Pietro Tiaraju Giavarina dos Santos

A386929 a(n) is the least base b in {2,...,10} such that the base-b expansion of n, when read as a decimal integer, is prime; a(n) = 0 if no such base exists.

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

Offset: 1

Pietro Tiaraju Giavarina dos Santos, Aug 08 2025

There are infinitely many zeros since if n is a multiple of 2520, then each base-b expansion ends with digit 0.

			a(10) = 3 since 10 in base 3 is "101" and 101 is prime; base 2 is "1010" -> 1010 composite.
a(11) = 4 since base 4 gives "23" -> 23 is prime; base 2 "1011" -> 1011 composite; base 3 "102" -> 102 composite.
a(23) = 2 since base 2 gives "10111" -> 10111 is prime.

Cf. A038537, A052026 (the zeros), A052033 (the tens).

a[n_] := Block[{m}, Do[m = FromDigits[IntegerDigits[n, b], 10]; If[PrimeQ[m], Return[b]], {b, 2, 10}]; 0]

a(n) = for(b=2, 10, if (isprime(fromdigits(digits(n, b))), return(b))); \\ Michel Marcus, Aug 09 2025

a(2520*n) = 0.

A384128 Number of iterations for the circular absolute first-difference on decimal digits to reach a repdigit.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 3, 5, 6, 5, 6

Offset: 1

Pietro Tiaraju Giavarina dos Santos, May 20 2025

a(n) = least t >= 0 such that the t-fold iteration of the circular absolute first-difference applied to the decimal digits of n yields a repdigit. The map sends n with decimal digits d_{k-1}...d_0 to the number whose digits are |d_{k-1}-d_{k-2}|, |d_{k-2}-d_{k-3}|, ..., |d_{0}-d_{k-1}|. Let F(n) = Sum_{i=0..k-1} |floor(n/10^i) mod 10 - floor(n/10^{(i+1) mod k}) mod 10|* 10^i. Then a(n) = min {t >= 0 : F^t(n) has all digits equal}.
a(n)=0 iff all decimal digits of n are equal.
a(n) <= 11 for n < 10000.

			The first value > 1 is a(100) = 3.
a(21) = |2-1| |1-2| = 11 -> repdigit at t = 1, so a(21) = 1.
a(109) = 109 -> 198 -> 817 -> 761 -> 156 -> 415 -> 341 -> 132 -> 211 -> 101 -> 110 -> 11 requires 11 steps, so a(109) = 11.

Cf. A010785.

SingleRepQ[x_Integer] := SameQ @@ IntegerDigits[x]
CAD[x_Integer] := FromDigits@Abs[IntegerDigits[x] - RotateLeft[IntegerDigits[x]]]
A384128[n_Integer] := Module[{x = n, cnt = 0}, While[! SingleRepQ[x], x = CAD[x]; cnt++]; cnt]
Table[A384128[n], {n, 1, 200}]

Data corrected by Sean A. Irvine, Jun 13 2025

A335332 Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 954, based on the 4-celled von Neumann neighborhood starting with a single black cell.

Original entry on oeis.org

1, 11, 81, 699, 5441, 43723, 349265, 2797243, 22368577, 178961099, 1431651409, 11453263547, 91625951553, 733007821515, 5864061944913, 46912496398011, 375299968667969, 3002399752698571, 24019198011524177, 192153584105615035, 1537228672804655425, 12297829382490929867

Offset: 1

Pietro Tiaraju Giavarina dos Santos, Jun 01 2020

Index entries for linear recurrences with constant coefficients, signature (8,4,-32,1,-8,-4,32).

Cf. A334340.

Table[DifferenceRoot[Function[{y, n}, {4872 + 32 y[n] + 28 y[1 + n] + 20 y[2 + n] + 21 y[3 + n] - 11 y[4 + n] - 7 y[5 + n] + y[6 + n] == 0, y[1] == 1, y[2] == 11, y[3] == 81, y[4] == 699, y[5] == 5441, y[6] == 43723, y[7] == 349265, y[8] == 2797243}]][n], {n, 1, 30}]

Vec(x*(1 + 3*x - 11*x^2 + 39*x^3 - 124*x^4 - 12*x^5 + 96*x^6 + 2304*x^7 - 7168*x^8) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 - 8*x)*(1 + x^2)) + O(x^40)) \\ Colin Barker, Jun 11 2020

G.f.: (-1 - 3*x + 11*x^2 - 39*x^3 + 124*x^4 + 12*x^5 - 96*x^6 - 2304*x^7 + 7168*x^8)/(-1 + 8*x + 4*x^2 - 32*x^3 + x^4 - 8*x^5 - 4*x^6 + 32*x^7).

a(n) = 8*a(n-1) + 4*a(n-2) - 32*a(n-3) + a(n-4) - 8*a(n-5) - 4*a(n-6) + 32*a(n-7) for n>9. - Colin Barker, Jun 11 2020

A334244 Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 950, based on the 4-celled von Neumann neighborhood starting with a single black cell.

Original entry on oeis.org

1, 15, 81, 959, 5185, 61391, 331857, 3929023, 21238849, 251457487, 1359286353, 16093279167, 86994326593, 1029969866703, 5567636901969, 65918071468991, 356328761726017, 4218756574015439, 22805040750465105, 270000420736988095, 1459522608029766721, 17280026927167238095

Offset: 1

Pietro Tiaraju Giavarina dos Santos, Apr 20 2020

nonn

a(n) is the decimal representation of the n-th step based on a simple initial condition, when a(1) = 1.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,64,0,1,0,-64).

Cf. A118171, A118173 (similar examples from elementary cellular automata).

