1..k-1} (-1)^(i+1)primepi((k-i)^2).

Original entry on oeis.org

1, 3, 8, 16, 36, 38, 70, 108, 116, 148, 251, 280, 1964

Offset: 1

Daniel Tisdale, Nov 28 2011

			For k = 3,  pi(3^2)-pi(2^2) = 2 = pi(2^2)-pi(1^2), so 3 is a term.

Cf. A000720.

isok(k) = sum(i=1, k-1, (-1)^(i+1)*primepi((k-i+1)^2)) == sum(i=1, k-1, (-1)^(i+1)*primepi((k-i)^2)); \\ Michel Marcus, Aug 16 2022

A_n = Sum_{i=1..n-1} (-1)^i * pi((n-i+1)^2);
B_n = Sum_{i=1..n-1} (-1)^i * pi((n-i)^2);
Sequence is S_n = {index(A_n - B_n) such that A_n - B_n = 0}.

New name and a(13) from Michel Marcus, Aug 16 2022

A199204 Irregular triangle read by rows: the n-th row being a probability distribution based on n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 3, 6, 6, 3, 2, 1, 1, 2, 3, 4, 6, 8, 12, 24, 24, 12, 8, 6, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120, 120, 60, 40, 30, 24, 20, 15, 12, 10, 8, 6, 5, 4, 3, 2, 1

Offset: 2

Daniel Tisdale, Nov 03 2011

Each row of triangle has 2^(n-1) terms.

The n-th row is built by multiplying the elements of the nonempty subsets of [n-1], sorting, removing the duplicates, and appending the reversed version. Equivalently, the n-th row is also given by the positive divisors of (n-1)! listed in increasing order (A079210), appending the reversed version. - Stefano Spezia, Aug 17 2022

			The first five rows are:
n=2: 1 1
n=3: 1 2 2 1
n=4: 1 2 3 6 6 3  2  1
n=5: 1 2 3 4 6 8 12 24 24 12 8 6 4 3 2 1

Cf. A079210.

Multiply[s_]:=Product[Part[s,i], {i,Length[s]}]; nmax=6; a={}; For[n=2, n<=nmax, n++, b=DeleteDuplicates[Sort[Table[Multiply[Part[Subsets[Drop[Range[n-1], 1]], i]], {i, 2^Length[Range[n-2]]}]]]; AppendTo[a, Join[b, Reverse[b]]]]; a (* or *)
nmax=6; a={};For[n=2,n<=nmax,n++, b=Divisors[(n-1)!]; AppendTo[a,Join[b,Reverse[b]]]]; a (* Stefano Spezia, Aug 17 2022 *)

A closed form for the sequence is: S_n =(1/n!) Sum {(n-1)!/{(n-1)^i(n-2)^j...2^q 1^r} (ij...qr = 00...00 to 11...11) where ij...qr is a base-2 number whose n-1 digits appear as exponents in the sum.

A197871 Irregular triangle T(n,k) of the number of numbers with k prime factors (repetitions allowed) less than n^2.

Original entry on oeis.org

0, 2, 1, 4, 3, 1, 6, 6, 2, 1, 9, 9, 4, 2, 11, 13, 7, 3, 1, 15, 17, 10, 4, 2, 18, 22, 13, 7, 2, 1, 22, 26, 19, 8, 4, 1, 25, 34, 22, 12, 4, 2, 30, 40, 28, 13, 7, 2, 34, 48, 32, 18, 7, 3, 1, 39, 56, 38, 21, 9, 4, 1, 44, 62, 48, 24, 11, 4, 2, 48, 75, 51, 29, 13, 6, 2

Offset: 1

Daniel Tisdale, Oct 18 2011

			In the third row, reading from the left, 6 is the number of primes <= 16, 6 is the number of semiprimes <= 16, 2 is the number of numbers with three prime divisors (repetitions allowed) <= 16, and 1 is the number of numbers with four divisors <= 16.
The triangle begins:
   0
   2  1
   4  3  1
   6  6  2  1
   9  9  4  2
  11 13  7  3  1
  15 17 10  4  2
  ...

G. J. O. Jameson, The Prime Number Theorem, Cambridge, 2004, p.145.

Similar to A052130.

