A077623 -id:A077623 - cp's OEIS frontend

A076497 Number of primes corresponding to n-th primeval number A072857(n).

0, 1, 3, 4, 5, 7, 11, 14, 19, 21, 26, 29, 31, 33, 35, 41, 53, 55, 60, 64, 89, 96, 106, 122, 153, 188, 248, 311, 349, 402, 421, 547, 705, 812, 906, 1098, 1162, 1268, 1662, 1738, 1953, 2418, 2920, 3133, 3457, 4483, 4517, 4917, 5174, 5953, 6552, 6799, 8938, 10219

Offset: 1

Lekraj Beedassy, Nov 08 2002

			a(3) = 3 because the primeval number A072857(3) = 13 can be used to create 3 prime numbers, namely 3, 13 and 31.
a(6) = 7 because the primeval number A072857(7) = 113 can be used to create 7 prime numbers, namely 3, 11, 13, 31, 113, 131 and 311. (The two primes 13 and 31 can be obtained in 2 ways, therefore A075053(113) = 9.)

Giovanni Resta, Table of n, a(n) for n = 1..100
C. K. Caldwell, The Prime Glossary, Primeval Number
M. Keith, Integers containing many embedded primes
G. Villemin's Almanach of Numbers, Nombre Primeval de Mike Keith
Wikipedia, Primeval number.

Cf. A072857, A077623, A076730.

Needs["DiscreteMath`Combinatorica`"]; f[n_] := Block[{a = Drop[ Sort[ Subsets[ IntegerDigits[n]]], 1], b = c = {}, k = 1, l}, l = Length[a] + 1; While[k < l, b = Append[b, Permutations[ a[[k]] ]]; k++ ]; b = Union[ Flatten[b, 1]]; l = Length[b] + 1; k = 1; While[k < l, c = Append[c, FromDigits[ b[[k]] ]]; k++ ]; Count[ PrimeQ[ Union[c]], True]]; d = -1; Do[ b = f[n]; If[b > d, Print[b]; d = b], {n, 1, 10^6}]

a(n) = A039993(A072857(n)). - M. F. Hasler, Mar 12 2014

Edited and extended by Robert G. Wilson v, Nov 12 2002
Links fixed by Charles R Greathouse IV, Aug 13 2009
a(40)-a(54) from Giovanni Resta, Nov 06 2013

A080096 a(1) = a(2) = 1, a(3) = 2, thereafter a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

1, 1, 2, 0, 3, 1, 2, 2, 1, 3, 0, 4, 1, 3, 2, 2, 3, 1, 4, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5

Offset: 1

Benoit Cloitre, Jan 28 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A028347, A028557, A077623, A077653, A079623, A079624, A088226.

a080096 n = a080096_list !! (n-1)
a080096_list = 1 : 1 : 2 : zipWith3 (\u v w -> abs (w - v - u))
               a080096_list (tail a080096_list) (drop 2 a080096_list)
-- Reinhard Zumkeller, Oct 11 2014

m:=120;
A080096:=[n le 3 select Floor((n+1)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A080096[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024

nxt[{a_,b_,c_}]:={b,c,Abs[c-b-a]}; NestList[nxt,{1,1,2},110][[All,1]] (* Harvey P. Dale, Nov 14 2021 *)

a(n)=local(k,m); if(n<1,0,k=sqrtint(n+4); m=n+4-k^2; if(m%2,m\2+1,k-m\2))

@CachedFunction
def a(n): # a = A080096
    if n<4: return int((n+1)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

For n>=3 Max( a(k) : 1<=k<=n ) = floor ( sqrt(n+4)).
Special cases: a(n^2 + 4*n - 1) = 0 and a(n^2 - 4) = n.
a(A028557(n)) = a(A028557(n+1)).
Sum_{k=(n-1)^2 .. n^2} a(k) = n^2.