Table[((1607 +311*(-1)^n)*8^n -1040 +130*(-1)^n -1008*Sqrt[2]*Cos[(2*n-1)*Pi/4] )/8190, {n, 25}] (* G. C. Greubel, May 29 2020 *)

Vec(x*(1 + 15*x + 17*x^2 - x^3) / ((1 - x)*(1 + x)*(1 - 8*x)*(1 + 8*x)*(1 + x^2)) + O(x^20)) \\ Colin Barker, Jun 10 2020

a(n) = (-1040 + 130*(-1)^n - (504 + 504*i)*(-i)^n - (504 - 504*i)*i^n + 1607*2^(3*n) + 311*(-1)^n*2^(3*n))/8190 where i = sqrt(-1).
G.f.: (1 + 15*x + 17*x^2 - x^3)/(1 - 64*x^2 - x^4 + 64*x^6).
From G. C. Greubel, May 29 2020: (Start)
a(n) = ( (1607 + 311*(-1)^n)*8^n - (1040 - 130*(-1)^n) - 1008*sqrt(2)*cos((2*n-1)*Pi/4) )/8190.
E.g.f.: (959*cosh(8*x) + 648*sinh(8*x) - 455*cosh(x) - 585*sinh(x) - 504*(cos(x) + sin(x)) )/4095.
(End)
a(n) = 64*a(n-2) + a(n-4) - 64*a(n-6) for n>6. - Colin Barker, Jun 10 2020

A334340 Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 434, based on the 4-celled von Neumann neighborhood starting with a single black cell.

Original entry on oeis.org

1, 11, 81, 699, 5441, 43723, 349201, 2796731, 22364481, 178965195, 1431573521, 11453377723, 91624653121, 733009857227, 5864040961041, 46912529804475, 375299632087361, 3002400290556619, 24019192622879761, 192153592724761787, 1537228586572923201, 12297829520450964171, 98382633680004977681

Offset: 1

Pietro Tiaraju Giavarina dos Santos, Apr 23 2020

Index entries for linear recurrences with constant coefficients, signature (8,16,-128,1,-8,-16,128).

Cf. A118171, A118173 (similar examples from elementary cellular automata).

a(n) = (1428 + 7*(-4)^n + 2278*(-1)^n + (1800 + 360*i)*(-i)^n + (1800 - 360*i)*i^n - 3*4^n + 85*8^n)/510 where i = sqrt(-1).

G.f.: (-1 - 3*x + 23*x^2 - 3*x^3 + 40*x^4 + 624*x^5 - 1856*x^6)/(-1 + 8*x + 16*x^2 - 128*x^3 + x^4 - 8*x^5 - 16*x^6 + 128*x^7).

A276805 a(n) = numerator((n^2 + 3*n + 2)/n^3).

Original entry on oeis.org

6, 3, 20, 15, 42, 7, 72, 45, 110, 33, 156, 91, 210, 30, 272, 153, 342, 95, 420, 231, 506, 69, 600, 325, 702, 189, 812, 435, 930, 124, 1056, 561, 1190, 315, 1332, 703, 1482, 195, 1640, 861, 1806, 473, 1980, 1035, 2162, 282, 2352, 1225, 2550, 663, 2756, 1431, 2970, 385

Offset: 1

Pietro Tiaraju Giavarina dos Santos, Sep 17 2016

			a(1) = numerator((n^2 + 3*n + 2)/n^3)  = 1^2+3*1+2/1^3 = 6.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,1).

Cf. A277542.

Table[Numerator[(n^2 + 3*n + 2)/n^3], {n, 1, 100}]

a(n) = numerator((n^2 + 3*n + 2)/n^3); \\ Michel Marcus, Sep 18 2016

Vec(x*(6 +3*x +20*x^2 +15*x^3 +42*x^4 +7*x^5 +72*x^6 +45*x^7 +92*x^8 +24*x^9 +96*x^10 +46*x^11 +84*x^12 +9*x^13 +56*x^14 +18*x^15 +30*x^16 +5*x^17 +12*x^18 +3*x^19 +2*x^20 +x^23) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^3*(1 +x^4)^3) + O(x^100)) \\ Colin Barker, Oct 20 2016

From Colin Barker, Sep 18 2016: (Start)
a(n) = 3*a(n-8)-3*a(n-16)+a(n-24) for n>24.
G.f.: x*(6 +3*x +20*x^2 +15*x^3 +42*x^4 +7*x^5 +72*x^6 +45*x^7 +92*x^8 +24*x^9 +96*x^10 +46*x^11 +84*x^12 +9*x^13 +56*x^14 +18*x^15 +30*x^16 +5*x^17 +12*x^18 +3*x^19 +2*x^20 +x^23) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^3*(1 +x^4)^3).
(End)
a(n) = a(n-8)*(n^2+3*n+2)/(n^2-13*n+42), for n>8. - Gionata Neri, Feb 25 2017

cp's OEIS Frontend

User: Pietro Tiaraju Giavarina dos Santos

A386929 a(n) is the least base b in {2,...,10} such that the base-b expansion of n, when read as a decimal integer, is prime; a(n) = 0 if no such base exists.

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A384128 Number of iterations for the circular absolute first-difference on decimal digits to reach a repdigit.

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A335332 Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 954, based on the 4-celled von Neumann neighborhood starting with a single black cell.

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A334244 Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 950, based on the 4-celled von Neumann neighborhood starting with a single black cell.

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A334340 Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 434, based on the 4-celled von Neumann neighborhood starting with a single black cell.

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A276805 a(n) = numerator((n^2 + 3*n + 2)/n^3).

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