A066272 -id:A066272 - cp's OEIS frontend

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

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

A192274 Numbers which are both Zumkeller numbers and anti-Zumkeller numbers.

Original entry on oeis.org

42, 70, 78, 88, 126, 160, 176, 228, 234, 258, 270, 280, 308, 342, 350, 368, 378, 380, 390, 396, 402, 438, 448, 462, 468, 490, 500, 522, 532, 540, 552, 558, 560, 572, 580, 588, 608, 618, 620, 630, 644, 650, 690, 702, 732, 756, 770, 780, 798, 812, 822, 852, 858

Offset: 1

Paolo P. Lava, Jun 28 2011

nonn

Numbers n whose sets of divisors and anti-divisors can each be partitioned into two disjoint sets whose sums are sigma(n)/2 for the sets in the divisors partition and sigma*(n)/2 for the anti-divisors partition, where sigma*(n) is the sum of the anti-divisors of n.

			270-> divisors: 1,2,3,5,6,9,10,15,18,27,30,45,54,90,135,270; sigma(270)/2=360; 1+2+3+5+6+9+10+15+18+27+30+45+54+135=90+270=360.
270-> anti-divisors: 4,7,11,12,20,36,49,60,77,108,180; sigma*(270)/2=282; 4+7+11+20+60+180=12+36+49+77+108=282.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A083207, A066272, A192273.

with(combstruct);
with(numtheory);
P:=proc(i)
local S,R,Stop,Comb,a,b,c,d,k,m,n,s;
for n from 3 to i do
  a:={};
  for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a:=a union {k}; fi; od;
  b:=nops(a); c:=op(a); s:=0;
   if b>1 then for k from 1 to b do s:=s+c[k]; od;
   else s:=c;
  fi;
  if (modp(s,2)=0 and 2*n<=s) then
     S:=1/2*s-n; R:=select(m->m<=S,[c]); Stop:=false; Comb:=iterstructs(Combination(R));
     while not (finished(Comb) or Stop) do Stop:=add(d,d=nextstruct(Comb))=S; od;
     if Stop then
        s:=sigma(n);
        if (modp(s,2)=0 and 2*n<=s) then
          S:=1/2*s-n; R:=select(m->m<=S,divisors(n)); Stop:=false;       Comb:=iterstructs(Combination(R));
          while not (finished(Comb) or Stop) do Stop:=add(d,d=nextstruct(Comb))=S; od;
          if Stop then print(n); fi;
        fi;
     fi;
  fi;
od;
end:
P(10000);

from sympy import divisors
from sympy.combinatorics.subsets import Subset
def antidivisors(n):
    return [2*d for d in divisors(n) if n > 2*d and n % (2*d)] + \
           [d for d in divisors(2*n-1) if n > d >=2 and n % d] + \
           [d for d in divisors(2*n+1) if n > d >=2 and n % d]
for n in range(1, 10**3):
    d = divisors(n)
    s = sum(d)
    if not s % 2 and max(d) <= s/2:
        for x in range(1, 2**len(d)):
            if sum(Subset.unrank_binary(x, d).subset) == s/2:
                d = antidivisors(n)
                s = sum(d)
                if not s % 2 and max(d) <= s/2:
                    for x in range(1, 2**len(d)):
                        if sum(Subset.unrank_binary(x, d).subset) == s/2:
                            print(n, end=', ')
                            break
                break
# Chai Wah Wu, Aug 14 2014

from sympy import divisors
import numpy as np
A192274 = []
for n in range(3,10**3):
    d = divisors(n)
    s = sum(d)
    if not s % 2 and 2*n <= s:
        d.remove(n)
        s2, ld = int(s/2-n), len(d)
        z = np.zeros((ld+1,s2+1),dtype=int)
        for i in range(1,ld+1):
            y = min(d[i-1],s2+1)
            z[i,range(y)] = z[i-1,range(y)]
            z[i,range(y,s2+1)] = np.maximum(z[i-1,range(y,s2+1)],z[i-1,range(0,s2+1-y)]+y)
            if z[i,s2] == s2:
                d2 = [2*x for x in d if n > 2*x and n % (2*x)] + \
                [x for x in divisors(2*n-1) if n > x >=2 and n % x] + \
                [x for x in divisors(2*n+1) if n > x >=2 and n % x]
                s, dmax = sum(d2), max(d2)
                if not s % 2 and 2*dmax <= s:
                    d2.remove(dmax)
                    s2, ld = int(s/2-dmax), len(d2)
                    z = np.zeros((ld+1,s2+1),dtype=int)
                    for i in range(1,ld+1):
                        y = min(d2[i-1],s2+1)
                        z[i,range(y)] = z[i-1,range(y)]
                        z[i,range(y,s2+1)] = np.maximum(z[i-1,range(y,s2+1)],z[i-1,range(0,s2+1-y)]+y)
                        if z[i,s2] == s2:
                            A192274.append(n)
                        break
                break
# Chai Wah Wu, Aug 19 2014

Corrected entries and comment by Chai Wah Wu, Aug 13 2014

A192282 Numbers n such that n and n+1 have same sum of anti-divisors.