Join[{0}, Flatten[Table[Transpose[Tally[Table[Plus @@ Last /@ FactorInteger[i], {i, 2, n^2}]]][[2]], {n, 2, 15}]]]

T(n,k) = #select(x->(bigomega(x) == k), [1..n^2]);
row(n) = my(v = vector(n, k, T(n,k))); my(pos); for (k=1, n, if (v[k], pos=k)); Vec(v, pos); \\ Michel Marcus, Aug 16 2022

A196448 Value of a variable (d) in an iterative system. See equations.

Original entry on oeis.org

0, 1, 2, 48, 10240, 738197504, 3026418949592973312, 57156803818751382534863703592054816768, 19223296064788821505866276808082953452203024798006734264363075470063703162880

Offset: 0

Daniel Tisdale, Oct 02 2011

nonn

In genetics, one could ask, given an initial population of genotypes xX and XY (x being a recessive sex-linked trait), how many afflicted males would result in successive generations if each generation mated only inter sese, and each pairing resulted in only the expected four hypothetical results.

In any system of this kind, provided the sex ratio is 1:1, the frequency of each allele is the same in each subsequent generation, and the genome ratios approach the product values fairly quickly. (If the sex ratios are not equal in the base generation, they will be from the first generation on.) In this case, this results in a limit of 1/3 of males affected, and 1/9 of females. - Franklin T. Adams-Watters, Oct 02 2011

			Let F_0 be: a=0,b=1,c=0,d=0,e=1; F_1 is {0,1,1,1,1}; F_2 = {1,4,3,2,6}.

Cf. A196197, A058891 (males in each generation), A000302 (total size of generations).

a = xx, b = xX, c = XX, d = xY, e = XY.
a' = 2ad + bd;
b' = 2ae + bd + be + 2cd;
c' = be + 2 ce;
d' = bd + be + 2ad + 2ae;
e' = bd + be + 2cd + 2ce.

a(5)-a(8) from Franklin T. Adams-Watters, Oct 03 2011

A196197 Successive values of variable in iterative system. See formula.

Original entry on oeis.org

1, 2, 46, 9524, 668167328, 2659622445121618432, 49010511918326168029213178565961711616, 16062797652514333892481251366398766288832009528055271611504951962161641422848

Offset: 1

Daniel Tisdale, Sep 29 2011

nonn

v=[1,1,0,1,1];for(n=1,10,a=v[1];b=v[2];c=v[3];d=v[4];e=v[5];print1(e", ");v=[2*a*d+b*d,2*a*e+b*d+b*e+2*c*d+c*e,b*e+c*e,2*a*d+2*a*e+b*d+b*e+c*e,b*d+b*e+2*c*d+c*e]) \\ Charles R Greathouse IV, Sep 29 2011

Successive values of e in iterative system (LHS subscript for a-e is _n+1, RHS subscript for a-e is _n): a = 2ad + bd; b = 2ae + bd + be + 2cd + ce; c = be + ce; d = 2ad + 2ae + bd + be + ce; e = bd + be + 2cd + ce. The initial values for the system are a=b=d=e=1, c=0.

A180302 Sequence of primes separated by [sequence of prime] elements. 2, [find 2nd prime after 2 = ] 5, [find 3rd prime after 5 =] 13, [find 5th prime after 13 =] 61, ..., etc.

Original entry on oeis.org

2, 5, 13, 31, 61, 109, 181, 277, 397, 547, 733, 947, 1213, 1499, 1831, 2207, 2633, 3083, 3583, 4133, 4751, 5407, 6073, 6793, 7589, 8513, 9397, 10313, 11353, 12409, 13451, 14713, 15889, 17299, 18593, 20129, 21613, 23167, 24851, 26561, 28387, 30203

Offset: 1

Daniel Tisdale, Aug 24 2010

nonn

Similar to A006450.

Cf. A006450, A014284.

NestList[n = 0; (n++; NextPrime[ #, Prime@ n]) &, 2, 41] (* Robert G. Wilson v, Aug 25 2010 *)
Prime[Accumulate[Join[{1}, Prime[Range[45]]]]] (* Alonso del Arte, Oct 09 2012 *)

a(8) onwards from Robert G. Wilson v, Aug 25 2010

A160757 Averages of the Fibonacci numbers which take integer values.

Original entry on oeis.org

1, 1, 5058, 262213938, 18124577012898, 187952389930860, 1409394295257361938, 116903055445824294157698, 10100618828005365858877129458, 81435914480042681825934186407384633298, 7505278652741640947693896415563573183203138, 700346071081054203480884565881868806176873272498

Offset: 1

Daniel Tisdale, May 25 2009

The numbers n such that F(1)+F(2)+...+F(n)=F(n+2)-1 is divisible by n are given in A111035. [From Max Alekseyev, Jun 04 2009]

Cf. A050248, integer average of n primes for some n, A000045.

lst = {}; Do[a = Sum[ Fibonacci@ j, {j, n}]/n; If[ IntegerQ@ a, AppendTo[lst, a]], {n, 250}]; lst

1/n*Sum {j=1..n} Fibonacci_j is an integer.

a(n) = (A000045(A111035(n)+2)-1) / A111035(n) = A000071(A111035(n)+2) / A111035(n) [From Max Alekseyev, Jun 04 2009]

Corrected and extended by Max Alekseyev and Robert G. Wilson v, Jun 04 2009

A160966 Starting from a(1)=2, a(n) = A028260(1+a(n-1)) if n is even, a(n) = A026424(a(n-1)) if n is odd.

Original entry on oeis.org

2, 6, 11, 25, 45, 94, 176, 361, 700, 1405, 2752, 5533, 10964, 22011, 43839, 87868, 175557, 351746, 703243, 1407705, 2814709, 5631461, 11261009, 22524901, 45044446, 90091251, 180165450, 360333977, 720640449, 1441293048, 2882532607

Offset: 1

Daniel Tisdale, May 31 2009

nonn

If a(1) were set to 3, the equivalent sequence would start 3, 9, 17, 38, 71, 146, 279,...

Alternatively starting from a(1)=4 we get 4, 10, 18, 39, 72, 150,..

			a(2)= 6, the second integer containing an even number of prime factors.
a(3)= 11, the sixth integer containing an odd number of primes.

Cf. A141436

Definition clarified, sequence extended beyond 25 by R. J. Mathar, May 31 2010

a(15)-a(31) from Donovan Johnson, Jul 02 2010

A161960 Numerators to accompany A161961.

Original entry on oeis.org

1, 2, 3, 3, 11, 15, 9, 11, 5, 6, 17, 1, 11, 8, 9, 61, 11, 9, 39, 85, 23, 11, 35, 38, 61, 43, 137, 73, 77, 162, 43, 181, 191, 40, 35, 73, 1, 20, 251, 263, 137, 283, 295, 17, 319, 7, 19, 17, 367, 42, 131, 409, 421, 217, 445, 457, 237, 487, 503, 173, 133, 61, 47, 289, 1, 609, 156

Offset: 2

Daniel Tisdale, Jun 22 2009

Cf. A161961.

Table[PrimePi[n^2]/(n*PrimePi[n]),{n,1,100}]

More terms from R. J. Mathar, Oct 05 2009

cp's OEIS Frontend

User: Daniel Tisdale

A209934 a(n) is the first value to occur consecutively in the sequence b_n defined by p_2k(b_n(k)) = p_k(n)^2, k=1,2,3,..., where p_k(n) is the n-th k-almost prime.

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A201250 Integers k such that Sum_{i=1..k-1} (-1)^(i+1)*primepi((k-i+1)^2) = Sum_{i=1..k-1} (-1)^(i+1)*primepi((k-i)^2).

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PARI

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A199204 Irregular triangle read by rows: the n-th row being a probability distribution based on n!.

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Mathematica

Formula

A197871 Irregular triangle T(n,k) of the number of numbers with k prime factors (repetitions allowed) less than n^2.

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Mathematica

PARI

A196448 Value of a variable (d) in an iterative system. See equations.

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A196197 Successive values of variable in iterative system. See formula.

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Programs

PARI

Formula

A180302 Sequence of primes separated by [sequence of prime] elements. 2, [find 2nd prime after 2 = ] 5, [find 3rd prime after 5 =] 13, [find 5th prime after 13 =] 61, ..., etc.

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Mathematica

Extensions

A160757 Averages of the Fibonacci numbers which take integer values.

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Mathematica

Formula

Extensions

A160966 Starting from a(1)=2, a(n) = A028260(1+a(n-1)) if n is even, a(n) = A026424(a(n-1)) if n is odd.

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A201250 Integers k such that Sum_{i=1..k-1} (-1)^(i+1)primepi((k-i+1)^2) = Sum_{i=1..k-1} (-1)^(i+1)primepi((k-i)^2).