A008578 -id:A008578 - cp's OEIS frontend

A280055 Nachos sequence based on 1 plus primes (A008578).

1, 2, 1, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 2, 3, 4, 2, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 2, 3, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 2, 3, 4, 3, 4, 5, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 2, 3, 4, 3, 4, 5, 3, 2, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 2, 3, 4, 3, 4, 5, 3, 2, 3, 4, 3

Offset: 1

N. J. A. Sloane, Jan 08 2017

nonn

Like A280053 but based on 1,2,3,5,7,11,... rather than squares. See that entry for further information.

Equivalently, greedily subtract terms of A014284 from n until reaching 0; a(n) = number of steps required.

			26 takes 4 phases to read 0:
subtract leaves
1   25
2   23
3   20
5   15
7   8
------
1   7
2   5
3   2
------
1   1
------
1   0
so a(26) = 4

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A280055, A014284.

For records see A280760.

A280055 := proc(n)
    local a,nres,i ;
    a := 0 ;
    nres := n;
    while nres > 0 do
        for i from 1 do
            if A014284(i) > nres then
                break;
            end if;
        end do:
        nres := nres-A014284(i-1) ;
        a := a+1 ;
    end do:
    a ;
end proc:
seq(A280055(n),n=1..80) ; # R. J. Mathar, Mar 05 2017

A327154 a(n) = Product_{d|n, d>1} A008578(1+A286561(sigma(n),d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 12, 1, 1, 1, 2, 1, 6, 1, 5, 2, 1, 1, 2, 1, 2, 1, 3, 1, 48, 1, 2, 1, 80, 1, 45, 1, 1, 2, 2, 1, 1, 1, 3, 1, 8, 1, 44, 1, 6, 2, 5, 1, 6, 1, 1, 3, 2, 1, 20, 1, 20, 1, 2, 1, 80, 1, 11, 1, 1, 1, 63, 1, 2, 2, 7, 1, 2, 1, 2, 1, 6, 1, 20, 1, 2, 1, 2, 1, 264, 1, 3, 2, 6, 1, 48, 2, 10, 1, 7, 2, 108, 1, 1, 2, 1, 1, 125, 1, 2, 2

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000040, A008578, A286561, A073802.

Cf. also A293514, A322312, A323155, A327155, A327156.

A327154(n) = { my(m=1,s=sigma(n),v); fordiv(n,d,if((d>1) && ((v = valuation(s,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|n, d>1} A008578(1+A286561(sigma(n),d)), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A073802(n).

A327155 a(n) = Product_{d|sigma(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 8, 1, 1, 1, 2, 1, 6, 1, 2, 2, 1, 1, 3, 1, 3, 1, 2, 1, 80, 1, 2, 1, 48, 1, 8, 1, 1, 2, 2, 1, 1, 1, 2, 1, 20, 1, 8, 1, 6, 3, 2, 1, 21, 1, 1, 2, 3, 1, 20, 1, 20, 1, 2, 1, 48, 1, 2, 1, 1, 1, 8, 1, 3, 2, 2, 1, 3, 1, 2, 1, 6, 1, 8, 1, 7, 1, 2, 1, 48, 1, 2, 2, 10, 1, 48, 2, 6, 1, 2, 2, 264, 1, 1, 3, 1, 1, 8, 1, 5, 2

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000040, A008578, A286561, A073802.

Cf. also A293514, A322312, A323155, A327154, A327156.

A327155(n) = { my(m=1,v); fordiv(sigma(n),d,if((d>1) && ((v = valuation(n,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|sigma(n), d>1} A008578(1+A286561(n,d)), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A073802(n).

A327156 a(n) = Product_{d|n, d>1} A008578(1+A286561(n,sigma(d))), where A286561(n,x) gives the highest exponent of x dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 8, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 5, 1, 8, 1, 1, 1, 16, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000040, A008578, A286561, A173441.

Cf. also A293514, A322312, A323155, A327154, A327155.

A327156(n) = { my(m=1,v); fordiv(n,d,if((d>1) && ((v = valuation(n,sigma(d)))>0), m *= prime(v))); (m); };

a(n) = Product_{d|n, d>1} A008578(1+A286561(n,sigma(d))), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A173441(n).