Original entry on oeis.org

1, 8, 17, 120, 717, 729, 957, 8097, 10785, 12057, 35817, 44817, 52863, 58677, 59757, 76759, 95397, 102957, 114117, 119337, 182157, 206097, 215997, 230037, 253977, 263877, 269277, 271797, 295377, 321417, 402657, 435477, 483117, 485637, 510837, 586797, 589317

Offset: 1

Paolo P. Lava, Jul 27 2011

nonn

Like A002961 but using anti-divisors.

Curiously 957 and 958 have same sum of divisors and same sum of anti-divisors.

			Anti-divisors of 717 are 2, 5, 6, 7, 35, 41, 205, 287, 478 and their sum is 1066.
Anti-divisors of 718 are  3, 4, 5, 7, 35, 41, 205, 287, 479 and their sum is 1066.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A002961, A066272, A192283.

with(numtheory);
P:=proc(n)
local a,b,i,k;
b:=2;
for i from 4 to n do
  a:=0;
  for k from 2 to i-1 do
    if abs((i mod k)- k/2) < 1 then a:=a+k; fi;
  od;
  if a=b then print(i-1); fi;
  b:=a;
od;
end:
P(200000);

Initial term a(1)=1 inserted, a(2)=9 through a(20)=119337 verified, and a(21)-a(28) added by John W. Layman, Aug 04 2011

A192284 Numbers n such that the average of the anti-divisors of n is an integer.

Original entry on oeis.org

3, 4, 6, 8, 9, 11, 15, 16, 18, 19, 20, 26, 29, 30, 34, 36, 37, 38, 47, 49, 51, 63, 64, 68, 69, 82, 93, 94, 95, 96, 106, 117, 120, 121, 131, 134, 135, 141, 155, 169, 174, 186, 187, 190, 191, 197, 200, 203, 206, 224, 226, 244, 252, 257, 271, 272, 274, 276, 289

Offset: 1

Paolo P. Lava, Jul 20 2011

nonn

Like A003601 but using anti-divisors.

			Anti-divisors of 38 are 7: 3, 4, 5, 7, 11, 15, 25. Their sum is 70 and 70/7=10 that is an integer.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A003601, A066272.

with(numtheory);
P:=proc(n)
local a,b,i,k;
for i from 3 to n do
  a:=0; b:=0;
  for k from 2 to i-1 do
    if abs((i mod k)- k/2) < 1 then a:=a+k; b:=b+1; fi;
  od;
  if trunc(a/b)=a/b then print(i); fi;
od;
end:
P(1000);

A192291 Couple of numbers a, b for which sigma(a)=b and sigma(b)-b=a, where sigma(n) is the sum of the anti-divisors of n.

Original entry on oeis.org

10, 14, 32, 58, 154, 182, 382, 758, 3830, 5962, 67815454, 94941602, 7172169026, 8196764584, 18624907238, 34790550682, 30033199624, 31387575416, 38857270202, 48571587730

Offset: 1

Paolo P. Lava, Jun 29 2011

a(21) > 10^11. - Hiroaki Yamanouchi, Sep 28 2015

			sigma*(10) = 3+4+7 = 14.
sigma(14)-14 = 1+2+7 = 10.
sigma*(32)= 3+5+7+9+13+21 = 58.
sigma(58)-58 = 1+2+29 = 32.

Cf. A063990, A066272, A192290, A192292, A192293.

with(numtheory);
P:= proc(n)
local a,b,c,i,ks;
for i from 3 to n do
   a:={};
   for k from 2 to i-1 do
     if abs((i mod k)- k/2) < 1 then
       a:=a union {k};
     fi;
   od;
   b:=nops(a); c:=op(a); s:=0;
   for k from 1 to b do
       s:=s+c[k];
   od;
   if sigma(s)-s=i then
      print(i,s);
   fi;
od;
end:
P(10000);

a(11)-a(20) from Hiroaki Yamanouchi, Sep 28 2015

A192292 Pairs of numbers a, b for which sigma(a)=b and sigma(b)-b-1=a, where sigma(n) is the sum of the anti-divisors of n.

