A070824 -id:A070824 - cp's OEIS frontend

A167131 Numbers k such that A002808(k) - A144925(k) is prime.

1, 8, 9, 12, 21, 24, 26, 30, 38, 44, 45, 49, 53, 61, 66, 81, 84, 86, 97, 100, 106, 109, 116, 121, 131, 140, 154, 165, 183, 189, 191, 198, 203, 205, 208, 216, 232, 245, 252, 257, 270, 283, 290, 305, 308, 310, 313, 323, 325, 330, 340, 342, 358, 363, 367, 377, 388

Offset: 1

Juri-Stepan Gerasimov, Oct 28 2009

nonn

Cf. A000040, A002808, A144925.

A070824 := proc(n) if n <=3 then 0; else numtheory[tau](n)-2 ; end if; end proc: A144925 := proc(n) A070824(A002808(n)) ; end proc: isA167131 := proc(n) isprime( A002808(n)-A144925(n)) ; end proc: for n from 1 to 1000 do if isA167131(n) then printf("%d,",n) ; end if; od: # R. J. Mathar, Nov 02 2009

Edited (but not checked) by N. J. A. Sloane, Nov 01 2009

105 replaced with 106 and sequence extended by R. J. Mathar, Nov 02 2009

A255429 Numbers with a prime number of nontrivial divisors.

Original entry on oeis.org

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 196

Offset: 1

Rory Glover, Feb 22 2015

nonn

Empirically, numbers in this sequence seem to have few divisors.

This sequence appears to be the union of A130763 and the squares of A225649. - Kellen Myers, Apr 21 2015

Cf. A063806, A070824, A130763, A225649.

[n: n in [1..200] | IsPrime(NumberOfDivisors(n)-2)]; // Vincenzo Librandi, Apr 21 2015

seq[n_] := Select[Range[n], PrimeQ[DivisorSigma[0, #] - 2] &] (* Kellen Myers, Apr 21 2015 *)

isok(m) = isprime(numdiv(m)-1); \\ Michel Marcus, Jan 13 2023

{n: A070824(n) in A000040}.

Terms fixed by Kellen Myers, Apr 21 2015

Name corrected by Michel Marcus, Jan 13 2023

A306307 Numbers that are divisible by the number of their nontrivial divisors.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 20, 22, 24, 25, 26, 28, 30, 32, 34, 38, 42, 44, 46, 48, 49, 52, 54, 58, 60, 62, 66, 68, 74, 76, 78, 80, 81, 82, 86, 90, 92, 94, 102, 106, 112, 114, 116, 118, 121, 122, 124, 134, 138, 140, 142, 146, 148, 150, 158, 160, 164, 166, 168, 169, 172, 174

Offset: 1

Todor Szimeonov, Feb 05 2019

nonn

We may define the number of divisors of a number n in four ways:
(1) A070824(n) = number of nontrivial or real divisors: 1 < d < n;
(2) variant of A032741(n) = number of small divisors: 1 and real divisors;
(3) A032741(n) = number of big or proper divisors: real divisors and n;
(4) A000005(n) = number of all divisors of n: 1, n and real divisors.
The case (1), divisibility through the number of nontrivial divisors, defines this sequence.

			1 and the prime numbers do not have any nontrivial divisors; A070824(n) is 0 for n=1 or a prime, and so they are not terms.
The only nontrivial divisor of 4 is 2, so A070824(4) = 1; 4 is divisible by 1, so 4 is a term.
A070824(15) = 2, and 15 is not divisible by 2, so 15 is not a term.

T. Szimeonov, A számok [The numbers], Budapest, 2019, VVMA, 124 p.

Cf. A070824, A054010, A033950.

seqQ[n_] := (nd = DivisorSigma[0, n] - 2) > 0 && Divisible[n, nd]; Select[Range[200], seqQ] (* Amiram Eldar, Mar 11 2019 *)

f(n) = if (n==1, 0, numdiv(n)-2); \\ A070824
isok(n) = f(n) && !frac(n/f(n)); \\ Michel Marcus, Feb 17 2019

More terms from Michel Marcus, Feb 17 2019

A317746 Irregular triangle read by rows in which row n lists the divisors k of n such that k^n + n^k == 0 (mod k + n).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 3, 6, 1, 7, 1, 2, 8, 1, 3, 9, 1, 10, 1, 11, 1, 4, 6, 12, 1, 13, 1, 7, 14, 1, 3, 5, 15, 1, 16, 1, 17, 1, 6, 9, 18, 1, 19, 1, 5, 20, 1, 3, 7, 21, 1, 22, 1, 23, 1, 3, 8, 12, 24, 1, 5, 25, 1, 13, 26, 1, 3, 9, 27, 1, 4, 28, 1, 29, 1, 6, 10, 15, 30

Offset: 1

Juri-Stepan Gerasimov, Aug 06 2018

Triangle read by rows in which row n lists the type-1 divisors of n. For each divisor k of n, call k a type-r divisor of n if (r*k)^n + n^(r*k) == 0 (mod r*k + n), r >= 1.
Triangle read by rows in which row n lists the smallest types r of divisor k of n such that (r*k)^n + n^(r*k) == 0 (mod r*k + n) begins:
  1;
  1,  1;
  1,  1;
  1,  2,  1;
  1,  1;
  1,  3,  1,  1;
  1,  1;
  1,  1,  2,  1;
  1,  1,  1;
  1,  3,  2,  1;
  1,  1;
  1,  2,  3,  1,  1,  1;
..., where the total number of type-1 divisors of n is the sum of the number of all trivial divisors of n and a certain number of nontrivial divisors of n, namely: 1+0, 2+0, 2+0, 2+0, 2+0, 2+1, 2+0, 2+1, 2+1, 2+0, 2+0, 2+2, ...

			Triangle begins:
  1;
  1,  2;
  1,  3;
  1,  4;
  1,  5;
  1,  3,  6;
  1,  7;
  1,  2,  8;
  1,  3,  9;
  1, 10;
  1, 11;
  1,  4,  6, 12;

Cf. A027750, A070824.

[[k: k in [ 1..n] | Denominator(n/k) eq 1 and Denominator((k^n+n^k)/(k+n)) eq 1]: n in [1..30]]

a[n_] := Select[ Divisors@ n, Mod[PowerMod[#, n, # + n] + PowerMod[n, #, # + n], # + n] == 0 &]; Array[a, 30] // Flatten (* Robert G. Wilson v, Aug 06 2018 *)

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A167131 Numbers k such that A002808(k) - A144925(k) is prime.

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A255429 Numbers with a prime number of nontrivial divisors.

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A306307 Numbers that are divisible by the number of their nontrivial divisors.

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A317746 Irregular triangle read by rows in which row n lists the divisors k of n such that k^n + n^k == 0 (mod k + n).

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Author

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Comments

Examples

Crossrefs

Programs

Magma

Mathematica