A086361 -id:A086361 - cp's OEIS frontend

A086353 Fixed point if nonzero-digit product of n! is iterated.

1, 2, 6, 8, 2, 4, 2, 8, 8, 8, 6, 1, 2, 8, 8, 8, 8, 6, 8, 8, 8, 8, 8, 2, 2, 8, 4, 8, 6, 2, 2, 6, 1, 8, 8, 8, 2, 2, 6, 8, 8, 8, 8, 8, 8, 6, 8, 6, 8, 8, 8, 6, 6, 1, 8, 8, 5, 8, 6, 6, 8, 6, 8, 2, 8, 8, 8, 6, 8, 2, 8, 8, 2, 6, 6, 8, 9, 6, 8, 8, 6, 2, 2, 8, 8, 8, 8, 4, 6, 8, 9, 6, 2, 2, 8, 2, 8, 8, 4, 4, 8, 8, 6, 2, 8

Offset: 1

Labos Elemer, Jul 21 2003

			n=10, 10!=362880, iteration list={3628800,2304,24,8},a(10)=8.

Cf. A051801, A051802, A086354-A086361, A029898, A010888.

prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]] Table[FixedPoint[prd, w! ], {w, 1, 128}]

a(n)=A051802[n! ]=fixed-point of A051801[n! ]

A023287 Primes that remain prime through 3 iterations of function f(x) = 6x + 1.

Original entry on oeis.org

61, 101, 1811, 3491, 4091, 5711, 5801, 6361, 7121, 10391, 10771, 11311, 13421, 15131, 17791, 18911, 19471, 20011, 24391, 25601, 25951, 30091, 35251, 41911, 45631, 47431, 55631, 58711, 62921, 67891, 70451, 70571, 72271, 74051, 74161, 75431, 80471, 86341

Offset: 1

David W. Wilson

nonn

Primes p such that s1=p, s2=6*s1+1, s3=6*s2+1 and s4=6*s3+1 are primes forming a special chain of four primes. A fifth term in such a chain cannot arise. See A085956, A086361, A086362.

Entries in chains are congruent to {1,7,3,9} mod 10.

			First chain is {61, 367, 2203, 13219};
319th chain is {1291391, 7748347, 46490083, 278940499}.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A085956, A086361, A086362, A023330, A059766.

Subsequence of A007693, A023256, and A024899.

[n: n in [1..150000] | IsPrime(n) and IsPrime(6*n+1) and IsPrime(36*n+7) and IsPrime(216*n+43)] // Vincenzo Librandi, Aug 04 2010

k=0; m=6; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; If[PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], k=k+1; Print[{k, n, s, s1, s2, s3}]], {n, 1, 100000}] (* edited by Zak Seidov, Feb 08 2011 *)
thrQ[n_]:=AllTrue[Rest[NestList[6#+1&,n ,3]],PrimeQ]; Select[Prime[Range[9000]],thrQ] (* Harvey P. Dale, Mar 03 2024 *)

{p, 6p+1, 36p+7, 216p+43} are all primes, where p is prime.

Additional comments from Labos Elemer, Jul 23 2003

A086362 a(n) = A085956(3n+1).

Original entry on oeis.org

5, 17, 239, 41, 131, 1889, 419, 89, 101, 113, 2543, 2789, 149, 881, 173, 9293, 491, 14249, 3191, 1973, 3539, 21377, 7103, 281, 5987, 38153, 317, 2789, 6971, 353, 214943, 42677, 3299, 11801, 2267, 27773, 29867, 10529, 461, 1181, 2663, 129209

Offset: 0

Labos Elemer, Jul 22 2003

nonn

A086361 and A086362 includes most probably the solvable [or small?] cases of A085956, where n=3k-1 has probably only two solutions, at [k,n,a(n)]=[1,2,13] and at [2,5,31].

			n=50:3n+1=151,6n+2=302, a(50)=93923 prime with (93922-1)/302=311, prime and with 302.93923+1=28364747 prime; so the {311,93923,28364747} "generalized [short] Cunningham-chain" is defined.

Michael De Vlieger, Table of n, a(n) for n = 0..1000

Cf. A085956, A086361.

Table[Function[m, p = 2; While[Nand @@ PrimeQ@ {2 m p + 1, (p - 1)/(2 m)}, p = NextPrime@ p]; p][3 n + 1], {n, 0, 41}] (* Michael De Vlieger, May 15 2017 *)

A023347 Primes which remain prime through 5 iterations of function f(x) = 8x + 1.

