A143207 -id:A143207 - cp's OEIS frontend

A174164 Numbers n such that 1 = abs(sum{p-1|p is prime and divisor of n} - product{p-1|p is prime and divisor of n}).

6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 68, 72, 74, 76, 80, 82, 86, 88, 90, 92, 94, 96, 98, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 134, 136, 142, 144, 146, 148, 150, 152, 158, 160, 162, 164, 166, 172

Offset: 1

Juri-Stepan Gerasimov, Mar 10 2010

nonn

			6 is a term because 6=2*3 and 1=abs((2-1)+(3-1)-(2-1)*(3-1)).
10 is a term because 10=2*5 and 1=abs((2-1)+(5-1)-(2-1)*(5-1)).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A055631, A173557.

Union of A100367 and A143207.

From R. J. Mathar, Apr 26 2010: (Start)
A055631 := proc(n) add(d-1, d= numtheory[factorset](n) ) ; end proc:
A173557 := proc(n) mul(d-1, d= numtheory[factorset](n) ) ; end proc:
isA174164 := proc(n) A055631(n)-A173557(n) ; abs(%) = 1 ; end proc:
for n from 2 to 200 do if isA174164(n) then printf("%d,",n) ; end if; end do: (End)

filterQ[n_] := With[{pp = FactorInteger[n][[All, 1]]}, 1 == Abs[Total[pp-1] - Times @@ (pp-1)]];
Select[Range[200], filterQ] (* Jean-François Alcover, Sep 17 2020 *)

Corrected (53 replaced by 52, 90 and 120 inserted) by R. J. Mathar, Apr 26 2010

A373559 Squares k such that rad(k) is a primorial number.

Original entry on oeis.org

1, 4, 16, 36, 64, 144, 256, 324, 576, 900, 1024, 1296, 2304, 2916, 3600, 4096, 5184, 8100, 9216, 11664, 14400, 16384, 20736, 22500, 26244, 32400, 36864, 44100, 46656, 57600, 65536, 72900, 82944, 90000, 104976, 129600, 147456, 176400, 186624, 202500, 230400, 236196, 262144

Offset: 1

David James Sycamore, Jun 09 2024

Squares k such that the squarefree kernel of k is primorial.
Intersection of A000290 and A055932.
1 is the only primorial term.
From Michael De Vlieger, Jun 09 2024: (Start)
Contains k^2 for k in each of A000079, A033845, A143207, A147571, A147572, etc.
Contains k^2 such that k is a product of primorials, i.e., A025487(i)^2, i >= 1, since A025487 is a proper subset of A055932.
(End)

			1 is a square, rad(1) = 1 = A002110(0).
4 is a square and rad(4) = 2 = A002110(1).
36 is a square and rad(36) = 6 = A002110(2).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000290, A002110, A007947, A055932,

{1}~Join~Select[Range[2, 512, 2], Or[# == {2}, Union@ Differences@ PrimePi[#] == {1}] &@ FactorInteger[#][[All, 1]] &]^2 (* Michael De Vlieger, Jun 09 2024 *)

a(n) = A055932(n)^2.

More terms from David A. Corneth, Jun 09 2024

A373737 a(n) is the smallest number k in the sorted sequence S(q) = {k : rad(k) = q}, q = A120944(n), such that tau(k) - A008479(k) is not positive, where rad = A007947 and tau = A000005.

Original entry on oeis.org

162, 250, 686, 1875, 7203, 2662, 4394, 750, 3993, 578, 12005, 722, 6591, 2058, 1058, 14739, 73205, 20577, 1682, 1922, 142805, 5346, 36501, 3430, 2738, 102487, 6318, 3362, 417605, 3698, 73167, 199927, 89373, 4418, 651605, 5202, 25725, 5618, 13310, 151959, 6498

Offset: 1

Michael De Vlieger, Jun 24 2024

Numbers k whose position i in S(n) is such that tau(k) <= i, i.e., that A372720(k) is not positive.
For k = p^m, m > 0, in S(p), p prime, tau(p^m) > A008479(p^m) since tau(p^m) = m + 1 and A008479(p^m) = m. Therefore we consider only composite squarefree q in this sequence.
a(n) is in A126706.
Conjecture: a(n) <= s*gpf(s)^floor(log_gpf(s) s^2), where gpf = A006530.

