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A384793 a(n) is the start of the first occurrence of exactly n consecutive zeroless primes (A038618).

461717, 162119, 75431, 81421, 19661, 5923, 4813, 1319, 2917, 1117, 1721, 521, 911, 613, 311, 11519, 25411, 7321, 7717, 8819, 9413, 5519, 9613, 2311, 2, 41213, 16319, 1423, 21121, 8219, 162221, 71233, 113, 68521, 148627, 192611, 86531, 48413, 269219, 13313, 275521, 11113, 111521

Offset: 1

Hugo Pfoertner, based on an idea by René-Louis Clerc, Jun 20 2025

			a(25) = 2 because the 25 primes 2, 3, ..., 97 don't have a zero in their decimal representation, terminated by 101.
a(1) = 461717 because it is the smallest zeroless prime, whose nearest lower and upper prime neighbors 461707 and 461801 both have at least one zero in their decimal representation.

Hugo Pfoertner, Table of n, a(n) for n = 1..95
René-Louis Clerc, Nombres premiers primaires et nombres premiers secondaires , 2025.
Hugo Pfoertner, Table of terms up to n=110, including the unknown terms, Jun 28 2025.

Cf. A038618, A052382, A384794.

A384306 Primes whose sum of digits in both base 8 and base 10 are recursively prime down to 2, 3, 5, or 7.

Original entry on oeis.org

2, 3, 5, 7, 131, 311, 887, 1013, 1949, 2399, 2621, 2957, 3251, 3323, 3701, 4289, 4919, 4973, 5099, 5101, 5477, 5927, 5981, 6359, 6599, 6779, 6863, 8069, 8447, 8573, 8627, 8669, 8951, 9677, 10141, 10181, 10211, 10589, 10631, 11399, 11597, 12101, 12479, 12659, 12983

Offset: 1

Jean-Louis Lascoux, May 25 2025

A prime p belongs to this sequence if in both bases 8 and 10 the iterative digit-sum process yields only prime values down to one of {2, 3, 5, 7}.

			a(5) = 131:
In base 8: 131 = 203_8 -> 2+0+3 = 5 -> 5 is prime -> ends in 5.
In base 10: 1+3+1 = 5 -> 5 is prime -> ends in 5.
All intermediate values for both bases are primes, and the last values are in {2,3,5,7}.
a(6) = 887:
In base 8: 887 = 1567_8 -> 1+5+6+7 = 19 -> 19 is prime -> 19 = 23_8 -> 2+3 = 5-> 5 is prime -> ends in 5.
In base 10: 8+8+7 = 23 -> 23 is prime -> 2+3 = 5 -> 5 is prime -> ends in 5.
All intermediate values for both bases are primes, and the last values are in {2,3,5,7}.

Alois P. Heinz, Table of n, a(n) for n = 1..11089 (first 1635 terms from Jean-Louis Lascoux)

Subsequence of A070027.

Cf. A000040, A007953.

q:= (n, k)-> isprime(n) and (n q(x, 8) and q(x, 10), [$1..20000])[];  # Alois P. Heinz, May 27 2025

isokb(p,b) = while(1, my(s=sumdigits(p, b)); if (! isprime(s), return(0)); if (sMichel Marcus, May 27 2025

from gmpy2 import digits, is_prime
def rp(n, b): return is_prime(n) and (n < b or rp(sum(map(int, digits(n, b))), b))
def ok(n): return rp(n, 10) and rp(n, 8)
print([k for k in range(2, 13000) if ok(k)]) # Michael S. Branicky, May 27 2025

A383919 Primes made up of 0's and seven 1's only.

Original entry on oeis.org

11110111, 11111101, 101101111, 101111011, 110111011, 111010111, 1001110111, 1010011111, 1011110011, 1100101111, 1101010111, 1101110011, 1110011101, 1110110011, 1111100101, 1111110001, 10010110111, 10011101011, 10011110101, 10100111101, 10111001011, 10111110001, 11001011101

Offset: 1

René-Louis Clerc, May 15 2025

Expression of the primes that are 0-successors of the preprime 1111111 (= 239*4649); they constitute the infinite set of secondary primes with seven 1's and zeros denoted {1111111} (Definitions 1, 2, 3, 4 of Clerc).

Alois P. Heinz, Table of n, a(n) for n = 1..12277
René-Louis Clerc, Nombres premiers primaires et nombres premiers secondaires, 2025.

Intersection of A020449 and A062337.