Original entry on oeis.org

831167, 1154567, 2502767, 3019787, 3675197, 5056577, 6352487, 14519177, 26724377, 43003577, 47378927, 47695607, 56406197, 86332457, 86611757, 99568757, 121967987, 126435527, 127990997, 128149127, 128975057, 145281557, 155715407

Offset: 1

David W. Wilson

nonn

			First chain is {831167, 6649337, 53194697, 425557577, 3404460617, 27235684937};
If p is congruent to {1,3,7,9} mod 10, then consecutive iterates are congruent to {9,5,7,3}, {3,1,7,5}, {5,9,7,1} respectively; so only 10k+7 may remain prime through five iterations, as sequence demonstrates nicely. - _Labos Elemer_, Jul 23 2003

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A085956, A086361, A086362, A023330, A059766, A023287, A000668, A076481, A086127.

Subsequence of A005123, A023228, A023260, A023291, and A023319.

k=0; m=8; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; s4=m*s3+1; s5=m*s4+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5], k=k+1; Print[s]], {n, 1, 1000000}]
it5Q[n_]:=AllTrue[Rest[NestList[8#+1&,n,5]],PrimeQ]; Select[Prime[Range[ 9*10^6]],it5Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 12 2014 *)

{p, 8p+1, 64p+9, 512p+73, 4096p+585, 32768p+4681} are all primes, where the initial p is prime.

a(n) == 197 (mod 210). - John Cerkan, Nov 04 2016

A086358 Digital root of n!.

Original entry on oeis.org

1, 1, 2, 6, 6, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Labos Elemer, Jul 21 2003

a(n) = 9 for n >= 6.

			n = 5, 5 != 120, iteration list = {120,3}, a(5) = 3.

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

Cf. A000142, A086353-A086361, A007953, A010888, A038194, A029898, A004152.

sud[x_] := Apply[Plus, DeleteCases[IntegerDigits[x], 0]]; Table[FixedPoint[sud, w!], {w, 1, 87}]

a(n) = A010888(n!) = fixed-point of A007953(n!). It equals n! modulo(9); at r = 0 use 9.

G.f.: (1 + x^2 + 4*x^3 - 3*x^5 + 6*x^6)/(1 - x). - Stefano Spezia, Jan 26 2023

a(0) = 1 prepended by Alois P. Heinz, Dec 05 2018

A086360 The n-th primorial number reduced modulo 9.

Original entry on oeis.org

1, 2, 6, 3, 3, 6, 6, 3, 3, 6, 3, 3, 3, 6, 6, 3, 6, 3, 3, 3, 6, 6, 6, 3, 6, 6, 3, 3, 6, 6, 3, 3, 6, 3, 3, 6, 6, 6, 6, 3, 6, 3, 3, 6, 6, 3, 3, 3, 3, 6, 6, 3, 6, 6, 3, 6, 3, 6, 6, 6, 3, 3, 6, 6, 3, 3, 6, 6, 6, 3, 3, 6, 3, 3, 3, 3, 6, 3, 3, 6, 6, 3, 3, 6, 6, 6, 3, 6

Offset: 0

Labos Elemer, Jul 21 2003

a(n) is the fixed point reached by decimal-digit-sum-function (A007953), when starting the iteration from the value of the n-th primorial, A002110(n). - The (edited) original definition of the sequence, which is equal to a simple definition a(n) = A002110(n) mod 9, because taking the decimal digit sum preserves congruence modulo 9. - Antti Karttunen, Nov 14 2024

Only a(0)=1 and a(1)=2; each subsequent term is either a 3 or a 6.

			For n=7, 7th primorial = 510510, list of iterated digit sums is {510510,12,3}, thus a(7)=3.

Antti Karttunen, Table of n, a(n) for n = 0..19683 (terms 1..10000 from Nathaniel Johnston)
Index entries for sequences related to primorial numbers

Cf. A002110, A007953, A010878, A010888, A029898, A038194, A086353-A086361.

Cf. also A377876, A377877.

A086360 := proc(n) option remember: if(n=1)then return 2:fi: return ithprime(n)*procname(n-1) mod 9: end: seq(A086360(n), n=1..100); # Nathaniel Johnston, May 04 2011

sud[x_] := Apply[Plus, DeleteCases[IntegerDigits[x], 0]] q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] Table[FixedPoint[sud, q[w]], {w, 1, 128}]

up_to = 19683;
A086360list(up_to_n) = { my(m=9, v=vector(1+up_to_n), pr=1); v[1] = 1; for(n=1, up_to_n, pr = (pr*prime(n))%m; v[1+n] = pr); (v); };
v086360 = A086360list(up_to);
A086360(n) = v086360[1+n]; \\ Antti Karttunen, Nov 14 2024

a(n) = A010878(A002110(n)) = A002110(n) mod 9.

a(n) = A010888(A002110(n)).