A077653 a(1)=1, a(2)=2, a(3)=2, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

1, 2, 2, 1, 3, 0, 4, 1, 3, 2, 2, 3, 1, 4, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7

Offset: 1

Benoit Cloitre, Dec 02 2002

nonn

Conjecture : let z(1)=x; z(2)=y; z(3)= z; z(n)=abs(z(n-1)-z(n-2)-z(n-3)) if z(n) is unbounded (i.e. x,y,z are such that z(n) doesn't reach a cycle of length 2), then there are 2 integers n(x,y,z) and w(x,y,z) such that M(n) = floor(sqrt(n+w(x,y,z))) for n>n(,x,y,z) where M(n) = Max ( a(k) : 1<=k<=n ). As example : w(1,2,2)=9 n(1,2,2)=4; w(1,2,4)=29 n(1,2,4)=4; w(1,2,8)=157 n(1,2,8)=9

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A077623, A079623, A079624, A080096, A088226.

a077653 n = a077653_list !! (n-1)
a077653_list = 1 : 2 : 2 : zipWith3 (\u v w -> abs (w - v - u))
               a077653_list (tail a077653_list) (drop 2 a077653_list)
-- Reinhard Zumkeller, Oct 11 2014

m:=120;
A077653:=[n le 3 select Floor((n+2)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A077653[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024

nxt[{a_,b_,c_}]:={b,c,Abs[c-b-a]}; NestList[nxt,{1,2,2},110][[All,1]] (* Harvey P. Dale, Sep 01 2020 *)

@CachedFunction
def a(n): # a = A077653
    if n<4: return int((n+2)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

a(n)/sqrt(n) is bounded. More precisely, let M(n) = Max ( a(k) : 1<=k<=n ); then M(n)= floor(sqrt(n+9)) for n>4

A079623 a(1) = a(2) = 1, a(3)=4, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

1, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 0, 11, 1, 10, 2, 9, 3, 8

Offset: 1

Benoit Cloitre, Jan 30 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A077623, A077653, A078108, A079624, A080096, A088226.

a079623 n = a079623_list !! (n-1)
a079623_list = 1 : 1 : 4 : zipWith3 (\u v w -> abs (w - v - u))
               a079623_list (tail a079623_list) (drop 2 a079623_list)
-- Reinhard Zumkeller, Oct 11 2014

m:=120;
A079623:=[n le 3 select 4^Floor((n-1)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A079623[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024

nxt[{a_,b_,c_}]:={b,c,Abs[c-b-a]}; NestList[nxt,{1,1,4},110][[All,1]] (* Harvey P. Dale, Aug 12 2020 *)

@CachedFunction
def a(n): # a = A079623
    if n<4: return 4^((n-1)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

a(n*(n-10)) = 0.

Max( a(k) : 1<=k<=n) = floor(sqrt(n+24)).

A079624 a(1) = a(2) = 1, a(3) = 6, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

1, 1, 6, 4, 3, 7, 0, 10, 3, 7, 6, 4, 9, 1, 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, 7, 6, 8, 5, 9, 4, 10, 3, 11, 2, 12, 1, 13, 0, 14, 1, 13, 2, 12, 3, 11, 4, 10, 5, 9, 6, 8, 7, 7, 8, 6, 9, 5, 10, 4, 11, 3, 12, 2, 13, 1, 14, 0, 15, 1, 14, 2, 13, 3, 12, 4, 11, 5, 10, 6, 9, 7, 8, 8, 7, 9, 6, 10, 5, 11, 4, 12, 3

Offset: 1

Benoit Cloitre, Jan 30 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A077623, A077653, A078108, A079623, A080096, A088226.

a079624 n = a079624_list !! (n-1)
a079624_list = 1 : 1 : 6 : zipWith3 (\u v w -> abs (w - v - u))
               a079624_list (tail a079624_list) (drop 2 a079624_list)
-- Reinhard Zumkeller, Oct 11 2014

m:=120;
A079624:=[n le 3 select 6^Floor((n-1)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A079624[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024

a[n_]:= a[n]= If[n<4, 6^Floor[(n-1)/2], Abs[a[n-1] -a[n-2] -a[n-3]]];
Table[a[n], {n,100}] (* G. C. Greubel, Sep 11 2024 *)

@CachedFunction
def a(n): # a = A079624
    if n<4: return 6^((n-1)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

For n >= 5, a(n^2 + 24*n - 13) = 0.

For n >= 38, Max( a(k) : 1<=k<=n) = floor(sqrt(n+156)).