Cf. A157711, A038447, A020449.

list(M) = for(i=3, M, for(j=2, i-1, for(k=1, j-1, for(r=1, k-1, for(l=1, r-1, for(m=1, l-1, my(p=10^i+10^j+10^k+10^r+10^l+10^m+1); isprime(p) && print1(p, ", ")))))))

from itertools import count, islice
from sympy import isprime
def A383919_gen(): # generator of terms
    for a in count(6):
        for b in range(5,a):
            for c in range(4,b):
                for d in range(3,c):
                    for e in range(2,d):
                        for f in range(1,e):
                            if isprime(p:=10**a+10**b+10**c+10**d+10**e+10**f|1):
                                yield(p)
A383919_list = list(islice(A383919_gen(),23)) # Chai Wah Wu, May 28 2025

A383918 Primes made up of 0's and five 1's only.

Original entry on oeis.org

101111, 10011101, 10101101, 10110011, 10111001, 11000111, 11100101, 100100111, 100111001, 101001011, 101100011, 110010101, 110101001, 111000101, 111001001, 1000011011, 1000110101, 1001000111, 1001001011, 1001010011, 1010000111, 1010001101, 1010010011, 1010100011, 1010110001

Offset: 1

René-Louis Clerc, May 15 2025

Expression of the primes that are 0-successors of the preprime 11111 (= 41*271); they constitute the infinite set of secondary primes with five 1's and zeros denoted {11111} (Definitions 1, 2, 3, 4 of Clerc).

Robert Israel, Table of n, a(n) for n = 1..10000
René-Louis Clerc, Nombres premiers primaires et nombres premiers secondaires, 2025.

Cf. A157711, A038447, A020449.

f:= proc(n) local R,c,i;
 sort(select(isprime, [seq(1+10^(n-1) + add(10^i,i=c), c=combinat:-choose(n-2,3))]))
end proc:
map(op,[seq(f(i),i=6..10)]); # Robert Israel, May 29 2025

list(M) = for(i=3, M, for(j=2, i-1, for(k=1, j-1, for(r=1, k-1, my(p=10^i+10^j+10^k+10^r+1); isprime(p) && print1(p, ", ")))))

from itertools import count, islice
from sympy import isprime
def A383918_gen(): # generator of terms
    for a in count(4):
        for b in range(3,a):
            for c in range(2,b):
                for d in range(1,c):
                    if isprime(p:=10**a+10**b+10**c+10**d|1):
                        yield(p)
A383918_list = list(islice(A383918_gen(),30)) # Chai Wah Wu, May 29 2025

A381257 Numbers k such that 6*k+1 divides 6^k+1.

Original entry on oeis.org

0, 1, 6, 30, 58, 70, 73, 90, 101, 105, 121, 125, 146, 153, 166, 170, 181, 182, 185, 210, 233, 241, 242, 266, 282, 290, 322, 373, 381, 385, 390, 397, 441, 445, 446, 450, 453, 530, 557, 562, 585, 593, 601, 602, 605, 606, 621, 646, 653, 670, 685, 710, 726, 805, 810, 817, 833, 837, 853, 866

Offset: 1

René-Louis Clerc, Feb 18 2025

nonn

The numbers are called Curzon numbers by Tattersall (p. 85, exercise 43).

			6*30+1 = 181 divides 6^30+1 = 221073919720733357899777.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Second Edition, Cambridge University Press, 2005, p. 85.

Giovanni Resta,Curzon numbers , Numbers Aplenty.

Cf. A224486, A222948, A230076, A381256, A381258.

Select[Range[0, 866], PowerMod[6, #, 6#+1]==6#&]  (* James C. McMahon, Apr 02 2025 *)

PARI
```
isok(n) = my(m=6*n+1); Mod(6, m)^n==-1
```

A381256 Numbers k such that 5*k+1 divides 5^k+1.

Original entry on oeis.org

0, 1, 625, 57057, 7748433, 30850281, 111494625, 393423745, 499088601, 519341361, 1051107705, 1329416385, 1616038425, 2215448001, 2433936225, 2852972265, 3399207273, 4344683849, 4961725281, 5454760185, 5485530369, 6578054145, 6678031745, 7701979761, 7807302825

Offset: 1

René-Louis Clerc, Feb 18 2025

nonn

The numbers are called Curzon numbers by Tattersall (p. 85, exercise 43).

			5*625+1 = 3126 divides 5^625+1.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Second Edition, Cambridge University Press, 2005, p. 85.

