A008578 -id:A008578 - cp's OEIS frontend

A158635 6n - A008578(n).

5, 10, 15, 19, 23, 25, 29, 31, 35, 37, 37, 41, 41, 43, 47, 49, 49, 49, 53, 53, 55, 59, 59, 61, 61, 59, 61, 65, 67, 71, 73, 65, 67, 67, 71, 67, 71, 71, 71, 73, 73, 73, 77, 73, 77, 79, 83, 77, 71, 73, 77, 79, 79, 83, 79, 79, 79, 79, 83, 83, 85, 89, 85, 77, 79, 83, 85, 77, 77, 73, 77

Offset: 1

Paul Curtz, Mar 23 2009

Sequence items become negative at index 191, since the prime numbers A008578 grow faster than linearly.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Join[{5},Table[6n-Prime[n-1],{n,2,80}]] (* Harvey P. Dale, Nov 18 2012 *)

a(n)=if(n>1,6*n-prime(n-1),5) \\ Charles R Greathouse IV, Mar 16 2016

Edited, offset changed by R. J. Mathar, Apr 04 2009

Comment clarified and modified by Harvey P. Dale, Nov 18 2012

A159081 Let d be the largest element of A008578 which divides n, then a(n) is the position of d in A008578.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 3, 4, 6, 3, 7, 5, 4, 2, 8, 3, 9, 4, 5, 6, 10, 3, 4, 7, 3, 5, 11, 4, 12, 2, 6, 8, 5, 3, 13, 9, 7, 4, 14, 5, 15, 6, 4, 10, 16, 3, 5, 4, 8, 7, 17, 3, 6, 5, 9, 11, 18, 4, 19, 12, 5, 2, 7, 6, 20, 8, 10, 5, 21, 3, 22, 13, 4, 9, 6, 7, 23, 4, 3, 14, 24, 5, 8, 15, 11, 6, 25, 4, 7, 10, 12

Offset: 1

Jaroslav Krizek, Apr 05 2009

Let p be the largest prime factor of n; if p = prime(k) then set a(n) = k + 1. a(n) = A061395(n) + 1.

			For n=30, the largest element of the set {1,2,3,5} (1 and prime divisors of 30) is 5, and 5 is a(n)=4th term of A008578, the extended set of primes.

Cf.: A061395, A049084, A006530, A008578.

a(n) = A049084(A006530(n)) + 1. A008578(a(n)) = A006530(n);

Edited by R. J. Mathar, Apr 06 2009

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

A166747 Numbers with n digits from the set {1,2,4}, requiring a nondecreasing sequence of digits and a digits sum of A008578(n).

Original entry on oeis.org

1, 11, 111, 1112, 11122, 122222, 1222222, 12222224, 122222224, 1222222244, 12222224444, 122222224444, 1222222444444, 12222224444444, 122222224444444, 1222222244444444, 12222224444444444, 122222444444444444, 1222222444444444444, 12222244444444444444

Offset: 1

Paul Curtz, Oct 21 2009

Or: a partitioning of p=A008578(n) into n parts, each part an element of {1,2,4}.

The representation is often not unique: p=11 could be represented by 111224 or 122222, p=13 by 1111144 or 1112224 or 1222222, p=17 by 11111444 or 11122244 or 12222224. a(n) selects the representation with the minimum number of 4's. - R. J. Mathar, Oct 25 2009

Cf. A134732.

A008578 := proc(n) if n = 1 then 1; else ithprime(n-1) ; fi; end:
A166747 := proc(n) local p,n1,n2,n4,i ; p := A008578(n) ; for n4 from 0 to n do n2 := p-n-3*n4 ; n1 := n-n2-n4 ; if n2 >= 0 and n1 >= 0 then a := 0 ; for i from 1 to n1 do a := 10*a+1 ; od: for i from 1 to n2 do a := 10*a+2 ; od: for i from 1 to n4 do a := 10*a+4 ; od: return a ; end if: end do: end:
seq(A166747(n),n=1..20) ; # R. J. Mathar, Oct 25 2009

Edited by R. J. Mathar, Oct 25 2009

A167136 a(n) = b(n)-th highest positive integer not equal to any a(k), 1 <= k <= n-1, where b(n) = noncomposite numbers = A008578(n).

