A176554 -id:A176554 - cp's OEIS frontend

A037278 Replace n with concatenation of its divisors.

1, 12, 13, 124, 15, 1236, 17, 1248, 139, 12510, 111, 1234612, 113, 12714, 13515, 124816, 117, 1236918, 119, 12451020, 13721, 121122, 123, 1234681224, 1525, 121326, 13927, 12471428, 129, 12356101530, 131, 12481632, 131133, 121734, 15735, 123469121836, 137

Offset: 1

N. J. A. Sloane

a(n) is the union of A176555(n) for n >= 1 and A176556(n) for n >= 2. See A176553 (numbers m such that concatenations of divisors of m are noncomposites) and A176554 (numbers m such that concatenations of divisors of m are nonprimes). [Jaroslav Krizek, Apr 21 2010]

a(n) is the concatenation of n-th row of the triangle in A027750.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A027750, A037283, A037277, A106708.

a037278 = read . concatMap show . a027750_row :: Integer -> Integer
-- Reinhard Zumkeller, Jul 13 2013, May 01 2012, Aug 07 2011

m=1;
for u=1:34 div=divisors(u); conc=str2num(strrep(num2str(div), ' ', ''));
   sol(m)=conc; m=m+1;
end
sol % Marius A. Burtea, Jun 01 2019

k:=1; sol:=[];
for u in [1..34] do D:=Divisors(u); conc:=D[1];
    for u1 in [2..#D] do a:=#Intseq(conc); a1:=#Intseq(D[u1]); conc:=10^a1*conc+D[u1];end for;
     sol[u]:=conc; k:=k+1;
end for;
sol; // Marius A. Burtea, Jun 01 2019

a[n_] := ToExpression[ StringJoin[ ToString /@ Divisors[n] ] ]; Table[ a[n], {n, 1, 34}] (* Jean-François Alcover, Dec 01 2011 *)
FromDigits[Flatten[IntegerDigits/@Divisors[#]]]&/@Range[40] (* Harvey P. Dale, Nov 09 2012 *)

a(n) = my(s=""); fordiv(n, d, s = concat(s, Str(d))); eval(s); \\ Michel Marcus, Jun 01 2019 and Sep 22 2022

from sympy import divisors
def a(n): return int("".join(str(d) for d in divisors(n)))
print([a(n) for n in range(1, 35)]) # Michael S. Branicky, Dec 31 2020

A134681(n) = A055642(a(n)). - Reinhard Zumkeller, Nov 06 2007

More terms from Erich Friedman

A176553 Numbers m such that concatenations of divisors of m are noncomposites.

Original entry on oeis.org

1, 3, 7, 9, 13, 21, 31, 37, 67, 73, 79, 97, 103, 109, 121, 151, 163, 181, 183, 193, 219, 223, 229, 237, 277, 283, 307, 363, 367, 373, 381, 409, 433, 439, 471, 487, 489, 499, 511, 523, 571, 601, 603, 607, 613, 619, 657, 669, 709, 733, 787, 811, 817, 819, 823, 841, 867

Offset: 1

Jaroslav Krizek, Apr 20 2010

Do all primes p > 5 have a multiple in this sequence? This holds at least for p < 10^4. - Charles R Greathouse IV, Sep 23 2016
Conjecture: this sequence is a subsequence of A003136 (Loeschian numbers). - Davide Rotondo, Jan 02 2022
If m is not in A003136, there is a prime p == 2 (mod 3) such that the exponent of p in the factorization of m is odd, then we have 3 | 1+p | 1+p+p^2+...+p^(2*r-1) | sigma(m), sigma = A000203 is the sum of divisors, so the concatenation of the divisors of m is also divisible by 3. - Jianing Song, Aug 22 2022

			a(6) = 21: the divisors of 21 are 1,3,7,21, and their concatenation 13721 is noncomposite.

Bill McEachen, Table of n, a(n) for n = 1..10000

Cf. A037278, A176554, A176555.

Subsequence of A045572.

Select[Range[10^3], ! CompositeQ@ FromDigits@ Flatten@ IntegerDigits@ Divisors@ # &] (* Michael De Vlieger, Sep 23 2016 *)

is(n)=my(d=divisors(n)); d[1]="1"; isprime(eval(concat(d))) || n==1 \\ Charles R Greathouse IV, Sep 23 2016

from sympy import divisors, isprime
def ok(m): return m==1 or isprime(int("".join(str(d) for d in divisors(m))))
print([m for m in range(1, 900) if ok(m)]) # Michael S. Branicky, Feb 05 2022

Edited and extended by Charles R Greathouse IV, Apr 30 2010

Data corrected by Bill McEachen, Nov 03 2021

A176556 Nonprime concatenations of divisors of some k, ordered by k.

Original entry on oeis.org

1, 12, 124, 15, 1236, 1248, 12510, 111, 1234612, 12714, 13515, 124816, 117, 1236918, 119, 12451020, 121122, 123, 1234681224, 1525, 121326, 13927, 12471428, 129, 12356101530, 12481632, 131133, 121734, 15735, 123469121836, 121938, 131339

Offset: 1

Jaroslav Krizek, Apr 20 2010

Intersection of A037278 and A018252.

			For n = 6; a(6) = 1248 because A176554(6) = 8 and divisors of 8 are 1, 2, 4, 8; concatenation of divisors A037278(8) = 1248 is nonprime number.

a(n) = A037278(A176554(n)).

Extended and edited by Charles R Greathouse IV, Apr 27 2010

A176590 Numbers k such that both concatenation of divisors of k and reverse concatenation of divisors of k are nonprime.

Original entry on oeis.org

1, 2, 5, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88

Offset: 1

Jaroslav Krizek, Apr 21 2010

Numbers k such that both A037278(k) and A176558(k) are nonprime.

			10 is a term; divisors of 10: 1, 2, 5, 10; both concatenation of divisors 12510 and reverse concatenation of divisors 10521 are nonprime numbers.
Sequence of corresponding values of concatenations of divisors of a(n): 1, 12, 15, 1236, 1248, 12510, 111, ...
Sequence of corresponding values of reverse concatenations of divisors of a(n): 1, 21, 51, 6321, 8421, 10521, 111, ...

Cf. A037278, A176558, A176554, A175354, A176589.

Intersection of A176554 and A175354. - Jason Yuen, Feb 08 2025

Extended and revised by Charles R Greathouse IV, Apr 24 2010

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A037278 Replace n with concatenation of its divisors.

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A176553 Numbers m such that concatenations of divisors of m are noncomposites.

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A176556 Nonprime concatenations of divisors of some k, ordered by k.

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Examples

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Extensions

A176590 Numbers k such that both concatenation of divisors of k and reverse concatenation of divisors of k are nonprime.

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