A000430 -id:A000430 - cp's OEIS frontend

A085724 Numbers k such that 2^k - 1 is a semiprime (A001358).

4, 9, 11, 23, 37, 41, 49, 59, 67, 83, 97, 101, 103, 109, 131, 137, 139, 149, 167, 197, 199, 227, 241, 269, 271, 281, 293, 347, 373, 379, 421, 457, 487, 523, 727, 809, 881, 971, 983, 997, 1061, 1063

Offset: 1

Jason Earls, Jul 20 2003

Subsequence of A000430. Apart from 4, 9, and 49 composites in this sequence are greater than 1.9e7. - Charles R Greathouse IV, Jun 05 2013
1427 and 1487 are also terms. 1277 is the only remaining unknown below them. - Charles R Greathouse IV, Jun 05 2013
Among the known terms only 11, 23, 83 and 131 are in A002515, that is, they are the only known values for n such that (2^n - 1)/(2*n + 1) is prime. - Jianing Song, Jan 22 2019
Either a(n) is a prime, or the square of a Mersenne prime exponent. - M. F. Hasler, Jun 23 2025

			11 is a member because 2^11 - 1 = 23*89.

J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 56-60. ASIN: B002ACVZ6O [From Jason Earls, Nov 22 2009]
J. Earls, "Cole Semiprimes," Mathematical Bliss, Pleroma Publications, 2009, pages 56-60. ASIN: B002ACVZ6O [From Jason Earls, Nov 25 2009]

S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Mersenne Number
Eric Weisstein's World of Mathematics, Semiprime

Cf. A092558, A092559, A092561, A092562.

SemiPrimeQ[n_]:=(n>1) && (2==Plus@@(Transpose[FactorInteger[n]][[2]])); Select[Range[100],SemiPrimeQ[2^#-1]&] (Noe)
Select[Range[1100],PrimeOmega[2^#-1]==2&] (* Harvey P. Dale, Feb 18 2018 *)
Select[Range[250], Total[Last /@ FactorInteger[2^# - 1, 3]] == 2 &] (* Eric W. Weisstein, Jul 28 2022 *)

issemi(n)=bigomega(n)==2
is(n)=if(isprime(n), issemi(2^n-1), my(q); isprimepower(n,&q)==2 && ispseudoprime(2^q-1) && ispseudoprime((2^n-1)/(2^q-1))) \\ Charles R Greathouse IV, Jun 05 2013

More terms from Zak Seidov, Feb 27 2004
More terms from Cunningham project, Mar 23 2004
More terms from the Cunningham project sent by Robert G. Wilson v and T. D. Noe, Feb 22 2006
a(41)-a(42) from Charles R Greathouse IV, Jun 05 2013

A058080 Numbers whose product of divisors exceeds their square.

Original entry on oeis.org

12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 144, 147

Offset: 1

N. J. A. Sloane, Nov 24 2000

nonn

Numbers with five or more divisors. - Lekraj Beedassy, Sep 11 2003

Called multiplicatively abundant numbers by Chau (2004). - Amiram Eldar, Jun 29 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
William Chau, The tau, sigma, rho functions, and some related numbers, Pi Mu Epsilon Journal, Vol. 11, No. 10 (Spring 2004), pp. 519-534; entire issue.

Complement of A007964.

Cf. A000430, A007422, A007955.

Select[Range[150], #^(DivisorSigma[0, #]/2) > #^2 &] (* Amiram Eldar, Jun 29 2022 *)
Select[Range[200],Times@@Divisors[#]>#^2&] (* Harvey P. Dale, Oct 20 2024 *)

is(n)=numdiv(n)>4 \\ Charles R Greathouse IV, Sep 18 2015

from sympy import divisor_count
def ok(n): return divisor_count(n) > 4
print([k for k in range(148) if ok(k)]) # Michael S. Branicky, Dec 16 2021

The number of terms not exceeding x is N(x) ~ x*(1 - log(log(x))/log(x)) (Chau, 2004). - Amiram Eldar, Jun 29 2022

A347705 Number of factorizations of n with reverse-alternating product > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 1, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 8, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 3, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 12 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			The a(n) factorizations for n = 2, 6, 8, 12, 24, 30, 48, 60:
  2   6     8       12      24        30      48          60
      2*3   2*4     2*6     3*8       5*6     6*8         2*30
            2*2*2   3*4     4*6       2*15    2*24        3*20
                    2*2*3   2*12      3*10    3*16        4*15
                            2*2*6     2*3*5   4*12        5*12
                            2*3*4             2*3*8       6*10
                            2*2*2*3           2*4*6       2*5*6
                                              3*4*4       3*4*5
                                              2*2*12      2*2*15
                                              2*2*2*6     2*3*10
                                              2*2*3*4     2*2*3*5
                                              2*2*2*2*3