Original entry on oeis.org

1, 3, 5, 8, 11, 16, 19, 24, 27, 32, 39, 42, 49, 54, 57, 62, 69, 76, 79, 86, 91, 94, 101, 106, 113, 122, 127, 130, 135, 138, 143, 158, 163, 170, 173, 184, 187, 194, 201, 206, 213, 220, 223, 234, 237, 242, 245, 258, 271, 276, 279, 284, 291, 294, 305, 312, 319, 326

Offset: 1

Jaroslav Krizek, Oct 28 2009

nonn

a(1) = 1, a(n) = A014688(n-1) = (n-1)-th prime + n - 1 for n >= 2. a(n) = A090178(n) - 1 = n-th noncomposite number + n - 1 for n >= 2.

			A008578(4) = 5, so a(4) = 8 = 5th highest positive integer not equal to 1, 3, or 5 (the values of a(k), 1 <= k <= 3).

a(1) = 1, a(n) = a(n-1) + A008578(n+1) - A008578(n) + 1 for n >= 2. a(1) = 1, a(2) = 3, a(n) = a(n-1) + A001223(n) + 1 for n >= 3. a(1) = 1, a(n) = n - 1 + A000040(n-1) = n - 1 + A008578(n) = n - 1 + A158611(n+1) for n >= 2.

A175249 Noncomposites (A008578) with nonprime (A018252) subscripts.

Original entry on oeis.org

1, 5, 11, 17, 19, 23, 31, 41, 43, 47, 59, 67, 71, 73, 83, 89, 97, 101, 103, 109, 127, 131, 137, 139, 149, 157, 163, 167, 179, 191, 193, 197, 211, 223, 227, 229, 233, 241, 251, 257, 263, 269, 277, 283, 293, 307, 311, 313, 331, 337, 347, 353, 367, 373, 379, 383, 389, 401, 409, 419

Offset: 1

Jaroslav Krizek, Mar 13 2010

nonn

			a(5) = 19 because a(5) = q(b(5)) = q(9) = 19, q = noncomposite, b = nonprime.

Cf. A008578, A018252, A088982.

Join[{1}, Table[Prime[ResourceFunction["Composite"][n] - 1], {n, 1, 100}]]

a(n) = noncomposite(nonprime(n)) = A008578(A018252(n)).

a(34)-a(60) from Terry D. Grant, Aug 16 2016

A180129 Expansion of log(1/(1-Prime(x))) where Prime(x) = Sum{n>=1} A008578(n)*x^n.

Original entry on oeis.org

1, 5, 16, 49, 136, 380, 1016, 2745, 7369, 19840, 53395, 143620, 386374, 1039386, 2796001, 7521561, 20233860, 54431237, 146425762, 393900764, 1059634634, 2850529093, 7668224519, 20628334404, 55492399461, 149280418500, 401580100348, 1080292905482, 2906102072179, 7817721667285

Offset: 1

Vladimir Kruchinin, Aug 12 2010

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A008578, A300662.

seq(n)={Vec(deriv(log(1/(1 - x - x*sum(k=1, n-1, prime(k)*x^k, O(x^n))))))} \\ Andrew Howroyd, Jan 04 2020

Logarithm g.f.: Sum_{n>0} a(n)*x^n/n = log(1/(1-Prime(x))).

Logarithmic derivative of A300662. - Andrew Howroyd, Jan 04 2020

Terms a(16) and beyond from Andrew Howroyd, Jan 04 2020

A181095 Permutation of positive integers created from sequence A008578(n) = noncomposites.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 11, 13, 17, 8, 19, 9, 23, 29, 31, 10, 37, 12, 41, 43, 47, 14, 53, 59, 61, 67, 71, 15, 73, 16, 79, 83, 89, 97, 101, 18, 103, 107, 109, 20, 113, 21, 127, 131, 137, 22, 139, 149, 151, 157, 163, 24, 167, 173, 179, 181, 191, 25, 193, 26, 197, 199, 211, 223

Offset: 1

Jaroslav Krizek, Oct 02 2010

nonn

Method of creation of sequence a(n) from b(n) = A008578(n):
Take b(1) and put in vacancy place a(1), swap this place and b(1), repeat with b(2), b(3), ...:
b(1) = 1, a(1) = 1,
b(2) = 2, a(2) = 2,
b(3) = 3, a(3) = 3,
b(4) = 5, a(4) = 5, a(5) = 4,
b(5) = 7, a(6) = 7, a(7) = 6,
...

a(n) = A163465(n) for n >= 4. a(n) = A181094(n) for n >= 4.