A116151 a(n) = smallest positive integer x satisfying the system of congruences { x == 1 (mod 2), x == 2 (mod 3), x == 3 (mod 5), x == 5 (mod 7), ..., x == A008578(n) (mod A008578(n+1)) }.

Original entry on oeis.org

1, 5, 23, 173, 2273, 2273, 452723, 6578843, 113275433, 3682761353, 10152454583, 5024164707833, 249908523156563, 5726413266646343, 345878207890067123, 15103232990013860963, 1905274424667036455303, 111502614383457156882293

Offset: 1

Christian Bjartli (cbjartli(AT)gmail.com), Apr 14 2007

nonn

Minimum Chinese Remainder Prime Modulus Ladder: for the n-th term, the number modulus a prime equals the previous prime for the first n primes (the initial term is defined to be 1). - Fred Schneider, Oct 21 2007

			a(3)=23 because that is the smallest number such that n==1 (mod 2), n==2 (mod 3) and n == 3 (mod 5).

Harvey P. Dale, Table of n, a(n) for n = 1..350 (First 100 terms from T. D. Noe)

Cf. A070198.

Primes:= [1,seq(ithprime(i),i=1..30)]:
seq(chrem(Primes[1..k],Primes[2..k+1]),k=1..30); # Robert Israel, Oct 26 2018

Table[ChineseRemainder[Join[{1},Prime[Range[n-1]]],Prime[Range[n]]],{n,20}] (* Harvey P. Dale, Mar 30 2018 *)

{ a(n) = lift(chinese(vector(n,i,Mod(if(i==1,1,prime(i-1)),prime(i))))) }; vector(30,n,a(n)) \\ Max Alekseyev, Apr 16 2007

my(z=Mod(1,2)); forprime(x=3,100,z=chinese(z,Mod(precprime(x-1),x)); print1(lift(z), ", ")); \\ Fred Schneider, Oct 21 2007

More terms from Max Alekseyev, Apr 16 2007
Edited by N. J. A. Sloane, May 05 2007
Further edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar and Christian Bjartli.

A162618 Triangle read by rows in which row n lists n consecutive natural numbers A000027, starting with A008578(n-1) - n + 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 3, 4, 5, 3, 4, 5, 6, 7, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 15, 16, 17, 11, 12, 13, 14, 15, 16, 17, 18, 19, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30

Offset: 1

Omar E. Pol, Jul 10 2009

Note that the last term of the n-th row is the noncomposite number A008578(n-1).

			Contribution from _Omar E. Pol_, Jul 15 2009: (Start)
Triangle begins:
   1;
   1,  2;
   1,  2,  3;
   2,  3,  4,  5;
   3,  4,  5,  6,  7;
   6,  7,  8,  9, 10, 11;
   7,  8,  9, 10, 11, 12, 13;
  10, 11, 12, 13, 14, 15, 16, 17;
  11, 12, 13, 14, 15, 16, 17, 18, 19;
  14, 15, 16, 17, 18, 19, 20, 21, 22, 23;
  19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29;
  20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31;
(End)

Cf. A000040, A008578, A159797, A159798, A162611, A162619, A162620.

Cf. A162614, A162622. - Omar E. Pol, Jul 15 2009

A243498 Records (and distinct values) of A243069: a(n) = A243069(A008578(n)).

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 12, 13, 15, 16, 19, 21, 22, 27, 28, 30, 31, 36, 37, 39, 42, 45, 46, 49, 51, 52, 57, 58, 60, 61, 66, 69, 72, 73, 75, 76, 81, 82, 87, 88, 91, 96, 97, 99, 102, 105, 109, 111, 112, 118, 120, 126, 127, 129, 132, 133, 135, 136, 142, 147, 148, 150

Offset: 1

Antti Karttunen, Jun 21 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A243069, A243488, A126917, A008578.

(define (A243498 n) (A243069 (A008578 n)))

a(n) = A243069(A008578(n)).

A329037 a(n) = Product_{d|n, d>1} A008578(1+A286561(A276086(n),d)), where A286561(x,d) gives the exponent of the highest power of d dividing x.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 12, 1, 1, 1, 1, 5, 2, 1, 1, 1, 21, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 12, 1, 1, 1, 2, 10, 2, 1, 1, 1, 7, 2, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 12, 1, 1, 1, 1, 48, 1, 3, 1, 1, 5, 2, 1, 1, 3, 7, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 1, 10, 2, 2, 1, 1, 1, 1, 720

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to primorial base

Cf. A008578, A276086, A286561, A327168.

