J.W.L. (Jan) Eerland - cp's OEIS frontend

A383746 Numbers k such that k divides the sum of the digits of k^(3k).

1, 2, 3, 6, 9, 11, 18, 38, 43, 87, 126, 670, 1098, 2421, 3588, 4201, 5114, 5877, 5922, 6048, 11799, 46119, 46419, 55098, 55945, 77439, 91541, 129624, 153229, 182402

Offset: 1

J.W.L. (Jan) Eerland, May 08 2025

			2 is a term since the sum of digits of 2^(3*2) is 64, which is divisible by 2.
3 is a term since the sum of digits of 3^(3*3) is 19683, which is divisible by 3.
1098 is a term since the sum of digits of 1098^(3*1098) is 45018, which is divisible by 1098.

Cf. A083282, A108859.

Do[If[Mod[Plus @@ IntegerDigits[n^(3*n)], n] == 0, Print[n]], {n, 1, 10000}]

from gmpy2 import digits, mpz
def ok(n): return n and sum(map(mpz, digits(n**(3*n))))%n == 0
print([k for k in range(1100) if ok(k)]) # Michael S. Branicky, May 08 2025

a(21)-a(23) and a(28)-a(30) from Michael S. Branicky, May 08 2025

A382666 Smallest k such that 7^(7^n) - k is prime.

Original entry on oeis.org

2, 2, 6, 512, 3918, 48966

Offset: 0

J.W.L. (Jan) Eerland, Apr 08 2025

This is to 7 as A058220 is to 2, A140331 is to 3 and A364454 is to 6.

a(6) > 10000. - Michael S. Branicky, Apr 15 2025

			a(2) = 6 because 7^(7^2) - 6 = 256923577521058878088611477224235621321601 is prime.

Cf. A058220, A140331, A364452, A364453, A364454.

lst={};Do[Do[p=7^(7^n)-k;If[PrimeQ[p],AppendTo[lst,k];Break[]],{k,2,11!}],{n,7}];lst
Table[k=1;Monitor[Parallelize[While[True,If[PrimeQ[7^(7^n)-k],Break[]];k++];k],k],{n,0,7}]
y[n_] := Module[{x = 7^(7^n)}, x - NextPrime[x, -1]]; Array[y, 7]

a(n) = my(x = 7^(7^n)); x - precprime(x-1);

from sympy import prevprime
def a(n):
    base = 7**(7**n)
    return base - prevprime(base)
# Jakub Buczak, May 04 2025

a(5) from Michael S. Branicky, Apr 14 2025

A382898 Beginning with 13, least prime such that concatenation of first n terms and its digit reversal both are primes.

Original entry on oeis.org

13, 151, 227, 2083, 887, 79, 2963, 1579, 6287, 1321, 6719, 54919, 26699, 8647, 4229, 3919, 102161, 42433, 1667, 192193, 11633, 186343, 47339, 3259, 65963, 14293, 29717, 61297, 28493, 231367, 43793, 145021, 566441, 475903, 92381, 80473, 139967, 882061, 72893, 709279, 6053, 114487, 1179389, 204331, 203351, 139831, 396239, 205327, 501173, 951589

Offset: 1

J.W.L. (Jan) Eerland, Apr 08 2025

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..150

Cf. A113584 (same for 3), A379761 (same for 7), A380227 (same for 11).

Cf. A111382, A111383, A379354, A379355.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
tcat:= proc(a,b)
  a*10^(1+ilog10(b))+b
end proc:
A:= 13: x:= 13:
for i from 1 to 50 do
   p:= 2:
   do
     p:= nextprime(p);
     y:= tcat(x,p);
     if isprime(y) and isprime(rev(y)) then
          A:= A,p;
          x:= y;
          break
     fi;
   od
od:
A; # after Robert Israel in A113584

w={13};Do[k=1;q=Monitor[Parallelize[While[True,If[PrimeQ[FromDigits[Join@@IntegerDigits/@Reverse[IntegerDigits[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]]]]]&&PrimeQ[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]],Break[]];k++];Prime[k]],{i,k}];w=Append[w,q],{i,2,50}];w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    s, r, an = "", "", 13
    while True:
        yield int(an)
        d = digits(an)
        s, r, p, sp = s+d, d[::-1]+r, 3, "3"
        while not is_prime(mpz(s+sp)) or not is_prime(mpz(sp[::-1]+r)):
            p = next_prime(p)
            sp = digits(p)
        an = p
print(list(islice(agen(), 40))) # after Michael S. Branicky in A113584