			a(1) = 162 since the 12th term in S(6) = A033845 = {6, 12, 18, 24, 36, 48, 54, ..., 162, ...} is the smallest k = S(6, i) such that tau(S(6, i)) <= i: tau(162) = 10 while i = 12.
a(2) = 250 since S(10, 9) = 250 gives tau(250) = 8, and 8 < 9.
a(3) = 686 since S(14, 10) = 686 is such that A372720(686) <= 0, etc.
Table of first and some notable terms:
       n        q     i         a(n) a(n)/q  A372720(a(n))
  --------------------------------------------------------
       1        6    12         162   3^3         -2
       2       10     9         250   5^2         -1
       3       14    10         686   7^2         -2
       4       15    11        1875   5^3         -1
       5       21    13        7203   7^3         -3
       6       22    12        2662   11^2        -4
       7       26    13        4394   13^2        -5
       8       30    16         750   5^2          0
      82      210    51       26250   5^3        -11
    1061     2310    99      635250   5^2 * 11    -3
   15013    30030   222    25375350   5 * 13^2   -30
  268015   510510   338   679488810   11^3       -18

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..268015 in blue, and A120944(n)^3 in red.
Michael De Vlieger, Diagram of S(6) = A033845 arranged such that k appears vertically in order of magnitude, with smallest at the bottom. Color function relates to A372720(k), with positive values from largest in light yellow grading to A372720(k) = 1 in orange, and negative values with smallest absolute value in dark blue to greatest in light blue. a(1) = 162 appears at right.
Michael De Vlieger, Diagram of S(30) = A143207 arranged such that k appears vertically in order of magnitude, with smallest at the bottom. Color function is as above, but with A372720(k) = 0 in red. a(2) = 750 appears at left in red.

Cf. A000005, A008479, A120944, A126706, A372720.

(* First, load function f from A162306 *)
Table[k = 1; s = f[n, n^3]; While[DivisorSigma[0, n*s[[k]]] - k > 0, k++]; s[[k]], {n, Select[Range[6, 120], And[SquareFreeQ[#], CompositeQ[#]] &]}]

A374284 a(1) = 1, a(2) = 2; a(n) = least k != a(m), m < n, such that omega(a(n-1)) >= omega(k) requires rad(k) | a(n-1), where omega = A001221 and rad = A007947.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 10, 5, 12, 8, 14, 7, 15, 25, 18, 16, 20, 30, 24, 27, 21, 42, 28, 32, 22, 11, 26, 13, 33, 60, 36, 48, 54, 64, 34, 17, 35, 49, 38, 19, 39, 66, 44, 70, 40, 50, 78, 52, 84, 56, 90, 45, 75, 81, 46, 23, 51, 102, 68, 105, 63, 110, 55, 114, 57, 120, 72

Offset: 1

Michael De Vlieger, Jul 31 2024

nonn

A greedy algorithm selects the smallest candidate k new to the sequence if either omega(k) > omega(a(n-1)) or if omega(k) <= omega(a(n-1)) and rad(k) | a(n-1).
Primes enter the sequence late.
For a(n) = p odd and prime and n <= 587, a(n-1) = 2*p. It seems for n > 1278, primes are not preceded by 2*p.
a(718) = 167 is followed by a(719) = 2*167. There are 13 primes that are followed by 2*p, with a(1364) = 293 apparently the largest. Normally, aside from p = 2 or 3, a(n) = p is such that a(n) and a(n+1) are coprime.
Prime powers a(n) = p^j follow a(n-1) = m*p, m >= 1. a(2, 3) = {2, 4}, a(5, 6) = {3, 9}, and these are the only primes immediately followed by their squares, since there are smaller candidates for a(n+1) for larger primes p. Only rarely is a(n+1) a multiple of p.
Let r be squarefree and let S_r = { k : rad(k) = r } = { m*r : rad(m) | r }. As consequence of the greedy algorithm and definition, k in S_r enter in order of magnitude. There are chains of consecutive terms in A033845 (e.g., a(31..33) = {36, 48, 54}) and A143207 (e.g., a(496..499) = {720, 750, 810, 900}).
Squarefree numbers k appear in trajectories according to omega(k).
Composite prime powers p^j, j > 1, appear relatively early, since rad(p^j) = p, and thus prime powers are minimally restricted after a(n-1) such that p | a(n-1).
Conjecture: this sequence is a permutation of natural numbers.

			a(3) = 4 since omega(2) = omega(3) = 1, but rad(3) does not divide 2, but omega(4) = omega(2) = 1 and rad(4) = 2.
a(4) = 6 since though both 3 and 5 have the same number of distinct prime factors but are pairwise coprime to 4, omega(6) > omega(4).
a(5) = 3 since omega(3) < omega(6) and 3 | 6.
a(6) = 9 since though 5, 7, and 8 have 1 distinct prime factor like 3, these are pairwise coprime to 3, while omega(9) = omega(3) = 1 and rad(9) = 3.
a(7) = 10 since though 5, 7, and 8 have 1 distinct prime factor like 9, these are pairwise coprime to 9, but omega(10) > omega(9).
a(8) = 5 since omega(5) < omega(10) and 5 | 10.
a(9) = 12 since though 7, 8, and 11 have 1 distinct prime factor like 5, these are pairwise coprime to 5, but omega(12) > omega(5), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple additionally represents powerful numbers that are not prime powers.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14 with a color function showing primes in red, a(n) with omega(a(n)) = 2 in yellow, a(n) with omega(a(n)) = 3 in green, etc.
Michael De Vlieger, Log log scatterplot of a(n)/n, n = 1..360000, showing fine striation detail associated with numbers that have a given prime signature. The sequence features a phase change near n = 9740.