Positions of 1's are A000430.
The weak version (>= instead of >) is A001055, non-reverse A347456.
The non-reverse version is A339890, strict A347447.
The version for reverse-alternating product 1 is A347438.
Allowing any integer reciprocal alternating product gives A347439.
The even-length case is A347440, also the opposite reverse version.
Allowing any integer rev-alt product gives A347442, non-reverse A347437.
The version for partitions is A347449, non-reverse A347448.
A001055 counts factorizations (strict A045778, ordered A074206).
A038548 counts possible rev-alt products of factorizations, integer A046951.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A292886 counts knapsack factorizations, by sum A293627.
A347707 counts possible integer reverse-alternating products of partitions.
Cf. A028983, A119620, A339846, A347441, A347443, A347450, A347463, A347466.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
revaltprod[q_]:=Product[q[[-i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],revaltprod[#]>1&]],{n,100}]

a(n) = A001055(n) - A347438(n).

A080256 Sum of numbers of distinct and of all prime factors of n.

Original entry on oeis.org

0, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 6, 2, 6, 4, 4, 4, 6, 2, 4, 4, 6, 2, 6, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 7, 2, 4, 5, 7, 4, 6, 2, 5, 4, 6, 2, 7, 2, 4, 5, 5, 4, 6, 2, 7, 5, 4, 2, 7, 4, 4, 4, 6, 2, 7, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6, 2, 6, 2, 6, 6

Offset: 1

Reinhard Zumkeller, Feb 10 2003

a(n) = 2 iff n is prime, A000040; a(n) > 2 iff n is composite, A002808; a(n) <= 3 iff n is prime or square of prime, A000430; a(n) = 3 iff n is square of prime, A001248; a(A080257(n)) > 3;

a(n) <= 4 iff product of proper divisors <= n^2, A007964; a(n) = 4 iff n has four divisors, A030513; a(n) > 4 iff product of proper divisors > n^2, A058080; a(A064598(n)) <= 5; a(A080258(n)) = 5.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000040, A000430, A002808, A001248, A007964, A058080, A064598, A080257, A080258.

Cf. A001221, A001222, A046660, A077761, A083342.

f[n_] := Plus @@ (Last /@ FactorInteger[n] + 1); Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 03 2005 *)

a(n) = {my(f = factor(n)); omega(f) + bigomega(f);} \\ Amiram Eldar, Sep 28 2023

a(n) = Omega(n) + omega(n) = A001221(n) + A001222(n).
Additive with a(p^e) = e + 1.
Sum_{k=1..n} a(k) = 2 * n * log(log(n)) + c * n + O(n/log(n)), where c = A077761 + A083342 = 1.29615109474508069537... . - Amiram Eldar, Sep 28 2023

A074583 Numbers k such that sopfr(k) = S(k), where sopfr = A001414 and S = A002034.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Jason Earls, Aug 24 2002

These are the prime powers p^e with e <= p. - Reinhard Zumkeller, Dec 15 2003

Complement to A192135 with respect to A000961;

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

Subsequence of A000961; A000040, A000430, and A051674 are subsequences.

Cf. A095874, A192135, A192188, A343983.

import Data.Set (singleton, deleteFindMin, insert)
a074583 n = a074583_list !! (n-1)
a074583_list = 1 : f (singleton 2) a000040_list where
  f s ps'@(p:p':ps)
    | m == p      = p : f (insert (p*p) $ insert p' s') (p':ps)
    | m < spf^spf = m : f (insert (m*spf) s') ps'
    | otherwise   = m : f s' ps'
      where spf = a020639 m  -- smallest prime factor of m, cf. A020639
            (m, s') = deleteFindMin s
-- Simpler version:
a074583_list = map a000961 a192188_list
-- Reinhard Zumkeller, Jun 05 2011, Jun 26 2011

sopfr[n_] := Total[Times @@@ FactorInteger[n]];
S[n_] := Module[{m = 1}, While[!IntegerQ[m!/n], m++]; m];
Select[Range[1000], sopfr[#] == S[#]&] (* Jean-François Alcover, Nov 09 2017 *)

isok(n) = my(f=factor(n)); n==1 || (#f~==1 && f[1, 1]>=f[1, 2]); \\ Seiichi Manyama, May 07 2021

a(n) = A000961(A192188(n)); A095874(a(n)) = A192188(n). - Reinhard Zumkeller, Jun 26 2011

A084114 Number of divisions when calculating A084110(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 0, 4, 0, 1, 1, 1, 1, 2, 0, 3, 1, 1, 0, 4, 1, 1, 1, 2, 0, 4, 1, 1, 1, 1, 1, 4, 0, 1, 1, 2, 0, 2, 0, 2, 2

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = A000005(n) - 1 - A084113(n) = A032741(n) - A084113(n) = (A032741(n)-A084115(n))/2;

a(n) = 0 iff n is prime or a square of prime (A000430).