A088226 a(1)=0, a(2)=0, a(3)=1; for n>3, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 1, 1, 2, 0, 3, 1, 2, 2, 1, 3, 0, 4, 1, 3, 2, 2, 3, 1, 4, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2

Offset: 1

Benoit Cloitre, Nov 04 2003

nonn

Conjecture: a(n) = m/2 where m is the smallest even distance from n to a square. - Ralf Stephan, Sep 23 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A077623, A077653, A079623, A079624, A080096.

a088226 n = a088226_list !! (n-1)
a088226_list = 0 : 0 : 1 : zipWith3 (\u v w -> abs (w - v - u))
               a088226_list (tail a088226_list) (drop 2 a088226_list)
-- Reinhard Zumkeller, Oct 11 2014

m:=120;
A088226:=[n le 3 select Floor((n-1)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A088226[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024

RecurrenceTable[{a[1]==a[2]==0,a[3]==1,a[n]==Abs[a[n-1]-a[n-2]-a[n-3]]},a,{n,110}] (* Harvey P. Dale, Apr 13 2012 *)

a(n)=t=sqrtint(n);if((n-t*t)%2==0,(n-t*t)/2,((t+1)^2-n)/2) \\ Ralf Stephan, Sep 23 2013

@CachedFunction
def a(n): # a = A088226
    if n<4: return int((n-1)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

a(k^2 + 2*m + 2) = k-m and a(k^2 + 2*m + 1) = m, for k >= 0 and 0 <= m <= k.

A078108 Let u(1)=u(2)=1, u(3)=2n, u(k) = abs(u(k-1)-u(k-2)-u(k-3)) and M(k) = Max_{i<=i<=k} u(i), then for any k >= A078109(n), M(k) = floor(sqrt(k + a(n))).

Original entry on oeis.org

4, 24, 156, 184, 324, 608, 940, 1784, 1844, 3104, 5996, 4600, 4484, 6128, 6220, 7208, 8244, 9424, 11740, 13560, 14836, 19264, 19756, 23344, 24524, 26224, 32940, 34912, 34548, 42808, 52428, 46120, 47492, 52280, 67908, 86120, 80084, 147152

Offset: 1

Benoit Cloitre, Dec 05 2002

nonn

It appears that (1) a(n) always exists, (2) a(n) is even, (3) a(n)/n^(5/2) -> infinity. If initial conditions are u(1)=u(2)=1, u(3)=2n+1, then u(k) reaches a 2-cycle for any k>m large enough (cf. A078098). - Benoit Cloitre, Jan 29 2006

Cf. A000196, A078109, A077623, A078098.

Typos in data corrected by Sean A. Irvine, Jun 16 2025

A078109 Let u(1)=u(2)=1, u(3)=2n, u(k) = abs(u(k-1)-u(k-2)-u(k-3)) and M(k) = Max_{1<=i<=k} u(i), then for any k >= a(n), M(k) = floor(sqrt(k + A078108(n))).

Original entry on oeis.org

3, 10, 38, 10, 35, 66, 19, 150, 90, 30, 243, 159, 138, 270, 19, 186, 35, 178, 358, 127, 46, 334, 123, 370, 438, 343, 182, 430, 46, 454, 470, 534, 30, 618, 734, 903, 570, 302, 571, 638, 30, 166, 822, 647, 426, 998, 75, 106, 606, 322, 82, 210, 1798, 330, 506

Offset: 1

Benoit Cloitre, Dec 05 2002

nonn

Conjecture : a(n) always exists, a(n)/n^2 is bounded. If initial conditions are u(1)=u(2)=1, u(3)=2n+1, then u(k) reaches a 2-cycle for any k>m large enough (cf. A078098)

Cf. A000196, A078108, A077623.

Typos in data corrected and more terms from Sean A. Irvine, Jun 16 2025

cp's OEIS Frontend

A076497 Number of primes corresponding to n-th primeval number A072857(n).

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A080096 a(1) = a(2) = 1, a(3) = 2, thereafter a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

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A077653 a(1)=1, a(2)=2, a(3)=2, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

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A079623 a(1) = a(2) = 1, a(3)=4, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

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A079624 a(1) = a(2) = 1, a(3) = 6, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

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A088226 a(1)=0, a(2)=0, a(3)=1; for n>3, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

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A078108 Let u(1)=u(2)=1, u(3)=2n, u(k) = abs(u(k-1)-u(k-2)-u(k-3)) and M(k) = Max_{i<=i<=k} u(i), then for any k >= A078109(n), M(k) = floor(sqrt(k + a(n))).

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