A380227 Beginning with 11, least prime such that concatenation of first n terms and its digit reversal both are primes.

Original entry on oeis.org

11, 3, 11, 31, 59, 463, 131, 103, 599, 3253, 7649, 439, 12791, 2953, 17321, 16651, 10007, 51787, 4871, 1483, 6857, 15649, 53051, 61441, 84449, 35533, 19913, 39097, 23081, 206527, 44939, 189517, 32369, 106657, 606899, 117703, 222977, 220903, 69779, 12007, 95063, 136471, 43973

Offset: 1

J.W.L. (Jan) Eerland, Jan 17 2025

Cf. A113584 (same for 3), A379761 (same for 7).

Cf. A111382, A111383, A379354, A379355.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
tcat:= proc(a,b)
  a*10^(1+ilog10(b))+b
end proc:
A:= 11: x:= 11:
for i from 1 to 50 do
   p:= 2:
   do
     p:= nextprime(p);
     y:= tcat(x,p);
     if isprime(y) and isprime(rev(y)) then
          A:= A,p;
          x:= y;
          break
     fi;
   od
od:
A; # after Robert Israel in A113584

w={11};Do[k=1;q=Monitor[Parallelize[While[True,If[PrimeQ[FromDigits[Join@@IntegerDigits/@Reverse[IntegerDigits[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]]]]]&&PrimeQ[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]],Break[]];k++];Prime[k]],{i,k}];w=Append[w,q],{i,2,50}];w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    s, r, an = "", "", 11
    while True:
        yield int(an)
        d = digits(an)
        s, r, p, sp = s+d, d[::-1]+r, 3, "3"
        while not is_prime(mpz(s+sp)) or not is_prime(mpz(sp[::-1]+r)):
            p = next_prime(p)
            sp = digits(p)
        an = p
print(list(islice(agen(), 40))) # after Michael S. Branicky in A113584

A380010 Beginning with 7, least prime such that concatenation of the first n terms is prime.

Original entry on oeis.org

7, 3, 3, 3, 31, 23, 13, 3, 167, 13, 137, 3, 73, 383, 499, 431, 13, 101, 61, 47, 67, 101, 13, 83, 1237, 107, 97, 467, 499, 677, 1423, 353, 73, 431, 331, 683, 487, 2141, 3, 1753, 1787, 31, 443, 139, 653, 1327, 17, 919, 173, 2851, 137, 547, 557, 5167, 347, 7867, 839, 19, 179, 19

Offset: 1

J.W.L. (Jan) Eerland, Jan 09 2025

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..1000

Cf. A111382, A111383, A113584, A379354, A379355, A379761, A380011.

w={7};Do[k=1;q=Monitor[Parallelize[While[True,If[PrimeQ[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]],Break[]];k++];Prime[k]],k];w=Append[w,q],{i,2,50}];w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    s, an = "", 7
    while True:
        yield int(an)
        s += digits(an)
        p = 3
        while not is_prime(mpz(s+digits(p))): p = next_prime(p)
        an = p
print(list(islice(agen(), 50))) # after Michael S. Branicky in A379354

A380011 Beginning with 7, least prime such that the reversal concatenation of the first n terms is prime.