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

Cf. A080257.

Cf. A027750, A084110, A084113.

a084114 = g 0 1 . tail . a027750_row where
   g c _ []     = c
   g c x (d:ds) = if r > 0 then g c (x * d) ds else g (c + 1) x' ds
                  where (x', r) = divMod x d
-- Reinhard Zumkeller, Jul 31 2014

A088434 Number of ways to write n as n = uvw with 1 <= u < v < w.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 4, 0, 1, 1, 2, 0, 4, 0, 2, 1, 1, 1, 4, 0, 1, 1, 4, 0, 4, 0, 2, 2, 1, 0, 6, 0, 2, 1, 2, 0, 4, 1, 4, 1, 1, 0, 8, 0, 1, 2, 3, 1, 4, 0, 2, 1, 4, 0, 8, 0, 1, 2, 2, 1, 4, 0, 6, 1, 1, 0, 8, 1, 1, 1, 4, 0, 8, 1, 2, 1, 1, 1, 9, 0, 2, 2, 4, 0, 4, 0, 4, 4, 1, 0, 8, 0, 4, 1, 6, 0, 4, 1, 2, 2, 1, 1, 14

Offset: 1

Reinhard Zumkeller, Oct 01 2003

nonn

a(n)=0 iff n=1 or n prime or n prime^2: a(A000430(n)) = 0.

The integers a(n)+1 equal A045778(n) for n < 120 and differ at all n that admit factorization into 4 or more distinct factors, the smallest ones being n = 120 = 2*3*4*5, n = 144 = 2*3*4*6, n = 168 = 2*3*4*7, n = 180 = 2*3*5*6, ..., later continuing n = 312 = 2*3*4*13, n = 320 = 2*4*5*8, n = 324 = 2*3*6*9, n = 330 = 2*3*5*11, ... Coincidentally, A068350(5) to A068350(19) start this list. - R. J. Mathar, Jul 19 2007

			n=12: (1,2,6), (1,3,4): therefore a(12)=2;
n=18: (1,2,9), (1,3,6): therefore a(18)=2.

Antti Karttunen and Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 2048 terms from Antti Karttunen.)
Michael De Vlieger, Records and first positions of records in A088434.

Cf. A034836, A088432, A088433, A122179, A122180.

Table[Length[Select[Cases[Subsets[Divisors[n],{3}],{x_,y_,z_}->x*y*z],#==n &]],{n,102}] (* Jayanta Basu, May 23 2013 *)

A088434(n) = { my(s=0); fordiv(n, u, for(v=u+1, n-1, for(w=v+1, n, if(u*v*w==n, s++)))); (s); }; \\ Antti Karttunen, Aug 24 2017

Data section extended to 120 terms by Antti Karttunen, Aug 24 2017

A325251 Numbers whose omega-sequence covers an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 82, 83, 84, 85, 86

Offset: 1

Gus Wiseman, Apr 16 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

The enumeration of these partitions by sum is given by A325260.

			The sequence of terms together with their omega sequences begins:
   1:              31: 1             63: 3 2 2 1
   2: 1            33: 2 2 1         65: 2 2 1
   3: 1            34: 2 2 1         67: 1
   4: 2 1          35: 2 2 1         68: 3 2 2 1
   5: 1            37: 1             69: 2 2 1
   6: 2 2 1        38: 2 2 1         71: 1
   7: 1            39: 2 2 1         73: 1
   9: 2 1          41: 1             74: 2 2 1
  10: 2 2 1        43: 1             75: 3 2 2 1
  11: 1            44: 3 2 2 1       76: 3 2 2 1
  12: 3 2 2 1      45: 3 2 2 1       77: 2 2 1
  13: 1            46: 2 2 1         79: 1
  14: 2 2 1        47: 1             82: 2 2 1
  15: 2 2 1        49: 2 1           83: 1
  17: 1            50: 3 2 2 1       84: 4 3 2 2 1
  18: 3 2 2 1      51: 2 2 1         85: 2 2 1
  19: 1            52: 3 2 2 1       86: 2 2 1
  20: 3 2 2 1      53: 1             87: 2 2 1
  21: 2 2 1        55: 2 2 1         89: 1
  22: 2 2 1        57: 2 2 1         90: 4 3 2 2 1
  23: 1            58: 2 2 1         91: 2 2 1
  25: 2 1          59: 1             92: 3 2 2 1
  26: 2 2 1        60: 4 3 2 2 1     93: 2 2 1
  28: 3 2 2 1      61: 1             94: 2 2 1
  29: 1            62: 2 2 1         95: 2 2 1

Positions of normal numbers (A055932) in A325248.
Cf. A000430, A055932, A056239, A112798, A181819, A323023, A325247, A325260, A325261, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number).

