A161917 -id:A161917 - cp's OEIS frontend

A161918 Numbers n such that the sum of the divisors minus the sum of the prime factors (counted with multiplicity) is equal to n+1.

6, 8, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jun 23 2009

Equals A006881 union {8}. - Franklin T. Adams-Watters, Jun 26 2009

			n=21: Sum_divisors (1,3,7,21) = 32; Sum_prime_factors (3,7) = 10 -> 32-10 = 22. n=55: Sum_divisors (1,5,11,55) = 72; Sum_prime_factors (5,11) = 16 -> 72-16 = 56.

Jayanta Basu, Table of n, a(n) for n = 1..10000

Cf. A161917, A006881, A151797, A030229.

with(numtheory); P:=proc(i) local b,c,j,s,n; for n from 2 by 1 to i do b:=(convert(ifactors(n),`+`)-1); c:=nops(b); j:=0; s:=0; for j from c by -1 to 1 do s:=s+convert(b[j],`*`); od; if n=sigma(n)-s-1 then print(n); fi; od; end: P(500);

Select[Range[203], DivisorSigma[1, #] - Total[Times @@@ FactorInteger[#]] == # + 1 &] (* Jayanta Basu, Aug 11 2013 *)

\\ from M. F. Hasler
isA161918(n)={ n+1 == sigma(n)-(n=factor(n))[,1]~*n[,2] }
for(n=1,500, isA161918(n)&print1(n","))

Edited by N. J. A. Sloane, Jun 27 2009 incorporating suggestions from R. J. Mathar, M. F. Hasler, Benoit Jubin and others.

A191580 Numbers n for which the sum of their prime factors (with repetition) divides the sum of their anti-divisors.

Original entry on oeis.org

5, 10, 40, 41, 129, 135, 140, 155, 182, 189, 200, 204, 206, 238, 375, 429, 435, 441, 455, 475, 546, 564, 574, 616, 625, 678, 722, 744, 765, 836, 856, 902, 1035, 1056, 1170, 1188, 1272, 1296, 1344, 1518, 1650, 1764, 1806, 1918, 1925

Offset: 1

Paolo P. Lava, Jun 07 2011

			40-> sum prime factors=2+2+2+5=11; sum anti-divisors=3+9+16+27=55; 55/11=5
129-> sum prime factors=3+43=46; sum anti-divisors=2+6+7+37+86=138; 138/46=3

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

Cf. A001414, A066272, A161917, A191581.

with(numtheory); P:=proc(i) local a,b,j,k,s,n;
for n from 3 to i do b:=ifactors(n)[2];
s:=add(b[k][1]*b[k][2],k=1..nops(b));
k:=0; j:=n; while j mod 2<>1 do k:=k+1; j:=j/2; od; a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
if type(a/s,integer) then print(n); fi; od; end: P(2000);

A191581 Numbers whose sum of their anti-divisors divides the sum of their divisors.

Original entry on oeis.org

3, 6, 11, 22, 30, 33, 65, 82, 117, 218, 354, 483, 508, 537, 3276, 6430, 21541, 117818, 130356, 753612, 1007328, 2113416, 2379540, 3589646, 7231219, 7346148, 8515767, 13050345, 20199648, 34424166, 44575896, 47245905, 50414595, 104335023, 217728002, 1217532421

Offset: 1

Paolo P. Lava, Jun 07 2011

A161917 is a subsequence of this sequence.

			6-> sum divisors=sigma(6)=12; sum anti-divisors=4; 12/4=3.
30-> sum divisors=sigma(30)=72; sum anti-divisors=4+12+20=36; 72/36=2.

Cf. A000203, A066272, A066417, A161917, A191580.

with(numtheory): P:=proc(i) local a, b, j, k, s, n;
for n from 3 to i do b:=divisors(n); s:=add(b[k],k=1..nops(b));
k:=0; j:=n; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
if type(s/a,integer) then print(n); fi; od; end: P(10^6);

f[n_] := Total@ Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; Select[Range[3, 10^3], Mod[DivisorSigma[1, #], f@ #] == 0 &] (* _Michael De Vlieger, Oct 08 2015 *)

{n: A066417(n) | A000203(n)}. - R. J. Mathar, Oct 01 2011

a(21)-a(36) from Donovan Johnson, Jun 24 2012

A257048 Numbers n for which the sum of their prime factors (with repetition) divides the Euler totient function.

Original entry on oeis.org

9, 15, 16, 25, 27, 35, 42, 49, 72, 95, 119, 121, 140, 143, 154, 168, 169, 200, 209, 220, 240, 256, 264, 287, 288, 289, 297, 315, 319, 323, 342, 343, 361, 364, 377, 378, 442, 483, 490, 520, 525, 527, 529, 540, 559, 585, 588, 616, 620, 624, 625, 648, 693, 702, 729

Offset: 1

Paolo P. Lava, Apr 15 2015

			The value of Euler totient function for n = 15 is 8. Prime factors of 15 are 3, 5 and their sum is 3 + 5 = 8. Finally, 8 / 8 = 1.
The value of Euler totient function for n = 140 is 48. Prime factors of 140 are 2, 2, 5, 7 and their sum is 2 + 2 + 5 + 7 = 16. Finally, 48 / 16 = 3.

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

Cf. A000010, A161917.

with(numtheory); P:=proc(q) local a,n;
for n from 1 to q do a:=ifactors(n)[2];
if type(phi(n)/add(a[k][1]*a[k][2],k=1..nops(a)),integer)
then print(n); fi; od; end: P(10^9);

Rest@ Select[Range@ 729, Mod[EulerPhi@ #, Total@ Flatten[Table[#1, {#2}] & @@@ FactorInteger@ #]] == 0 &] (* Michael De Vlieger, Apr 15 2015 *)

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A161918 Numbers n such that the sum of the divisors minus the sum of the prime factors (counted with multiplicity) is equal to n+1.

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A191580 Numbers n for which the sum of their prime factors (with repetition) divides the sum of their anti-divisors.

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A191581 Numbers whose sum of their anti-divisors divides the sum of their divisors.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A257048 Numbers n for which the sum of their prime factors (with repetition) divides the Euler totient function.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica