A004277 -id:A004277 - cp's OEIS frontend

A191612 Image of A008578 (the noncomposite numbers) under the "forming" transformation.

1, 2, 3, 4, 6, 8, 12, 16, 18, 20, 24, 30, 36, 40, 42, 44, 48, 54, 60, 66, 68, 72, 78, 80, 84, 96, 100, 102, 104, 108, 112, 126, 128, 132, 138, 140, 150, 156, 162, 164, 168, 174, 180, 190, 192, 196, 198, 204, 216, 224, 228

Offset: 1

Jaroslav Krizek, Oct 16 2011

We define a transformation T_f [b(n)] = [c(n)] - the index f means "forming" - of an increasing sequence b(n) of integers b(1), b(2), b(3), ..., b(k) which produces an increasing sequence c(n) of the same length, c(1), c(2), c(3), ..., c(k) such that c(1) = b(1), and for j>1, c(j) is the only integer b(j-1) < c(j) <= b(j), with (b(j)-b(j-1)) | c(j). We say b(n) is forming c(n).
An increasing sequence c(n) is called formed from the increasing sequence b(n) by T_f [b(n)] when there is an increasing sequence b(n) such that b(1) = c(1), for j > 1, b(j) is an integer c(j) <= b(j) < c(j+1) such that difference b(j) - b(j-1) divides c(j).
This transformation T_invf [c(n)] is an inverse of T_f [b(n)], but this inversion of c(n) back to b(n) may not be unique, and there are also increasing sequences c(n) which do not have an image T_invf [c(n)]. We call the latter sequences c(n) "unformed."
Each increasing sequences b(n) can be transform by transformation T_f [b(n)] but this does not apply to transformation T_invf [b(n)]. An increasing sequence c(n) is called totally formed if c(n) = T_f [c(n)] = T_invf [c(n)]. Each totally formed sequence is formed.
There are infinitely many formed, totally formed and unformed increasing sequences.
Examples of totally formed sequences: A047229, A004277, A002808, A000079, A000027.
Examples of formed, but not totally formed, sequences: A000225, A000295, A018252.
Examples of unformed sequences: A000040, A008578, A005117, A005408.

			a(10) = 20 because 20 is the only integer such that 19 = A008578(9) < 20 <= A008578(10) = 23 and simultaneously is multiple of difference A008578(10) - A008578(9) = 4.

Tf := proc(L)
        local a,j,c ;
        a := [op(1,L)] ;
        while nops(a) < nops(L)-1 do
                j := nops(a)+1 ;
                for c from op(j-1,L)+1 to op(j,L) do
                        if (c mod ( op(j,L)-op(j-1,L) )) = 0 then
                                a := [op(a),c] ;
                                break;
                        end if;
                end do:
        end do:
        a ;
end proc:
nonc := [seq(A008578(n),n=1..80)] ;
Tf(nonc) ; # R. J. Mathar, Oct 27 2011

For n > 3, a(n) = A113709(n-2).

A229489 Conjecturally, possible differences between prime(k)^2 and the next prime for some k.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 40, 42, 46, 48, 52, 54, 58, 60, 64, 66, 70, 72, 76, 78, 82, 84, 88, 90, 94, 96, 100, 102, 106, 108, 112, 114, 118, 120, 124, 126, 130, 132, 136, 138, 142, 144, 148, 150, 154, 156, 160, 162, 166, 168, 172, 174

Offset: 1

T. D. Noe, Oct 21 2013

nonn

Are there any missing terms? The first 10^7 primes were examined. All these differences occur for some k < 10^5. Note that the first differences of these terms is 1, 2, or 4.

The similarity to A047233 is understood by a comment in A091666. - R. J. Mathar, Oct 28 2013

Cf. A000040 (primes), A001248 (primes squared).
Cf. A004277 (conjecturally, possible gaps between adjacent primes).
Cf. A091666 (prime greater than prime(n)^2).
Cf. A229488 (possible differences between prime(k)^2 and the previous prime).

t = Table[p2 = Prime[k]^2; NextPrime[p2] - p2, {k, 100000}]; Take[Union[t], 60]
NextPrime[#]-#&/@(Prime[Range[100000]]^2)//Union (* Harvey P. Dale, Apr 19 2020 *)

A238398 Numerators of inverse binomial transform of A198631(n)/A006519(n+1) with -1 instead of A198631(1)=1.

Original entry on oeis.org

1, -3, 2, -11, 4, -11, 6, -39, 8, -49, 10, 647, 12, -5487, 14, 929329, 16, -3202325, 18, 221930505, 20, -4722116563, 22, 968383680643, 24, -14717667114201, 26, 2093660879252563, 28, -86125672563201239, 30, 129848163681107300961, 32

Offset: 0

Paul Curtz, Feb 26 2014

sign

From modified fractional Euler numbers.
Inverse binomial transform:
1, -3/2, 2, -11/4, 4, -11/2, 6, -39/8, 8, -49/2, 10, 647/4, 12, -5487/2,... = a(n)/b(n).  b(2n) = A004277(n).
Difference table of c(n) = 1, -1/2, 0, -1/4,... :
1,    -1/2,    0,  -1/4,     0,    1/2,      0,...
-3/2,  1/2, -1/4,   1/4,   1/2,   -1/2,  -17/8,...
2,    -3/4,  1/2,   1/4,    -1,  -13/8,   17/4,...
-11/4, 5/4, -1/4,  -5/4,  -5/8,   47/8,   73/8,...
4,    -3/2,   -1,   5/8,  13/2,   13/4, -107/2,...
-11/2, 1/2, 13/8,  47/8, -13/4, -227/4, -227/8,
6,     9/8, 17/4, -73/8, -107/2, 227/8, 2957/4,...
etc.
c(n) + a(n)/b(n) = 2, -2, 2, -3, 4, -5, 6, -7, 8, -9,... = A233583(n+1) signed. (a(n) discovered in 2013)

Cf. A235774.

max = 40;(* b = A198631 *) b[0] = 1; b[1] = -1; b[n_] := Numerator[EulerE[n, 1]/(2^n-1)]; bb = Table[b[n]/2^IntegerExponent[n+1, 2], {n, 0, max}]; a[n_] := Differences[bb, n] // First // Numerator ; Table[a[n], {n, 0, max}]

A274922 a(n) = (-1)^n * n if n>0, a(0) = 1.

Original entry on oeis.org

1, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59

Offset: 0

Michael Somos, Dec 28 2016

This is a divisibility sequence.

			G.f. = 1 - x + 2*x^2 - 3*x^3 + 4*x^4 - 5*x^5 + 6*x^6 - 7*x^7 + 8*x^8 + ...

Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Cf. A004277, A005408, A028310, A060576, A106510.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + x+x^2)/(1+2*x+x^2))); // G. C. Greubel, Jul 29 2018

a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n n];
a[ n_] := SeriesCoefficient[ (1 + x + x^2) / (1 + 2*x + x^2), {x, 0, n}];
LinearRecurrence[{-2,-1},{1,-1,2},60] (* Harvey P. Dale, Mar 30 2019 *)

PARI
```
{a(n) = if( n<1, n==0, (-1)^n * n)};
```

{a(n) = if( n<0, 0, polcoeff( (1 + x + x^2) / (1 + 2*x + x^2) + x * O(x^n), n))};

Euler transform of length 3 sequence [-1, 2, -1].
a(n) = -b(n) where b() is multiplicative with b(2^e) = -(2^e) if e>0, b(p^e) = p^e otherwise.
E.g.f.: 1 - x * exp(-x).
G.f.: (1 + x + x^2) / (1 + 2*x + x^2).
G.f.: (1 - x) * (1 - x^3) / (1 - x^2)^2.
G.f.: 1 / (1 + x / (1 + x / (1 - x / (1 + x)))).
G.f.: 1 - x / (1 + x)^2 = 1 - x /(1 - x)^2 + 4*x^2 / (1 - x^2)^2.
a(n) = (-1)^n * A028310(n). a(2*n) = A004277(n). a(2*n + 1) = - A005408(n).
Convolution inverse of A106510.
A060576(n) = Sum_{k=0..n} binomial(n, k) * a(k).
A028310(n) = Sum_{k=0..n} binomial(n+1, k+1) * a(k).
a(n) = A038608(n), n>0. - R. J. Mathar, May 25 2020

A338329 First differences of A326118.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 3, 5, 7, 9, 5, 7, 9, 11, 7, 9, 11, 13, 9, 11, 13, 15, 11, 13, 15, 17, 13, 15, 17, 19, 15, 17, 19, 21, 17, 19, 21, 23, 19, 21, 23, 25, 21, 23, 25, 27, 23, 25, 27, 29, 25, 27, 29, 31, 27, 29, 31, 33, 29, 31, 33, 35, 31, 33, 35, 37, 33, 35, 37, 39

Offset: 0

Stefano Spezia, Oct 23 2020

It includes exclusively all the odd numbers (A005408). Except for 1, 3 and 5 that appear three times, each other odd number appears four times.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A004277 (averages of the increasing runs), A005408, A326118.

