A106708 -id:A106708 - cp's OEIS frontend

A131984 Where records occur in A106708.

1, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 84, 90, 120, 168, 180, 240, 336, 360, 420, 480, 540, 600, 660, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 4200, 4620, 5040, 7560, 9240, 10080, 12600, 13860, 15120, 18480, 20160, 25200, 27720

Offset: 1

Klaus Brockhaus, Aug 05 2007

Numbers n such that concatenation of proper divisors of n exceeds that of all smaller numbers. Empty concatenation is regarded as 0.

Sequence has many terms in common with A034090 (numbers n such that sum of proper divisors of n exceeds that of all smaller numbers), A034287 (numbers n such that product of divisors of n is larger than for any number less than n), A034288 (product of proper divisors is larger than for any smaller number), A067128 (Ramanujan's largely composite numbers, defined to be n such that d(n) >= d(k) for k = 1 to n-1).

Klaus Brockhaus, Table of n, a(n) for n=1..144

Cf. A106708, A131983 (records), A034090, A034287, A034288, A067128.

DeleteDuplicates[Table[{n,If[CompositeQ[n],FromDigits[Flatten[IntegerDigits/@Rest[ Most[ Divisors[n]]]]],0]},{n,30000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Apr 27 2024 *)

{map(n) = local(d); d=divisors(n); if(#d<3, 0, d[1]=""; eval(concat(vecextract(d, concat("..", #d-1)))))} {m=28000; r=-1; for(n=1, m, if(r<(a=map(n)), r=a; print1(n, ",")))}

A131983 Records in A106708.

Original entry on oeis.org

0, 2, 23, 24, 25, 2346, 2369, 24510, 2346812, 23561015, 234691218, 23468121624, 234561012152030, 234671214212842, 235691015183045, 2345681012152024304060, 2346781214212428425684, 23456910121518203036456090

Offset: 1

Klaus Brockhaus, Aug 05 2007

Cf. A106708, A131984 (where records occur).

{map(n) = local(d); d=divisors(n); if(#d<3, 0, d[1]=""; eval(concat(vecextract(d, concat("..", #d-1)))))} {m=200; r=-1; for(n=1, m, if(r<(a=map(n)), r=a; print1(r, ",")))}

A037278 Replace n with concatenation of its divisors.

Original entry on oeis.org

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

A037279 If n is composite, replace n with the concatenation of its nontrivial divisors, otherwise a(n) = n.

Original entry on oeis.org

1, 2, 3, 2, 5, 23, 7, 24, 3, 25, 11, 2346, 13, 27, 35, 248, 17, 2369, 19, 24510, 37, 211, 23, 2346812, 5, 213, 39, 24714, 29, 23561015, 31, 24816, 311, 217, 57, 234691218, 37, 219, 313, 24581020, 41, 23671421, 43, 241122, 35915, 223, 47, 23468121624, 7

Offset: 1

N. J. A. Sloane

			Divisors of 12 are 1,2,3,4,6,12, so a(12)=2346.

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

Cf. A037278, A120712, A106708.

A037279 := proc(n) local dvs ; if isprime(n) or n = 1 then n; else dvs := [op(numtheory[divisors](n) minus {1,n} )] ; dvs := sort(dvs) ; cat(op(dvs)) ; fi ; end: seq(A037279(n),n=1..80) ; # R. J. Mathar, Jul 23 2007

f[n_]:=If[PrimeQ[n],n,FromDigits[Flatten[IntegerDigits/@Rest[ Most[ Divisors[n]]]]]]; Join[{1},Array[f,50,2]] (* Harvey P. Dale, Sep 24 2012 *)

More terms from Erich Friedman

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A120712 Numbers k with the property that the concatenation of the nontrivial divisors of k (i.e., excluding 1 and k) is a prime.

Original entry on oeis.org

4, 6, 9, 21, 22, 25, 33, 39, 46, 49, 51, 54, 58, 78, 82, 93, 99, 111, 115, 121, 133, 141, 142, 147, 153, 154, 159, 162, 166, 169, 174, 177, 186, 187, 189, 201, 205, 219, 226, 235, 237, 247, 249, 253, 262, 267, 274, 286, 289, 291, 294, 301, 318

Offset: 1

Eric Angelini, Jul 19 2007

			   k |    divisors    | concatenation
  ---+----------------+--------------
   4 | (1) 2      (4) |        2
   6 | (1) 2, 3   (6) |       23
   9 | (1) 3      (9) |        3
  21 | (1) 3, 7  (21) |       37
  22 | (1) 2, 11 (22) |      211
  25 | (1) 5     (25) |        5
  33 | (1) 3, 11 (33) |      311
  39 | (1) 3, 13 (39) |      313

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A037274, A037278, A037279, A106708, A120713, A120716.