Giovanni Resta, Curzon numbers, Numbers Aplenty.

Cf. A224486, A222948, A230076, A381257, A381258.

PARI
```
isok(n) = my(m=5*n+1); Mod(5, m)^n==-1
```

A381258 Numbers k such that 7*k+1 divides 7^k+1.

Original entry on oeis.org

0, 1, 135, 5733, 11229, 42705, 50445, 117649, 131365, 168093, 636405, 699825, 1269495, 2528155, 4226175, 6176709, 6502545, 9365265, 9551115, 13227021, 14464485, 14912625, 20859435, 26903605, 28251265, 30589905, 32660901, 37597329, 41506875, 42766465, 55452075, 56192535, 111898605

Offset: 1

René-Louis Clerc, Feb 18 2025

nonn

The numbers are called Curzon numbers by Tattersall (p. 85, exercise 43).

James J. Tattersall, Elementary Number Theory in Nine Chapters, Second Edition, Cambridge University Press, 2005, p. 85.

Giovanni Resta, Curzon numbers, Numbers Aplenty.

Cf. A224486, A222948, A230076, A381256, A381257.

Select[Range[0,10^7],PowerMod[7,#,7#+1]==7#&] (* James C. McMahon, Mar 05 2025 *)

PARI
```
isok(n) = my(m=7*n+1); Mod(7, m)^n==-1
```

A379767 Triangle read by rows: row n lists numbers which are the n-th powers of their digit sum.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 81, 0, 1, 512, 4913, 5832, 17576, 19683, 0, 1, 2401, 234256, 390625, 614656, 1679616, 0, 1, 17210368, 52521875, 60466176, 205962976, 0, 1, 34012224, 8303765625, 24794911296, 68719476736, 0, 1, 612220032, 10460353203, 27512614111, 52523350144, 271818611107, 1174711139837, 2207984167552, 6722988818432

Offset: 1

René-Louis Clerc, Jan 02 2025

Each row begins with 0, 1. Solutions can have no more than R(n) digits, since (R(n)*9)^n < 10^R(n), hence, for each n, there are a finite number of solutions (Property 1 and table 1 of Clerc).

			Triangle begins:
  1 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
  2 | 0, 1, 81;
  3 | 0, 1, 512, 4913, 5832, 17576, 19683;
  4 | 0, 1, 2401, 234256, 390625, 614656, 1679616;
  5 | 0, 1, 17210368, 52521875, 60466176, 205962976;
  6 | 0, 1, 34012224, 8303765625, 24794911296, 68719476736;
  7 | 0, 1, 612220032, 10460353203, 27512614111, 52523350144, 271818611107, 1174711139837, 2207984167552, 6722988818432;
  8 | 0, 1, 20047612231936, 72301961339136, 248155780267521;
  9 | 0, 1, 3904305912313344, 45848500718449031, 150094635296999121;
  ...

René-Louis Clerc, Rows n=1...270 of triangle, flattened
René-Louis Clerc, Nombres de Niven-Harshad égaux à une puissance de la somme de leurs chiffres, 2024.

Rows 3..6 are A061209, A061210, A254000, A375343.
Row lengths are 1 + A046019(n).
Cf. A001014, A007953, A061211 (largest terms), A133509.
Cf. A152147.

R(n) = for(j=2,oo, if((j*9)^n <10^j, return(j)));
row(n) = my(L=List()); for (k=0, sqrtnint(10^R(n),n), if (k^n == sumdigits(k^n)^n, listput(L, k^n))); Vec(L)

A378947 Number of row states in an automaton for the enumeration of the number of fixed polyominoes with bounding box of width n.

Original entry on oeis.org

1, 2, 6, 16, 40, 99, 247, 625, 1605, 4178, 11006, 29292, 78652, 212812, 579672, 1588242, 4374282, 12103404, 33628824, 93786966, 262450878, 736710357, 2073834417, 5853011847, 16558618507, 46949351272, 133390812252, 379708642286, 1082797114046, 3092894319075, 8848275403639

Offset: 0

Louis Marin, Dec 11 2024

The states track the non-crossing partitions of the connected components and whether each side of the bounding rectangle has been reached.

a(n) is an upper bound on the order of the generating function of row n of A292357.

Louis Marin, Counting Polyominoes in a Rectangle b x h, arXiv:2406.16413 [cs.DM], 2024; EPTCS 403, 2024, pp. 145-149.