LinearRecurrence[{1,0,0,1,-1},{1,1,3,1,3,5,7,3},71]

O.g.f.: (1 + 2*x^2 - 2*x^3 + x^4 + 2*x^5 - 2*x^7)/((1 - x)^2*(1 + x + x^2 + x^3)).
E.g.f.: (3*exp(-x) + exp(x)*(7 + 2*x) - 6*cos(x) + 6*sin(x))/4 - 2*x - x^2.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 7.
a(n) = (7 + 2*n - 6*cos(n*Pi/2) + 3*(-1)^n + 6*sin(n*Pi/2))/4 for n > 2.

A340928 Least image of A001222 applied to the prime indices of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Gus Wiseman, Jan 28 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 4277 are {4,6,15} with images {2,2,2}, so a(4277) = 2.
The prime indices of 8303 are {8,8,9} with images {3,3,2}, so a(8303) = 2.

Positions of 0's are A000079.
Positions of first appearances are A033844.
The version for maximum is A340691.
A003963 multiplies together the prime indices.
A026794 counts partitions by sum and minimum.
A056239 adds up the prime indices.
A061395 selects the greatest prime index.
A112798 lists the prime indices of each positive integer.
Cf. A001222, A004277, A039900, A062447, A302540, A303975, A324522, A340610.

Table[If[n==1,0,Min@@PrimeOmega/@PrimePi/@First/@FactorInteger[n]],{n,100}]

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2677, 73327, 3578339, 329652351

Offset: 1

Thomas Scheuerle, Aug 19 2022

This sequence can be calculated by a recursive algorithm:
Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').
The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).
There exist some transformation pairs of infinite sequences in the database:
  A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;
  A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;
  A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;
  A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;
  A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;
  A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;
  A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;
  A279312 <--> A------; A031923 <--> A------.
These transformation pairs are conjectured:
  A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;
  A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;
  A349079 <--> A------.
("A------" means not yet in the database.)
Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.
If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.

			a(4) = 17. The 17 transformation pairs of length 4 are:
  {1, 2, 3, 4}  = S({1, 2, 6, 24}).
  {1, 2, 3, 5}  = S({1, 2, 6, 15}).
  {1, 2, 3, 6}  = S({1, 2, 6, 12}).
  {1, 2, 3, 9}  = S({1, 2, 6, 9}).
  {1, 2, 3, 12} = S({1, 2, 6, 8}).
  {1, 2, 3, 21} = S({1, 2, 6, 7}).
  {1, 2, 4, 5}  = S({1, 2, 4, 20}).
  {1, 2, 4, 6}  = S({1, 2, 4, 12}).
  {1, 2, 4, 8}  = S({1, 2, 4, 8}).
  {1, 2, 4, 12} = S({1, 2, 4, 6}).
  {1, 2, 4, 20} = S({1, 2, 4, 5}).
  {1, 2, 6, 7}  = S({1, 2, 3, 21}).
  {1, 2, 6, 8}  = S({1, 2, 3, 12}).
  {1, 2, 6, 9}  = S({1, 2, 3, 9}).
  {1, 2, 6, 12} = S({1, 2, 3, 6}).
  {1, 2, 6, 15} = S({1, 2, 3, 5}).
  {1, 2, 6, 24} = S({1, 2, 3, 4}).
b(1) = 1 by definition, b(2) = 1+1 as 1 has only 1 as divisor.
a(3) = A000005(b(2)*b(2)) = 3.
The divisors of b(2) are 1,2,4. So b(3) can be b(2)+1, b(2)+2 and b(2)+4.
a(4) = A000005((b(2)+1)*(b(2)+4)) + A000005((b(2)+2)*(b(2)+2)) + A000005((b(2)+4)*(b(2)+1)) = 17.