Cf. A130139, A130140, A130141, A130142.

with(numtheory):
for k from 2 to 1000 do:
v0:=divisors(k):
nn:=nops(v0):
if nn > 2 then
v1:=[seq(v0[j],j=2..nn-1)]:
v2:=cat(seq(convert(v1[n],string),n=1..nops(v1))):
v3:=parse(v2):
if isprime(v3) = true then lprint(k,v3) fi:
fi:
od: # Simon Plouffe

fQ[n_] := PrimeQ@ FromDigits@ Most@ Rest@ Divisors@ n; Select[ Range[2, 320], fQ]

from sympy import divisors, isprime
def ok(n):
    s = "".join(str(d) for d in divisors(n)[1:-1])
    return s != "" and isprime(int(s))
print([k for k in range(319) if ok(k)]) # Michael S. Branicky, Oct 01 2024

Name edited by Michel Marcus, Mar 09 2023

A037285 Replace n with concatenation of its nontrivial odd divisors.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 3, 5, 0, 3, 0, 7, 35, 0, 0, 39, 0, 5, 37, 11, 0, 3, 5, 13, 39, 7, 0, 3515, 0, 0, 311, 17, 57, 39, 0, 19, 313, 5, 0, 3721, 0, 11, 35915, 23, 0, 3, 7, 525, 317, 13, 0, 3927, 511, 7, 319, 29, 0, 3515, 0, 31, 37921, 0, 513, 31133, 0, 17, 323, 5735, 0, 39, 0

Offset: 1

N. J. A. Sloane

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

Cf. A182469, A209229, A010051, A106708, A037283, A037284.

import Data.List (delete)
a037285 n
| a209229 n == 1 = 0
| a010051 n == 1 = 0
| otherwise = read $ concat $ (map show) $ delete n $ tail $ a182469_row n
-- Reinhard Zumkeller, May 01 2012

from sympy import divisors
def a(n):
  nontrivial_odd_divisors = [d for d in divisors(n)[1:-1] if d%2 == 1]
  if len(nontrivial_odd_divisors) == 0: return 0
  else: return int("".join(str(d) for d in nontrivial_odd_divisors))
print([a(n) for n in range(1, 70)]) # Michael S. Branicky, Dec 31 2020

More terms from Erich Friedman

A037277 Replace n with concatenation of its divisors >1.

Original entry on oeis.org

0, 2, 3, 24, 5, 236, 7, 248, 39, 2510, 11, 234612, 13, 2714, 3515, 24816, 17, 236918, 19, 2451020, 3721, 21122, 23, 234681224, 525, 21326, 3927, 2471428, 29, 2356101530, 31, 2481632, 31133, 21734, 5735, 23469121836, 37, 21938, 31339

Offset: 1

N. J. A. Sloane

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

Cf. A027750, A037284, A037278, A106708.

a037277 1 = 0
a037277 n = read $ concat $ map show $ tail $ a027750_row n
-- Reinhard Zumkeller, May 01 2012, Feb 07 2011

FromDigits[Flatten[IntegerDigits/@Rest[Divisors[#]]]]&/@Range[40] (* Harvey P. Dale, Nov 06 2011 *)

from sympy import divisors
def a(n):
  divisors_gt1 = divisors(n)[1:]
  if len(divisors_gt1) == 0: return 0
  else: return int("".join(str(d) for d in divisors_gt1))
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Dec 31 2020

More terms from Erich Friedman

A130846 Replace n with the concatenation of its anti-divisors.

