A176553 -id:A176553 - 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

A176554 Numbers n such that concatenations of divisors of n are nonprime.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 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

Offset: 1

Jaroslav Krizek, Apr 20 2010

See A037278(n) = concatenation of divisors of n. See A176556 for corresponding values of concatenations. Complement of A176553(n) for n >= 2.

			a(6) = 8: divisors of 8 are 1,2,4,8 and their concatenation 1248 is nonprime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Select[Range[100],!PrimeQ[FromDigits[Flatten[IntegerDigits/@ Divisors[ #]]]]&] (* Harvey P. Dale, Jul 09 2021 *)

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

A176555 Noncomposite concatenations of divisors of some k, ordered by k.

Original entry on oeis.org

1, 13, 17, 139, 113, 13721, 131, 137, 167, 173, 179, 197, 1103, 1109, 111121, 1151, 1163, 1181, 1361183, 1193, 1373219, 1223, 1229, 1379237, 1277, 1283, 1307, 131133121363, 1367, 1373, 13127381, 1409, 1433, 1439, 13157471, 1487, 13163489

Offset: 1

Jaroslav Krizek, Apr 20 2010

Intersection of A037278 and A018252.

			For n = 6; a(6) = 13721 because A176553(6) = 21, and divisors of 21 are 1, 3, 7, 21; concatenation of divisors A037278(21) = 13721 is noncomposite number.

a(n) = A037278(A176553(n)).

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

A176589 Numbers k such that both concatenation of divisors of k and reverse concatenation of divisors of k are noncomposite.

Original entry on oeis.org

1, 3, 7, 13, 31, 97, 103, 109, 151, 181, 193, 367, 409, 439, 487, 523, 571, 601, 613, 733, 811, 823, 1069, 1117, 1279, 1483, 1489, 1579, 1597, 1789, 1831, 1867, 1897, 1939, 2161, 2203, 2239, 2251, 2269, 2281, 2437, 2503, 2509, 2539, 2659, 2671, 2689, 2953

Offset: 1

Jaroslav Krizek, Apr 21 2010

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

			31 is a term; divisors of 31: 1, 31; both concatenation of divisors 131 and reverse concatenation of divisors 311 are noncomposite.
1897 is a term; divisors of 1897: 1, 7, 271, 1897; both concatenation of divisors 172711897 and reverse concatenation of divisors 189727171 are noncomposite.
Sequence of corresponding values of concatenations of divisors of a(n): 1, 13, 17, 113, 131, 197, ...
Sequence of corresponding values of reverse concatenations of divisors of a(n): 1, 31, 71, 131, 311, 971, ...

Cf. A037278, A176558, A176553, A089374, A176590.

Intersection of A176553 and A089374. - Jason Yuen, Feb 08 2025

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

A272141 Numbers such that the concatenation of their aliquot parts, in ascending order, are prime numbers.

Original entry on oeis.org

9, 14, 21, 26, 27, 34, 35, 46, 49, 55, 57, 58, 62, 74, 98, 115, 118, 143, 155, 158, 161, 166, 169, 178, 183, 187, 194, 201, 202, 209, 214, 215, 218, 219, 221, 226, 245, 265, 279, 287, 295, 298, 309, 314, 323, 326, 327, 329, 335, 341, 355, 371, 374, 377, 381

Offset: 1

Paolo P. Lava, Apr 21 2016

			Aliquot parts of 9 are 1, 3 and concat(1,3) = 13 is prime;
aliquot parts of 3127 are 1, 53, 59 and concat(1,53,59) = 15359 is prime.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A176553, A272142.

with(numtheory): P:= proc(q) local a,b,k,n;
for n from 1 to q do a:=sort([op(divisors(n))]); b:=0;
for k from 1 to nops(a)-1 do b:=b*10^(ilog10(a[k])+1)+a[k]; od;
if isprime(b) then print(n); fi; od; end: P(10^9);

Select[Range@ 384, PrimeQ@ FromDigits@ Flatten@ IntegerDigits@ Most@ Divisors@ # &] (* Michael De Vlieger, Apr 21 2016 *)

A272142 Numbers such that the concatenation of their aliquot parts, in descending order, are prime numbers.

Original entry on oeis.org

8, 9, 10, 26, 34, 35, 49, 55, 56, 57, 62, 63, 75, 76, 77, 94, 95, 115, 122, 125, 142, 144, 146, 161, 169, 183, 194, 196, 203, 206, 219, 226, 235, 238, 254, 262, 265, 274, 275, 278, 290, 299, 302, 304, 305, 309, 320, 322, 332, 336, 338, 346, 355, 358, 361, 362

Offset: 1

Paolo P. Lava, Apr 21 2016

			Aliquot parts of 8 are 1, 2, 4 and concat(4,2,1) = 421 is prime;
aliquot parts of 1822 are 1, 2, 911 and concat(911,2,1) = 91121 is prime.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A176553, A272141.

with(numtheory): P:= proc(q) local a,b,k,n;
for n from 1 to q do a:=sort([op(divisors(n))]); b:=0;
for k from nops(a)-1 by -1 to 1 do b:=b*10^(ilog10(a[k])+1)+a[k]; od;
if isprime(b) then print(n); fi; od; end: P(10^9);

Select[Range@ 362, PrimeQ@ FromDigits@ Flatten@ IntegerDigits@ Reverse@ Most@ Divisors@ # &] (* Michael De Vlieger, Apr 21 2016 *)

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A037278 Replace n with concatenation of its divisors.

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A176554 Numbers n such that concatenations of divisors of n are nonprime.

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A176555 Noncomposite concatenations of divisors of some k, ordered by k.

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A176589 Numbers k such that both concatenation of divisors of k and reverse concatenation of divisors of k are noncomposite.

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A272141 Numbers such that the concatenation of their aliquot parts, in ascending order, are prime numbers.

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A272142 Numbers such that the concatenation of their aliquot parts, in descending order, are prime numbers.

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Maple

Mathematica