Cf. A000005.
Cf. A000027, A000045, A000079, A000142, A001405.
Cf. A001654, A004277, A006501, A008233, A019442.
Cf. A010551, A019464, A026549, A031923, A038754.
Cf. A057895, A058295, A062112, A066332, A100071.
Cf. A079352, A082458, A087811, A093968, A098011.
Cf. A098558, A100538, A111286, A166447, A137326.
Cf. A138278, A171647, A205825, A208147, A264557.
Cf. A264635, A308546, A329227, A336496, A349079.
Cf. A349080, A359039.

A086461 Symmetric version of square array A086460.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 1, 1, 4, 6, 6, 4, 1, 1, 5, 8, 9, 8, 5, 1, 1, 6, 10, 12, 12, 10, 6, 1, 1, 7, 12, 15, 16, 15, 12, 7, 1, 1, 8, 14, 18, 20, 20, 18, 14, 8, 1, 1, 9, 16, 21, 24, 25, 24, 21, 16, 9, 1, 1, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 1, 1, 11, 20, 27, 32, 35, 36

Offset: 0

Paul Barry, Jul 21 2003

Rows include A028310, A004277, A008486, A008574, A008706, A008458. Main diagonal is n^2+0^n (A000290, preceded by extra 1).

			Rows begin
  1 1 1 1 1 ...
  1 1 2 3 4 ...
  1 2 4 6 8 ...
  1 3 6 9 12 ...
  1 4 8 12 16 ...
As a triangle:
  {1},
  {1, 1},
  {1, 1, 1},
  {1, 2, 2, 1},
  {1, 3, 4, 3, 1},
  {1, 4, 6, 6, 4, 1},
  {1, 5, 8, 9, 8, 5, 1},
  {1, 6, 10, 12, 12, 10, 6, 1},
  {1, 7, 12, 15, 16, 15, 12, 7, 1},
  {1, 8, 14, 18, 20, 20, 18, 14, 8, 1},
  {1, 9, 16, 21, 24, 25, 24, 21, 16, 9, 1}

t[n_, m_] = If[ n == 0 || n == m || m == 0, 1, n - m]*If[n == m || n == 0 || m == 0, 1, m]; Table[t[n, m], {n, 0, 11}, {m, 0, n}] // Flatten (* Roger L. Bagula, Sep 06 2008 *)

T(0, k)=T(n, 0)=1, T(n, k)=nk+0^n, n, k>0

Alternatively, triangle read by rows with formula t(n,m)=If[n == 0 || n == m || m == 0, 1, n - m]*If[n == m || n == 0 || m == 0, 1, m]. - Roger L. Bagula, Sep 06 2008

A130028 A129686 * A054523.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 2, 0, 1, 6, 0, 1, 0, 1, 4, 3, 1, 1, 0, 1, 10, 0, 0, 0, 1, 0, 1, 6, 4, 1, 1, 0, 1, 0, 1, 12, 0, 2, 0, 0, 0, 1, 0, 1, 8, 6, 0, 1, 1, 0, 0, 1, 0, 1

Offset: 1

Gary W. Adamson, May 02 2007

Row sums = A004277, 1 followed by even terms: (1, 2, 4, 6, 8, 10, ...).

			First few rows of the triangle:
   1;
   1, 1;
   3, 0, 1;
   3, 2, 0, 1;
   6, 0, 1, 0, 1;
   4, 3, 1, 1, 0, 1;
  10, 0, 0, 0, 1, 0, 1;
   6, 4, 1, 1, 0, 1, 0, 1;
  ...

Cf. A129686, A054523, A004277.

A129686 * A054523 as infinite lower triangular matrices, where A129686 = the alternate term operator.

cp's OEIS Frontend

A191612 Image of A008578 (the noncomposite numbers) under the "forming" transformation.

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A229489 Conjecturally, possible differences between prime(k)^2 and the next prime for some k.

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A238398 Numerators of inverse binomial transform of A198631(n)/A006519(n+1) with -1 instead of A198631(1)=1.

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A274922 a(n) = (-1)^n * n if n>0, a(0) = 1.

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PARI

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A338329 First differences of A326118.

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A340928 Least image of A001222 applied to the prime indices of n.

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A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

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A086461 Symmetric version of square array A086460.

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A130028 A129686 * A054523.

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