Original entry on oeis.org

2, 3, 23, 4, 235, 35, 26, 347, 237, 58, 2359, 349, 2610, 311, 235711, 45712, 2313, 3813, 2614, 345915, 235915, 716, 2371017, 3417, 2561118, 3581119, 2319, 41220, 237921, 35791321, 2561322, 3423, 23101423, 824, 2351525, 3457111525, 2671126, 391627

Offset: 3

Jonathan Vos Post, Jul 20 2007, Jul 22 2007

Number of anti-divisors concatenated to form a(n) is A066272(n). We may consider prime values of the concatenated anti-divisor sequence and we may iterate it, i.e. n, a(n), a(a(n)), a(a(a(n))) which leads to questions of trajectory, cycles, fixed points.
See A066272 for definition of anti-divisor.
Primes in this sequence are at n=3,4,5,10,14,16,40,46,100,145,149,... - R. J. Mathar, Jul 24 2007

			3: 2, so a(3) = 2.
4: 3, so a(4) = 3.
5: 2, 3, so a(5) = 23.
6: 4, so a(6) = 4.
7: 2, 3, 5, so a(7) = 235.
17: 2, 3, 5, 7, 11, so a(17) = 235711

Jon Perry, The Anti-Divisor, cached copy.
Jonathan Vos Post, Factors of first 62 terms

Cf. A037278, A066272, A120712, A106708, A130799.

antiDivs := proc(n) local resul,odd2n,r ; resul := {} ; for r in ( numtheory[divisors](2*n-1) union numtheory[divisors](2*n+1) ) do if n mod r <> 0 and r> 1 and r < n then resul := resul union {r} ; fi ; od ; odd2n := numtheory[divisors](2*n) ; for r in odd2n do if ( r mod 2 = 1) and r > 2 then resul := resul union {2*n/r} ; fi ; od ; RETURN(resul) ; end: A130846 := proc(n) cat(op(antiDivs(n))) ; end: seq(A130846(n),n=3..80) ; # R. J. Mathar, Jul 24 2007

from sympy.ntheory.factor_ import antidivisors
def A130846(n): return int(''.join(str(s) for s in antidivisors(n))) # Chai Wah Wu, Dec 08 2021

More terms from R. J. Mathar, Jul 24 2007

A191647 Numbers n with property that the concatenation of their anti-divisors is a prime.

Original entry on oeis.org

3, 4, 5, 10, 14, 16, 40, 46, 100, 145, 149, 251, 340, 373, 406, 424, 439, 466, 539, 556, 571, 575, 617, 619, 628, 629, 655, 676, 689, 724, 760, 779, 794, 899, 901, 941, 970, 989, 1019, 1055, 1070, 1076, 1183, 1213, 1226, 1231, 1258, 1270, 1285, 1331, 1340

Offset: 1

Paolo P. Lava, Jun 10 2011

Similar to A120712 which uses the proper divisors of n.

			The anti-divisors of 40 are 3, 9, 16, 27, and 391627 is prime, hence 40 is in the sequence.

Klaus Brockhaus Table of n, a(n) for n = 1..1000 (first 250 terms from Paolo P. Lava)

Cf. A037279, A066272, A106708, A120712, A120716, A191648.

P:=proc(i) local a,b,c,d,k,n,s,v; v:=array(1..200000);
for n from 3 by 1 to i do k:=2; b:=0;
while k0 and (2*n mod k)=0 then b:=b+1; v[b]:=k; fi;
     else
     if (n mod k)>0 and (((2*n-1) mod k)=0 or ((2*n+1) mod k)=0) then
b:=b+1; v[b]:=k; fi; fi; k:=k+1; od; a:=v[1];
for s from 2 to b do a:=a*10^floor(1+evalf(log10(v[s])))+v[s]; od;
if isprime(a) then print(n); fi;
od; end: P(10^6);

antiDivisors[n_Integer] := Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]; a191647[n_Integer] := Select[Range[n],
PrimeQ[FromDigits[Flatten[IntegerDigits /@ antiDivisors[#]]]] &]; a191647[1350] (* Michael De Vlieger, Aug 09 2014, "antiDivisors" after Harvey P. Dale at A066272 *)

from sympy import isprime
[n for n in range(3,10**4) if isprime(int(''.join([str(d) for d in range(2,n) if n%d and 2*n%d in [d-1,0,1]])))] # Chai Wah Wu, Aug 08 2014

a(618) corrected in b-file by Paolo P. Lava, Feb 28 2018

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A131984 Where records occur in A106708.

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A131983 Records in A106708.

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A037278 Replace n with concatenation of its divisors.

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A037279 If n is composite, replace n with the concatenation of its nontrivial divisors, otherwise a(n) = n.

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A120712 Numbers k with the property that the concatenation of the nontrivial divisors of k (i.e., excluding 1 and k) is a prime.

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A037285 Replace n with concatenation of its nontrivial odd divisors.

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A037277 Replace n with concatenation of its divisors >1.

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A130846 Replace n with the concatenation of its anti-divisors.

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