Original entry on oeis.org

7, 10, 14, 16, 45, 86, 2379, 2324, 4213, 5866, 27323, 33604, 1303227, 1737628, 3722831, 4208308, 15752651, 18706108, 6094085371, 8114352508, 30090695519, 40119052564

Offset: 1

Paolo P. Lava, Jun 29 2011

Betrothed numbers mixed with anti-divisors.

a(23) > 10^11. - Hiroaki Yamanouchi, Sep 28 2015

			sigma*(45)= 2+6+7+10+13+18+30 = 86.
sigma(86)-86-1 = 2+43 = 45.
sigma*(2379) = 2+6+26+67+71+78+122+366+1586 = 2374.
sigma(2324)-2324-1 = 2+4+7+14+28+83+166+332+581+1162 = 2379.

Cf. A005276, A066272, A192290, A192291, A192293.

with(numtheory); P:= proc(n) local b,c,i,j,k;
for i from 3 to n do k:=0; j:=i;
while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
b:=sigma(2*i+1)+sigma(2*i-1)+sigma(i/2^k)*2^(k+1)-6*i-2;
if sigma(b)-b-1=i then print(i); print(b); fi;
od; end: P(10^9);

a(13)-a(14) from Paolo P. Lava, Dec 03 2014

a(7)-a(8) swapped and a(15)-a(22) added by Hiroaki Yamanouchi, Sep 28 2015

A195150 Number of divisors d of n such that d-1 does not divide n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 3, 3, 1, 3, 1, 3, 3, 2, 1, 4, 2, 2, 3, 4, 1, 4, 1, 4, 3, 2, 3, 5, 1, 2, 3, 5, 1, 4, 1, 4, 5, 2, 1, 6, 2, 4, 3, 4, 1, 5, 3, 5, 3, 2, 1, 6, 1, 2, 5, 5, 3, 5, 1, 4, 3, 6, 1, 7, 1, 2, 5, 4, 3, 5, 1, 7, 4, 2, 1, 7, 3, 2

Offset: 1

Omar E. Pol, Sep 19 2011

Define "subdivisor" of n to be the positive integer b such that b = d - 1, if d divides n and b does not divide n. For the list of subdivisors of n see A195153.

First occurrence of k is given in A173569. - Robert G. Wilson v, Sep 23 2011

			a(24) = 4 since the divisors of 24 are 1,2,3,4,6,8,12,24, so the subdivisors of 24 are 5,7,11,23 because 6-1 = 5, 8-1 = 7, 12-1 = 11 and 24-1 = 23. Note that the positive integers 1,2,3 are not subdivisors of 24 because they are divisors of 24.

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

Cf. A000005, A001620, A027750, A066272, A195153, A137921, A173569.

a195150 n = length [d | d <- [3..n], mod n d == 0, mod n (d-1) /= 0]
-- Reinhard Zumkeller, Sep 23 2011

f[n_] := Module[{d = Divisors[n]}, Length[Select[Rest[d-1], Mod[n, #] > 0 &]]]; Table[f[n], {n, 100}] (* T. D. Noe, Sep 22 2011 *)

a(n) = A137921(n) - 1. - Robert G. Wilson v, Sep 23 2001

Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 18 2024

A210732 Numbers n for which sigma(n)=sigma(x)+sigma(y), where n=x+y and sigma(n) is the sum of the anti-divisors of n.

Original entry on oeis.org

6, 9, 15, 18, 21, 24, 27, 30, 31, 33, 37, 39, 43, 44, 46, 47, 53, 56, 57, 62, 65, 66, 70, 73, 74, 75, 76, 78, 81, 83, 86, 88, 90, 91, 92, 93, 97, 99, 102, 103, 106, 107, 109, 110, 114, 116, 117, 118, 119, 121, 122, 123, 125, 126, 127, 129, 131, 133, 135, 136

Offset: 3

Paolo P. Lava, May 10 2012

nonn

Similar to A211223 but using anti-divisors.

			sigma*(127)=sigma*(45)+sigma*(82) that is 212=86+126.
In more than one way:
sigma*(133)=sigma*(50)+sigma*(83)=sigma*(52)+sigma*(81) that is
204=80+124=94+110.