Original entry on oeis.org

7, 3, 3, 13, 3, 2, 13, 47, 43, 47, 37, 41, 109, 41, 139, 149, 109, 263, 73, 563, 163, 41, 19, 797, 61, 107, 31, 821, 43, 149, 37, 953, 211, 89, 547, 353, 337, 167, 67, 239, 1009, 449, 97, 23, 349, 41, 31, 911, 61, 929, 229, 797, 331, 191, 463, 107, 463, 809, 2887, 971

Offset: 1

J.W.L. (Jan) Eerland, Jan 09 2025

"Reverse concatenation" here refers to the decimal concatenation R(a(n)) || R(a(n-1)) || ... || R(a(3)) || R(a(2)) || R(a(1)) where R(k) means "reverse digits of k".

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..982

Cf. A111382, A111383, A113584, A379354, A379355, A379761, A380010.

w={7};Do[k=1;q=Monitor[Parallelize[While[True,If[PrimeQ[FromDigits[Join@@IntegerDigits/@Reverse[IntegerDigits[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]]]]],Break[]];k++];Prime[k]],k];w=Append[w,q],{i,2,50}];w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    r, an = "", 7
    while True:
        yield int(an)
        r = digits(an)[::-1] + r
        p = 2
        while not is_prime(mpz(digits(p)[::-1]+r)): p = next_prime(p)
        an = p
print(list(islice(agen(), 50))) # after Michael S. Branicky in A379355

A379761 Beginning with 7, least prime such that concatenation of first n terms and its digit reversal both are primes.

Original entry on oeis.org

7, 3, 3, 31, 389, 1021, 2243, 1831, 5849, 15361, 9887, 3877, 4157, 919, 22637, 14449, 27617, 80221, 5039, 51043, 14009, 126079, 24443, 68311, 49193, 47059, 13049, 253681, 271409, 221227, 138869, 116953, 146297, 21841, 1211549, 322501, 212633, 281791, 216071, 1901749, 38747, 116437

Offset: 1

J.W.L. (Jan) Eerland, Jan 02 2025

			31 is a term because the concatenation of {7,3,3,31} and {13,3,3,7} are respectively 73331 and 13337 which are both prime.
2243 is a term because the concatenation of {7,3,3,31,389,1021,2243} and {3422,1201,983,13,3,3,7} are respectively 7333138910212243 and 3422120198313337 which are both prime.

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..100

Cf. A111382, A111383, A113584, A379354, A379355.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
tcat:= proc(a,b)
  a*10^(1+ilog10(b))+b
end proc:
A:= 7: x:= 7:
for i from 1 to 50 do
   p:= 2:
   do
     p:= nextprime(p);
     y:= tcat(x,p);
     if isprime(y) and isprime(rev(y)) then
          A:= A,p;
          x:= y;
          break
     fi;
   od
od:
A; # after Robert Israel in A113584

w={7};Do[k=1;q=Monitor[Parallelize[While[True,If[PrimeQ[FromDigits[Join@@IntegerDigits/@Reverse[IntegerDigits[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]]]]]&&PrimeQ[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]],Break[]];k++];Prime[k]],{i,k}];w=Append[w,q],{i,2,50}];w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    s, r, an = "", "", 7
    while True:
        yield int(an)
        d = digits(an)
        s, r, p, sp = s+d, d[::-1]+r, 3, "3"
        while not is_prime(mpz(s+sp)) or not is_prime(mpz(sp[::-1]+r)):
            p = next_prime(p)
            sp = digits(p)
        an = p
print(list(islice(agen(), 40))) # Michael S. Branicky, Jan 02 2025

A379355 Beginning with 3, least prime such that the reversal concatenation of first n terms is prime.

Original entry on oeis.org

3, 2, 2, 13, 2, 13, 59, 31, 263, 73, 23, 31, 449, 31, 59, 313, 2, 3, 211, 317, 31, 449, 241, 887, 349, 911, 853, 887, 313, 173, 1777, 179, 967, 503, 331, 113, 163, 359, 1153, 281, 97, 1823, 13, 23, 1657, 269, 223, 3623, 2017, 233, 61, 1361, 367, 1031, 79, 389, 577, 2963, 1741, 59, 13, 1439, 463, 797

Offset: 1

J.W.L. (Jan) Eerland, Dec 21 2024

"Reverse concatenation" here seems to refer to the decimal concatenation R(a(n)) || R(a(n-1)) || ... || R(a(3)) || R(a(2)) || R(a(1)) where R(k) means "reverse digits of k". - N. J. A. Sloane, Jan 03 2025

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Cf. A111382, A111383, A113584, A379354.

