A037276 -id:A037276 - cp's OEIS frontend

A349705 Numbers k such that the concatenation in increasing order of their prime factors, with multiplicity, is congruent to 1 (mod k).

36, 39, 66, 1435, 5714, 6410, 13861, 22564, 27346, 33137, 45542, 79260, 171860, 268218, 442068, 486127, 675423, 2287527, 3710027, 9610766, 14318290, 26293568, 29361702, 49703324, 227358366, 433100023, 442960845, 479174118, 1221238938, 1243718114, 4053362596, 8620689655

Offset: 1

J. M. Bergot and Robert Israel, Nov 25 2021

			a(3) = 66 is a term because the concatenation of its prime factors is 2311 and 2311 == 1 (mod 66).

Martin Ehrenstein, Table of n, a(n) for n = 1..39

Cf. A037276, A259047.

filter:= proc(n) local L,t;
  lcat(map(t -> t[1]$t[2], sort( ifactors(n)[2], (a,b) -> a[1] < b[1]))) mod n = 1;
end proc:
select(filter, [$1..10^7]);

upto=10^5;a={};Do[If[Mod[FromDigits[Flatten[Map[IntegerDigits[ConstantArray[First[#],Last[#]]]&,FactorInteger[k]]]],k]==1,AppendTo[a,k]],{k,upto}];a (* Paolo Xausa, Nov 26 2021 *)

from sympy import factorint
def ok(k): return int("".join(map(str, factorint(k, multiple=True))))%k == 1
print([k for k in range(2, 10**5) if ok(k)]) # Michael S. Branicky, Nov 26 2021

from itertools import count, islice
from sympy import factorint
def A349705_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        c = 0
        for d in sorted(factorint(k,multiple=True)):
            c = (c*10**len(str(d)) + d) % k
        if c == 1:
            yield k
A349705_list = list(islice(A349705_gen(),10)) # Chai Wah Wu, Feb 28 2022

a(28)-a(32) from Martin Ehrenstein, Nov 27 2021

A364050 Palindromes that have at least two distinct prime factors and whose prime factors, listed (with multiplicity) in ascending order and concatenated, form a palindrome in base 10.

Original entry on oeis.org

10001, 36763, 1037301, 1226221, 9396939, 12467976421, 14432823441, 93969696939, 119092290911, 1030507050301, 1120237320211, 1225559555221, 1234469644321, 1334459544331, 100330272033001, 101222252222101, 103023070320301, 121363494363121, 134312696213431

Offset: 1

Vitaliy Kaurov, Jul 03 2023

Palindromes p in A024619 such that A037276(p) is a palindrome.

Terms are coprime to 10. - David A. Corneth, Jul 05 2023

			  10001 = 73 * 137
  36763 = 97 * 379
1037301 = 3 * 29 * 11923
1226221 = 1021 * 1201
9396939 = 3 * 101 * 31013

Subsequence of A002113 and A024619. Cf. A037276.

Similar to A364023.

(* generate palindromes with even n *)
poli[n_Integer?EvenQ]:=FromDigits[Join[#,Reverse[#]]]&/@
DeleteCases[Tuples[Range[0,9],n/2],{0..,_}]
(* generate palindromes with odd n *)
poli[n_Integer?OddQ]:=Flatten[Table[FromDigits[Join[#,{k},Reverse[#]]]&/@
DeleteCases[Tuples[Range[0,9],(n-1)/2],{0..,_}],{k,0,9}]]
(* find ascending factor sequence *)
ascendFACTOR[n_Integer]:=
PalindromeQ[StringJoin[ToString/@Flatten[Table[#1,#2]&@@@#]]]&&
Length[#]>1&@FactorInteger[n]
(* example for palindromes of size 7 *)
Parallelize@Select[poli[7],ascendFACTOR]//Sort//AbsoluteTiming

nextpal(n, b) = {my(m=n+1, p = 0); while (m > 0, m = m\b; p++; ); if (n+1 == b^p, p++); n = n\(b^(p\2))+1; m = n; n = n\(b^(p%2)); while (n > 0, m = m*b + n%b; n = n\b; ); m; }
ispal(n) = my(d=digits(n)); Vecrev(d) == d;
g(f) = my(s=""); for (i=1, #f~, for (j=1, f[i,2], s = concat(s, Str(f[i,1])))); eval(s);
isok(k) = my(f=factor(k)); if (#f~>=2, ispal(g(f)));
lista(nn) = {my(k=0); while (k <= nn, if (ispal(k) && isok(k), print1(k, ", ")); k = nextpal(k,10););} \\ Michel Marcus, Jul 11 2023

A038524 Numbers k such that the k-th Fibonacci number is composite and the concatenation of its prime factors (written in base 10 in ascending order) is prime.