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

Original entry on oeis.org

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).

A238256 A060308 begins with one 2, one 3, one 5, two 7's, one 11, two 13's, i.e., d(n) = 1, 1, 1, 2, 1, 2, 1, 2, 3, 1,... times the primes (A000040). a(n) uses this distribution with noncomposites (A008578).

Original entry on oeis.org

1, 2, 3, 5, 5, 7, 11, 11, 13, 17, 17, 19, 19, 19, 23, 29, 29, 29, 31, 31, 37, 41, 41, 43, 43, 43, 47, 47, 47, 53, 59, 59, 59, 61, 61, 67, 71, 71, 71, 73, 73, 79, 79, 79, 83, 83, 83, 83, 89, 89, 97, 101, 101, 103, 107, 107, 109, 109, 109, 109, 109, 109, 109

Offset: 1

Paul Curtz, Mar 10 2014

nonn

Paul Curtz, Table of n, a(n) for n = 1..63

Cf. A224911.

lista(nn) = {nn = nn\2; v = vector(nn, i, precprime(2*i)); vnc = concat(1, vector(nn, i, prime(i))); nv = vector(1, i, vnc[i]); ivnc = 1; for (i=2, #v, if (v[i] == v[i-1], nv = concat(nv, nv[#nv]), ivnc++; nv = concat(nv, vnc[ivnc]));); for (i=1, #nv, print1(nv[i], ", "));} \\ Michel Marcus, Mar 20 2014

Conjecture: a(n) is the greatest noncomposite (A008578) dividing A238691(n-1).

A354003 Inverse Stirling transform of A008578 (1 together with the primes).

Original entry on oeis.org

1, 1, -1, 3, -14, 84, -604, 5020, -47144, 492408, -5653004, 70681706, -955450018, 13878511166, -215521103888, 3562431678650, -62439880637498, 1156609714838858, -22575425757129216, 463085375385002432, -9959296414838153618, 224079866356625633070, -5264190202707104532482

Offset: 1

Ilya Gutkovskiy, May 13 2022

sign

Cf. A008578, A307772, A307773, A351681, A354002.

nmax = 23; CoefficientList[Series[Log[1 + x] + Sum[Prime[k - 1] Log[1 + x]^k/k!, {k, 2, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] If[k == 1, 1, Prime[k - 1]], {k, 1, n}], {n, 1, 23}]

E.g.f.: log(1 + x) + Sum_{k>=2} prime(k-1) * log(1 + x)^k / k!.

a(n) = Sum_{k=1..n} Stirling1(n,k) * A008578(k).

cp's OEIS Frontend

A158635 6n - A008578(n).

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A159081 Let d be the largest element of A008578 which divides n, then a(n) is the position of d in A008578.

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A166747 Numbers with n digits from the set {1,2,4}, requiring a nondecreasing sequence of digits and a digits sum of A008578(n).

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A167136 a(n) = b(n)-th highest positive integer not equal to any a(k), 1 <= k <= n-1, where b(n) = noncomposite numbers = A008578(n).

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A175249 Noncomposites (A008578) with nonprime (A018252) subscripts.

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Mathematica

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A180129 Expansion of log(1/(1-Prime(x))) where Prime(x) = Sum{n>=1} A008578(n)*x^n.

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PARI

Formula

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A181095 Permutation of positive integers created from sequence A008578(n) = noncomposites.

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

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Formula

A238256 A060308 begins with one 2, one 3, one 5, two 7's, one 11, two 13's, i.e., d(n) = 1, 1, 1, 2, 1, 2, 1, 2, 3, 1,... times the primes (A000040). a(n) uses this distribution with noncomposites (A008578).

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