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],normQ[omseq[#]]&]

A084190 Least common multiple of {d-1: d > 1 and d divides n}.

Original entry on oeis.org

1, 1, 2, 3, 4, 10, 6, 21, 8, 36, 10, 330, 12, 78, 28, 105, 16, 680, 18, 684, 60, 210, 22, 53130, 24, 300, 104, 702, 28, 36540, 30, 3255, 160, 528, 204, 157080, 36, 666, 228, 62244, 40, 31980, 42, 9030, 616, 990, 46, 2497110, 48, 3528, 400, 5100, 52, 468520

Offset: 1

Reinhard Zumkeller, May 18 2003

nonn

Considering the set of divisors > 1 of n reduced by 1, a(n) is the smallest number whose divisors contain this set;
a(n) < n iff n=p^k, p prime and 1 <= k <= 2: a(A001248(n)) < A001248(n), a(A000430(n)) < A000430(n), a(A080257(n))> A080257(n);
a(n) is odd iff n=2^k.

			n=35: divisors > 1 of 35 = {5,7,35}, a(35) = lcm(4,6,34) = 204;
n=37: divisors > 1 of 37 = {37}, a(37) = lcm(36) = 36.

Carl R. White, Table of n, a(n) for n = 1..10000

Cf. A084191(n) = a(a(n)), A007955.
Cf. A027750.
Cf. A258409.

a084190 1 = 1
a084190 n = foldl1 lcm $ map (subtract 1) $ tail $ a027750_row' n
-- Reinhard Zumkeller, May 08 2012

Join[{1}, Table[LCM @@ (Rest[Divisors[n]] - 1), {n, 2, 100}]] (* T. D. Noe, Apr 25 2012 *)

a(n)=if(n>2,lcm(apply(k->k-1,vecextract(divisors(n),"2.."))),1) \\ Charles R Greathouse IV, Apr 25 2012

from math import lcm
from sympy import divisors
def A084190(n): return lcm(*(d-1 for d in divisors(n,generator=True) if d > 1)) # Chai Wah Wu, Jun 25 2022

a(45) was erroneously split into 61 and 6; repaired by Carl R. White, Apr 25 2012

A088433 Number of ways to write n as n = uvw with 1<=u<=v

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 4, 2, 2, 1, 6, 1, 2, 2, 4, 1, 5, 1, 4, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 1, 3, 2, 4, 1, 6, 2, 6, 2, 2, 1, 10, 1, 2, 4, 5, 2, 5, 1, 4, 2, 5, 1, 11, 1, 2, 3, 4, 2, 5, 1, 9, 3, 2, 1, 10, 2, 2, 2, 6, 1, 10, 2, 4, 2, 2, 2, 12, 1, 3, 4, 6

Offset: 1

Reinhard Zumkeller, Oct 01 2003

nonn

a(n) = 1 iff n prime or n prime^2: a(A000430(n))=1.

			n=12: (1,1,12), (1,2,6), (1,3,4), (2,2,3): therefore a(12)=4;
n=18: (1,1,18), (1,2,9), (1,3,6): therefore a(18)=3.

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A034836, A088432, A088434, A122179, A122180.

A088433(n) = { my(s=0); fordiv(n, u, for(v=u, n-1, for(w=v+1, n, if(u*v*w==n, s++)))); (s); }; \\ Antti Karttunen, Aug 24 2017

cp's OEIS Frontend

A085724 Numbers k such that 2^k - 1 is a semiprime (A001358).

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A058080 Numbers whose product of divisors exceeds their square.

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A347705 Number of factorizations of n with reverse-alternating product > 1.

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A080256 Sum of numbers of distinct and of all prime factors of n.

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A074583 Numbers k such that sopfr(k) = S(k), where sopfr = A001414 and S = A002034.

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A084114 Number of divisions when calculating A084110(n).

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A088434 Number of ways to write n as n = u*v*w with 1 <= u < v < w.

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A088434 Number of ways to write n as n = uvw with 1 <= u < v < w.

A088433 Number of ways to write n as n = uvw with 1<=u<=v