Original entry on oeis.org

8, 24, 28, 34, 50, 52, 58, 71, 77, 163, 197, 235, 238, 319, 570, 669, 949, 958, 1211, 1305

Offset: 1

Jeff Burch

			F(24) = 46368 = 2^5 * 3^2 * 7 * 23 is composite and the concatenation 2222233723 is prime, so 24 is in the sequence.

Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A000045, A037276, A038526.

More terms from Jens Kruse Andersen, Jan 05 2003
a(17)-a(18) from Ryan Propper, May 29 2006
a(19)-a(20) from Sean A. Irvine, Jan 16 2021

A038525 Numbers n with property that either the n-th Fibonacci number is prime or the concatenation of its prime factors is prime.

Original entry on oeis.org

3, 4, 5, 7, 8, 11, 13, 17, 23, 24, 28, 29, 34, 43, 47, 50, 52, 58, 71, 77, 83, 131, 137, 163, 197, 235, 238, 319, 359, 431, 433, 449, 509, 569, 570, 571, 669, 949, 958, 1211, 1305

Offset: 1

Jeff Burch

Up to and including a(17)=52, the sequence are the n such that A038526(n) is prime. Extrapolating, one would expect it to continue 83,86,93,95,131,137,139,174,182,... - R. J. Mathar, Oct 12 2007

Cf. A000045, A037276, A038526.

More terms from Thomas Baruchel, Oct 11 2003

More terms from Sean A. Irvine, Jan 16 2021

A342999 a(n) is always followed by the concatenation of a(n)'s distinct prime factors in increasing order. If this concatenation is already in the sequence, a(n+1) is the smallest term not yet present.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 23, 7, 8, 9, 10, 25, 11, 12, 13, 14, 27, 15, 35, 57, 319, 1129, 16, 17, 18, 19, 20, 21, 37, 22, 211, 24, 26, 213, 371, 753, 3251, 28, 29, 30, 235, 547, 31, 32, 33, 311, 34, 217, 731, 1743, 3783, 31397, 36, 38, 219, 373, 39, 313, 40, 41, 42, 237, 379, 43, 44, 45, 46, 223, 47

Offset: 1

Eric Angelini and Carole Dubois, Apr 02 2021

This is a permutation of the positive terms.

			a(6) is not = 5, though the only prime factor of a(5) is precisely 5; but as 5 is already in the sequence we must take a(6) = 6, the smallest term not yet present in the sequence.
a(7) = 23 as the prime factors of a(6) = 6 are 2 and 3, which, concatenated in increasing order, give 23;
a(8) is not = 23, though the only prime factor of a(7) is precisely 23; but as 23 is already in the sequence we must take a(8) = 7, the smallest term not yet present in the sequence; etc.

Index entries for sequences that are permutations of the natural numbers

Cf. A084317 (concatenation of the prime factors of n, in increasing order), A037276 (replace n with the concatenation of its prime factors in increasing order).

a[1]=1;a[n_]:=a[n]=(g=FromDigits@Flatten[IntegerDigits@*First/@FactorInteger@a[n-1]];If[FreeQ[k=Array[a,n-1],g],g,Min@Complement[Range@Max[k+1],k]])
Array[a,100] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

from sympy import primefactors
def aupton(terms):
  alst, aset = [1, 2], {1, 2}
  while len(alst) < terms:
    an = int("".join(map(str, primefactors(alst[-1]))))
    if an in aset:
      an = 1
      while an in aset: an += 1
    alst.append(an); aset.add(an)
  return alst[:terms]
print(aupton(100)) # Michael S. Branicky, Apr 02 2021

cp's OEIS Frontend

A349705 Numbers k such that the concatenation in increasing order of their prime factors, with multiplicity, is congruent to 1 (mod k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Python

Extensions

A364050 Palindromes that have at least two distinct prime factors and whose prime factors, listed (with multiplicity) in ascending order and concatenated, form a palindrome in base 10.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A038524 Numbers k such that the k-th Fibonacci number is composite and the concatenation of its prime factors (written in base 10 in ascending order) is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A038525 Numbers n with property that either the n-th Fibonacci number is prime or the concatenation of its prime factors is prime.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A342999 a(n) is always followed by the concatenation of a(n)'s distinct prime factors in increasing order. If this concatenation is already in the sequence, a(n+1) is the smallest term not yet present.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python