Cf. also A327167.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329037(n) = { my(m=1,x=A276086(n),v); fordiv(n,d,if((d>1) && ((v = valuation(x,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|n, d>1} A008578(1+A286561(A276086(n),d)).

1+A001222(a(n)) = A327168(n).

A329042 a(n) = Product_{d|n, d>1} A008578(1+A286561(A122111(n),d)), where A286561(x,d) gives the exponent of the highest power of d dividing x.

Original entry on oeis.org

1, 2, 1, 1, 1, 8, 1, 1, 6, 3, 1, 2, 1, 5, 3, 1, 1, 2, 1, 48, 3, 7, 1, 2, 1, 11, 1, 10, 1, 128, 1, 1, 3, 13, 1, 2, 1, 17, 3, 6, 1, 12, 1, 21, 3, 19, 1, 2, 1, 2, 3, 33, 1, 1, 1, 320, 3, 23, 1, 8, 1, 29, 1, 1, 1, 20, 1, 65, 3, 8, 1, 2, 1, 31, 48, 85, 1, 28, 1, 6, 1, 37, 1, 3072, 1, 41, 3, 42, 1, 8, 1, 133, 3, 43, 1, 2, 1, 1, 1, 1, 1, 44, 1, 66, 12

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A001222, A008578, A122111, A286561, A329036, A329043.

Cf. also A329037.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A329042(n) = { my(m=1,x=A122111(n),v); fordiv(n,d,if((d>1) && ((v = valuation(x,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|n, d>1} A008578(1+A286561(A122111(n),d)).

1+A001222(a(n)) = A329036(n).

A329043 a(n) = Product_{d|A122111(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the exponent of the highest power of d dividing n.

Original entry on oeis.org

1, 2, 1, 1, 1, 8, 1, 1, 6, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 48, 2, 2, 1, 5, 1, 2, 1, 6, 1, 128, 1, 1, 2, 2, 1, 3, 1, 2, 2, 10, 1, 8, 1, 6, 2, 2, 1, 7, 1, 3, 2, 6, 1, 1, 1, 320, 2, 2, 1, 12, 1, 2, 1, 1, 1, 8, 1, 6, 2, 8, 1, 3, 1, 2, 48, 6, 1, 8, 1, 21, 1, 2, 1, 3072, 1, 2, 2, 20, 1, 8, 1, 6, 2, 2, 1, 11, 1, 1, 1, 1, 1, 8, 1, 20, 8

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A001222, A008578, A122111, A286561, A329036, A329042.

Cf. also A327167.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A329043(n) = { my(m=1,v); fordiv(A122111(n),d,if((d>1) && ((v = valuation(n,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|A122111(n), d>1} A008578(1+A286561(n,d)).

1+A001222(a(n)) = A329036(n).

cp's OEIS Frontend

A280055 Nachos sequence based on 1 plus primes (A008578).

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A327154 a(n) = Product_{d|n, d>1} A008578(1+A286561(sigma(n),d)), where A286561(n,d) gives the highest exponent of d dividing n.

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A327155 a(n) = Product_{d|sigma(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

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Formula

A327156 a(n) = Product_{d|n, d>1} A008578(1+A286561(n,sigma(d))), where A286561(n,x) gives the highest exponent of x dividing n.

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A116151 a(n) = smallest positive integer x satisfying the system of congruences { x == 1 (mod 2), x == 2 (mod 3), x == 3 (mod 5), x == 5 (mod 7), ..., x == A008578(n) (mod A008578(n+1)) }.

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A162618 Triangle read by rows in which row n lists n consecutive natural numbers A000027, starting with A008578(n-1) - n + 1.

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A243498 Records (and distinct values) of A243069: a(n) = A243069(A008578(n)).

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Scheme

Formula

A329037 a(n) = Product_{d|n, d>1} A008578(1+A286561(A276086(n),d)), where A286561(x,d) gives the exponent of the highest power of d dividing x.

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PARI

Formula

A329042 a(n) = Product_{d|n, d>1} A008578(1+A286561(A122111(n),d)), where A286561(x,d) gives the exponent of the highest power of d dividing x.

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