Term a(0)=1 prepended, old definition moved to comments and replaced with one of the formulas, keyword:base removed because not really base-dependent - Antti Karttunen, Nov 14 2024

A086354 Fixed point if (nonzero-digit product)-function at initial value 2^n is iterated.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 21 2003

			n=20, 2^20=1048576, iteration list={1048576,6720,84,32,6}, so a(20)=6.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A051801, A051802, A086353-A086361, A029898, A010888.

A051801 := proc(n) local d,j: d:=convert(n,base,10): return mul(`if`(d[j]=0,1,d[j]), j=1..nops(d)): end: A086354 := proc(n) local m: m:=2^n: while(length(m)>1)do m:=A051801(m): od: return m: end: seq(A086354(n),n=1..100); # Nathaniel Johnston, May 04 2011

prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]] Table[FixedPoint[prd, 2^w], {w, 1, 128}]

a(n) = A051802(2^n) = fixed point of A051801(2^n).

A086127 Numbers k such that k remains prime after five iteration of function f(j) = 14*f(j)+1, starting at f(1) = prime.

Original entry on oeis.org

4889, 18059, 62639, 225527, 557093, 604973, 700703, 804077, 806903, 837077, 1341203, 1363403, 1932197, 2004269, 2062703, 2284637, 2797463, 3157379, 3493103, 3746399, 3995687, 4155413, 4227893, 4493297, 5534939, 5708603

Offset: 1

Labos Elemer, Jul 23 2003

nonn

{p, 14p+1, 196p+15, 2744p+211, 38416p+2955, 537824p+41371} are all primes, where p is prime.

			First chain is: {4889,68447,958259,13415627,187818779,2629462907}.
10th chain is {837077,11719079,164067107,2296939499,32157152987,450200141819}.

Cf. A085956, A086361, A086362, A023330, A059766, A023287, A000668, A076481.

k=0; m=14; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; s4=m*s3+1; s5=m*s4+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5], k=k+1; Print[s]], {n, 1, 1000000}]
Select[Range[6000000],And@@PrimeQ[NestList[14#+1&,#,5]]&] (* Harvey P. Dale, Sep 17 2012 *)

A086355 Fixed point if [nonzero-digit product]-function at initial-value=prime(n) is iterated.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 21 2003

			n=100, prime(100)=541, iteration list={541,20,2}, a(100)=2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A051801, A051802, A086353-A086361, A029898, A010888, A038194.

prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]]; Table[FixedPoint[prd, Prime[w]], {w, 1, 128}]

a(n) = A051802(A000040(n)) = fixed-point of A051801(n-th prime).

A086356 Fixed point if [nonzero-digit product]-function at initial-value=C[2n,n]=central binomial coefficient is iterated.

Original entry on oeis.org

2, 6, 2, 7, 2, 4, 4, 2, 2, 6, 6, 6, 8, 6, 5, 8, 8, 4, 8, 2, 9, 8, 6, 8, 6, 2, 8, 8, 2, 8, 6, 2, 6, 6, 8, 2, 6, 6, 6, 8, 9, 2, 2, 8, 2, 8, 2, 8, 6, 4, 2, 2, 8, 8, 2, 8, 6, 8, 2, 8, 6, 8, 9, 6, 6, 2, 6, 2, 2, 2, 8, 6, 8, 6, 8, 2, 8, 8, 8, 8, 8, 8, 6, 2, 6, 2, 8, 6, 8, 8, 8, 8, 8, 6, 8, 8, 6, 8, 2, 8, 2, 8, 6, 8, 8

Offset: 1

Labos Elemer, Jul 21 2003

			n=10, C[20,10]=184756, iteration list={184756,7560,210,2},
a(100)=2.

Cf. A051801, A051802, A086353-A086361, A029898, A010888, A038194, A000984.

prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]] Table[FixedPoint[prd, Binomial[2*n, n]], {w, 1, 128}]

a(n)=A051802[A000984(n)]=fixed-point of A051801[C(2n, n)]

cp's OEIS Frontend

A086353 Fixed point if nonzero-digit product of n! is iterated.

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A023287 Primes that remain prime through 3 iterations of function f(x) = 6x + 1.

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A086362 a(n) = A085956(3n+1).

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A023347 Primes which remain prime through 5 iterations of function f(x) = 8x + 1.

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A086358 Digital root of n!.

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A086360 The n-th primorial number reduced modulo 9.

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A086354 Fixed point if (nonzero-digit product)-function at initial value 2^n is iterated.

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