The primes produced are in A379782.

w = {3};
Do[k = 1;
  q = Monitor[
    Parallelize[
     While[True,
      If[PrimeQ[
         FromDigits[
          Join @@ IntegerDigits /@
            Reverse[
             IntegerDigits[
              FromDigits[
               Join @@ IntegerDigits /@ Append[w, Prime[k]]]]]]], Break[]]; k++];
     Prime[k]], k];
  w = Append[w, q], {i, 2, 57}];
w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    r, an = "", 3
    while True:
        yield int(an)
        r = digits(an)[::-1] + r
        p = 2
        while not is_prime(mpz(digits(p)[::-1]+r)): p = next_prime(p)
        an = p
print(list(islice(agen(), 57))) # Michael S. Branicky, Dec 21 2024

A379354 Beginning with 3, least prime such that concatenation of first n terms is prime.

Original entry on oeis.org

3, 7, 3, 3, 7, 29, 43, 11, 61, 71, 19, 191, 43, 53, 7, 239, 31, 173, 43, 137, 79, 53, 13, 557, 619, 47, 271, 797, 463, 83, 211, 467, 229, 131, 199, 359, 1249, 887, 853, 641, 109, 257, 1153, 1031, 613, 953, 607, 641, 499, 359, 1297, 1031, 2137, 401, 283, 29, 1321, 1499, 547, 83, 397, 2153, 1759, 1277

Offset: 1

J.W.L. (Jan) Eerland, Dec 21 2024

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Cf. A111382, A111383, A113584, A379355.

w = {3};
Do[k = 1;
  q = Monitor[
    Parallelize[
     While[True,
      If[PrimeQ[FromDigits[
          Join @@ IntegerDigits /@ Append[w, Prime[k]]]], Break[]]; k++];
     Prime[k]], k];
  w = Append[w, q], {i, 2, 57}];
w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    s, an = "", 3
    while True:
        yield int(an)
        s += digits(an)
        p = 3
        while not is_prime(mpz(s+digits(p))): p = next_prime(p)
        an = p
print(list(islice(agen(), 57))) # Michael S. Branicky, Dec 21 2024

A378512 Numbers k such that 6^sigma(k) - k is a prime.

Original entry on oeis.org

1, 7, 13, 77, 395, 2867, 3959, 5023

Offset: 1

J.W.L. (Jan) Eerland, Nov 29 2024

a(9) > 10^5. - Michael S. Branicky, Dec 01 2024

			7 is in the sequence because 6^sigma(7) - 7 = 6^8 - 7 = 1679609 is prime.

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851, A368651, A367460, A377927, A377786.

[n: n in[1..10000] | IsPrime((6^SumOfDivisors(n)) - n)];

a[n_] := Select[Range@ n, PrimeQ[6^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[6^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

isok(k) = ispseudoprime(6^sigma(k) - k); \\ Michel Marcus, Dec 09 2024

cp's OEIS Frontend

User: J.W.L. (Jan) Eerland

A383746 Numbers k such that k divides the sum of the digits of k^(3k).

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A382666 Smallest k such that 7^(7^n) - k is prime.

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A382898 Beginning with 13, least prime such that concatenation of first n terms and its digit reversal both are primes.

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A380227 Beginning with 11, least prime such that concatenation of first n terms and its digit reversal both are primes.

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A380010 Beginning with 7, least prime such that concatenation of the first n terms is prime.

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A380011 Beginning with 7, least prime such that the reversal concatenation of the first n terms is prime.

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A379761 Beginning with 7, least prime such that concatenation of first n terms and its digit reversal both are primes.

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Maple

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A379355 Beginning with 3, least prime such that the reversal concatenation of first n terms is prime.

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