Cf. A001221, A007947.

nn = 120; c[_] := False; a[1] = 1; j = r = a[2] = 2; c[1] = c[2] = True; u = 3;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
Do[k = u;
  While[Set[s, PrimeNu[k]];
    Or[c[k], If[s <= r, ! Divisible[j, rad[k]] ] ], k++];
  Set[{a[n], c[k], j, r}, {k, True, k, s}];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn]

A309944 Numbers m such that if m = p_1^e_1 * ... * p_k^e_k, where p_1 < ... < p_k are primes, then for all i < k, p_i = A000720(p_{i+1}).

Original entry on oeis.org

6, 12, 15, 18, 24, 30, 36, 45, 48, 54, 55, 60, 72, 75, 90, 96, 108, 119, 120, 135, 144, 150, 162, 165, 180, 192, 216, 225, 240, 270, 275, 288, 300, 324, 330, 341, 360, 375, 384, 405, 432, 450, 480, 486, 495, 533, 540, 576, 600, 605, 648, 660, 675, 720, 750, 768

Offset: 1

Michel Lagneau, Aug 24 2019

nonn

Numbers m such that for all k, d(k) = prime(d(k-1)), where d(k) is the k-th prime factor of m.
The primitive subsequence b(k), k = 1, 2, ... begins with 6, 15, 30, 55, 110, 165, 330, 341, 533, ... because if d(i) is the i-th prime factor of b(k), so b(k)*d(i)^m is in the sequence, m = 0, 1, 2, ...
Numbers m such that if m = p_1^e_1 * ... * p_k^e_k, p_1 < ... < p_k primes, then for all i > 1, p_i = A000040(p_{i-1}). - Antti Karttunen, Aug 24 2019

			330 is in the sequence because the prime factors are {2, 3, 5, 11} with 3 = prime(2), 5 = prime(3) and 11 = prime(5).
1299210 is in the sequence because the prime factors are {2, 3, 5, 11, 31, 127} with 3 = prime(2), 5 = prime(3), 11 = prime(5), 31 = prime(11) and 127 = prime(31).

Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A006450, A027746, A027748

Contains A033845, A033849, A143207.

sol:=[]; s:=1; for m in [2..1000] do v:=PrimeDivisors(m);  if #v ge 2 then nr:=0; for k in [2..#v] do  if v[k] eq NthPrime(v[k-1])  then nr:=nr+1;  end if; end for; if nr eq #v-1 then sol[s]:=m;s:=s+1; end if; end if; end for;  sol; // Marius A. Burtea, Aug 24 2019

with(numtheory):nn:=10^3:
for n from 1 to nn do:
d:=factorset(n):n0:=nops(d):it:=0:
  if n0>1
  then
  for i from 2 to n0 do :
   if d[i]=ithprime(d[i-1])
    then
    it:=it+1:
    else fi:
   od:
    if it=n0-1
    then
    printf(`%d, `,n):
    else fi:fi:
od:

aQ[n_] := (m = Length[(p = FactorInteger[n][[;; , 1]])]) > 1 && NestList[Prime@# &, p[[1]], m - 1] == p; Select[Range[770], aQ] (* Amiram Eldar, Aug 24 2019 *)

isok(m) = {my(f=factor(m)[,1]~); if (#f < 2, return(0)); for (i=2, #f, if (f[i] != prime(f[i-1]), return (0));); return (1);} \\ Michel Marcus, Aug 25 2019

Edited by N. J. A. Sloane, Oct 05 2019, using definition suggested by Antti Karttunen, Aug 24 2019

cp's OEIS Frontend

A174164 Numbers n such that 1 = abs(sum{p-1|p is prime and divisor of n} - product{p-1|p is prime and divisor of n}).

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A373559 Squares k such that rad(k) is a primorial number.

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A373737 a(n) is the smallest number k in the sorted sequence S(q) = {k : rad(k) = q}, q = A120944(n), such that tau(k) - A008479(k) is not positive, where rad = A007947 and tau = A000005.

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A374284 a(1) = 1, a(2) = 2; a(n) = least k != a(m), m < n, such that omega(a(n-1)) >= omega(k) requires rad(k) | a(n-1), where omega = A001221 and rad = A007947.

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A309944 Numbers m such that if m = p_1^e_1 * ... * p_k^e_k, where p_1 < ... < p_k are primes, then for all i < k, p_i = A000720(p_{i+1}).

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