René-Louis Clerc - cp's OEIS frontend

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.

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)

A375344 First term p1 of octuplets of consecutive prime numbers pi with given successive gaps pi-p1, i=2, ...,8 (6, 8, 18, 24, 30, 36, 38).

Original entry on oeis.org

233, 2721413, 154670903, 200559053, 232777673, 273788363, 299267663, 459117353, 527326403, 1015923113, 1563572243, 1688692763, 2426018723, 2918492243, 3743134523, 4445599853, 4458163943, 4697619593, 5493835013, 5546977823, 5930389313, 6131660663, 6470661143, 7598587943

Offset: 1

René-Louis Clerc, Aug 12 2024

nonn

The choice of successive gaps (6, 8, 18, 24, 30, 36, 38) is such that the sum of the eight prime numbers beginning with 233 is 2024. The next year being the sum of analogous octuplet is 21771464 (21772nd millenium).

			233, 239, 241, 251, 257, 263, 269, 271 (sum = 2024).
2721413, 2721419, 2721421, 2721431, 2721437, 2721443, 2721449, 2721451 (sum = 21771464).

René-Louis Clerc and Jean-Baptiste Hiriart-Urruty, The octuplet of the year 2024 and its relatives, pp. 1-10, hal-04666530, 2024.

Cf. A000040, A001223, A002476, A151800, A000720, A065706, A022011.

uplet(p)= {n=0;for(i=p, p+38, if(isprime(i), n+=1));n}
octo(m)={for(p=3,p=10^m,if(isprime(p) && isprime(p+6) && isprime(p+8) && isprime(p+18) && isprime(p+24) && isprime(p+30) && isprime(p+36) && isprime(p+38) && uplet(p)==8,print1(p,", ")))}
listocto(p1)=print1(p1,", ", p1+6,", ", p1+8,", ", p1+18,", ", p1+24,", ", p1+30,", ", p1+36", ", p1+38)

A375343 Numbers which are the sixth powers of their digit sum.

Original entry on oeis.org

0, 1, 34012224, 8303765625, 24794911296, 68719476736

Offset: 1

René-Louis Clerc, Aug 12 2024

Solutions can have no more than 13 digits, since (13*9)^6 < 10^13.

			68719476736 = (6+8+7+1+9+4+7+6+7+3+6)^6 = 64^6.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, 2023.

Cf. A001014, A007953, A061209, A061210, A254000, A061211.

Cf. A055577, A152147.

for (k=0, sqrtnint(10^13,6), if (k^6 == sumdigits(k^6)^6, print1(k^6, ", ")); )

{ k : k = A007953(k)^6}.

a(n) = A055577(n)^6. - Alois P. Heinz, Aug 24 2024

A371338 Numbers k>0 such that k = |(product of nonzero digits of k^2) - (sum of digits of k^2)|.

Original entry on oeis.org

161, 198, 1701, 604755, 629810, 4354506, 100018736, 411505847, 14869757951891, 2239397044538572646, 40766979086355529727820, 6289762487609138872319999999757

Offset: 1

René-Louis Clerc, Mar 19 2024

Most often P-S is strictly positive but to always have an application of N* in N* we prefer to use |P-S| (cf. Clerc).

			1701^2 = 2893401, |(2*8*9*3*4*1) - (2+8+9+3+4+1)| = 1728 - 27 = 1701.

René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, pp. 1-15, 2024.

Cf. A038364, A061763.

SmP(k,r)=my(d=select(x->(x>0),digits(k^r))); abs(vecsum(d)- vecprod(d)) == k;
 resuSmP(p,r)={for(k=1,10^p,if(SmP(k,r)==1, print1(k,";")))}

a(9)-a(12) from Chai Wah Wu, Apr 20 2024

cp's OEIS Frontend

User: René-Louis Clerc

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

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

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PARI

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

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Maple

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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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PARI

A375344 First term p1 of octuplets of consecutive prime numbers pi with given successive gaps pi-p1, i=2, ...,8 (6, 8, 18, 24, 30, 36, 38).

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