Cf. A000108, A001006, A069010, A140662, A292357.

a:= proc(n) option remember; `if`(n<3, [1, 2, 6][n+1],
       ((3*n^2+2*n-12)*a(n-1)+(n^2-13*n+15)*a(n-2)
        -3*(n-3)*(n-1)*a(n-3))/((n-2)*(n+3)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 20 2024

a[n_] := a[n] = If[n < 3, {1, 2, 6}[[n+1]],
   ((3*n^2 + 2*n - 12)*a[n-1] + (n^2 - 13*n + 15)*a[n-2]
   - 3*(n-3)*(n-1)*a[n-3])/((n-2)*(n+3))];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 26 2025, after Alois P. Heinz *)

b(n) = (1 + (hammingweight(bitxor(n, n>>1)))) >> 1;
C(n) = binomial(2*n, n)/(n+1);
a(n) = 1 + sum(m=1, 2^n-1, C(b(m)) * 2^((m % 2)==0) * 2^(m<2^(n-1))); \\ Michel Marcus, Dec 12 2024

a(n) = {1 + sum(k=1, (n+1)\2, (binomial(n+1, 2*k)+2*binomial(n,2*k)+binomial(n-1,2*k))*binomial(2*k, k)/(k+1))} \\ Andrew Howroyd, Dec 17 2024

a(n) = 1 + Sum_{m=1..2^n-1} A000108(A069010(m)) * 2^[m=0 mod 2] * 2^[m<2^(n-1)], where [] is the Iverson bracket.
From Andrew Howroyd, Dec 17 2024: (Start)
a(n) = 1 + Sum_{k=1..floor((n+1)/2)} (binomial(n+1, 2*k) + 2*binomial(n,2*k) + binomial(n-1,2*k)) * binomial(2*k, k)/(k+1).
a(n) = A001006(n+1) + 2*A001006(n) + A001006(n-1) - 3 for n > 0. (End)

More terms from Michel Marcus, Dec 12 2024

a(26) onwards from Andrew Howroyd, Dec 17 2024

A377482 Iterated integer log of n: denote A001414(n) by b(n). a(n) = n if b(n) = n. Otherwise, a(n) = b(n) + b(b(n)) + ... + b^k(n), where k is the smallest integer such that b^k(n) is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 11, 11, 7, 11, 7, 13, 20, 19, 19, 17, 19, 19, 20, 17, 13, 23, 20, 17, 34, 20, 11, 29, 17, 31, 17, 34, 19, 19, 17, 37, 38, 35, 11, 41, 19, 43, 34, 11, 42, 47, 11, 34, 19, 40, 17, 53, 11, 35, 13, 35, 31, 59, 19, 61, 67, 13, 19, 37, 35, 67

Offset: 1

Louis-Simon Cyr, Oct 29 2024

Can be understood as an exotic way of measuring how far a number is from being prime, since omitting n = 1, 4, |a(n) - n| = 0 if and only if n is prime. Note that A274718(n) = k - 1 when a(n) = b(n) + b(b(n)) + ... + b^k(n). The scatter plot for n >= 10000 shows intriguing regularities.

			a(24) is computed as follows: 24 = (2^3) * 3, 2 * 3 + 3 = 9. 9 = (3^2), 3 + 3 = 6. 6 = 2 * 3, 2 + 3 = 5. Since 5 is prime, we stop and take the sum: 9 + 6 + 5 = 20.

Cf. A001414, A274718.

from sympy import*
def a(n):
 t=0
 while n not in (1,4) and not isprime(n):
  n=sum(p*e for p,e in factorint(n).items());t+=n
 return t or n

cp's OEIS Frontend

User: Louis

A384793 a(n) is the start of the first occurrence of exactly n consecutive zeroless primes (A038618).

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A384306 Primes whose sum of digits in both base 8 and base 10 are recursively prime down to 2, 3, 5, or 7.

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Maple

PARI

Python

A383919 Primes made up of 0's and seven 1's only.

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PARI

Python

A383918 Primes made up of 0's and five 1's only.

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Maple

PARI

Python

A381257 Numbers k such that 6*k+1 divides 6^k+1.

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Mathematica

PARI

A381256 Numbers k such that 5*k+1 divides 5^k+1.

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PARI

A381258 Numbers k such that 7*k+1 divides 7^k+1.

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Mathematica

PARI

A379767 Triangle read by rows: row n lists numbers which are the n-th powers of their digit sum.

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