Cf. A066272, A066417, A083207, A204830, A204831, A211223-A211225.

with(numtheory);
A210732:=proc(q)
local a,b,c,i,j,k,n;
for n from 3 to q do
  a:=0;
  for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+k; fi; od;
  for i from 1 to trunc(n/2) do
   b:=0; c:=0;
   for k from 2 to i-1 do if abs((i mod k)-k/2)<1 then b:=b+k; fi; od;
   for k from 2 to n-i-1 do if abs(((n-i) mod k)-k/2)<1 then c:=c+k; fi; od;
   if a=b+c then print(n); break; fi;
  od;
od; end:
A210732(10000);

A240979 Sum of unitary anti-divisors of n.

Original entry on oeis.org

0, 0, 2, 3, 5, 0, 10, 8, 2, 10, 12, 5, 19, 12, 2, 14, 28, 12, 18, 16, 2, 32, 34, 7, 29, 20, 18, 38, 24, 0, 42, 58, 20, 26, 28, 0, 50, 66, 20, 39, 41, 22, 56, 32, 22, 54, 60, 24, 58, 56, 2, 86, 88, 0, 42, 40, 30, 92, 90, 35, 57, 74, 32, 46, 48, 26, 132, 104, 2

Offset: 1

Paolo P. Lava, Aug 06 2014

nonn

For unitary anti-divisors of n are intended all the anti-divisors of n which are coprime to n.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A066272, A066417, A242029.

P:=proc(q) local a,k,n,v; v:=[]; for n from 1 to q do a:=0;
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then if gcd(n,k)=1
then a:=a+k; fi; fi; od; v:=[op(v),a]; od; print(op(v)); end: P(69);
# corrected by Paolo P. Lava, Aug 17 2024

antiDivisors[n_Integer] := Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; a240979[n_Integer] := Total[Select[antiDivisors[n], CoprimeQ[#, n] &]]; a240979 /@ Range[120] (* _Michael De Vlieger, Aug 17 2014 *)

Anti-divisors of 14 are 3, 4, 9. Anti-divisors coprime to 14 are 3 and 9 and therefore a(14) = 3 + 9 = 12.

A242965 Numbers whose anti-divisors are all primes.

Original entry on oeis.org

3, 4, 5, 7, 8, 11, 16, 17, 19, 29, 43, 47, 61, 64, 71, 79, 89, 101, 107, 109, 151, 191, 197, 223, 251, 271, 317, 349, 359, 421, 439, 461, 521, 569, 601, 631, 659, 673, 691, 701, 719, 811, 821, 881, 911, 919, 947, 971, 991, 1009, 1024, 1051, 1091, 1109, 1153

Offset: 1

Paolo P. Lava, May 28 2014

			The anti-divisors of 191 are all primes: 2, 3, 127.
The same for 1024: 3, 23, 89, 683.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A066272, A242966.

P := proc(q) local k,ok,n; for n from 3 to q do ok:=1;
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then
if not isprime(k) then ok:=0; break; fi; fi; od;
if ok=1 then print(n); fi; od; end: P(10^3);

from sympy import divisors, isprime
for n in range(3, 10**4):
    for d in [2*d for d in divisors(n) if n > 2*d and n % (2*d)] + \
             [d for d in divisors(2*n-1) if n > d >= 2 and n % d] + \
             [d for d in divisors(2*n+1) if n > d >= 2 and n % d]:
        if not isprime(d):
            break
    else:
        print(n, end=', ')
# Chai Wah Wu, Aug 15 2014

cp's OEIS Frontend

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

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A192274 Numbers which are both Zumkeller numbers and anti-Zumkeller numbers.

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A192282 Numbers n such that n and n+1 have same sum of anti-divisors.

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A192284 Numbers n such that the average of the anti-divisors of n is an integer.

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A192291 Couple of numbers a, b for which sigma*(a)=b and sigma(b)-b=a, where sigma*(n) is the sum of the anti-divisors of n.

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A192292 Pairs of numbers a, b for which sigma*(a)=b and sigma(b)-b-1=a, where sigma*(n) is the sum of the anti-divisors of n.

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A195150 Number of divisors d of n such that d-1 does not divide n.

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A192291 Couple of numbers a, b for which sigma(a)=b and sigma(b)-b=a, where sigma(n) is the sum of the anti-divisors of n.

A192292 Pairs of numbers a, b for which sigma(a)=b and sigma(b)-b-1=a, where sigma(n) is the sum of the anti-divisors of n.

A210732 Numbers n for which sigma(n)=sigma(x)+sigma(y), where n=x+y and sigma(n) is the sum of the anti-divisors of n.