A000005 -id:A000005 - cp's OEIS frontend

A328956 Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 104, 106, 111, 112, 115, 116, 117

Offset: 1

Gus Wiseman, Nov 01 2019

nonn

First differs from A084227 in having 60.

			The sequence of terms together with their prime indices begins:
   6: {1,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  35: {3,4}
  38: {1,8}
  39: {2,6}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Zeros of A328958.
The complement is A328957.
Prime signature is A124010.
Omega-sequence is A323023.
omega(n) * Omega(n) is A113901(n).
(Omega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - omega(n) * Omega(n) is A328958(n).
sigma_0(n) - 2 - (Omega(n) - 1) * omega(n) is A328959(n).
Cf. A000040, A005117, A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.

Select[Range[100],DivisorSigma[0,#]==PrimeOmega[#]*PrimeNu[#]&]

is(k) = {my(f = factor(k)); numdiv(f) == omega(f) * bigomega(f);} \\ Amiram Eldar, Jul 28 2024

A000005(a(n)) = A001222(a(n)) * A001221(a(n)).

A328958 a(n) = d(n) - (omega(n) * bigomega(n)), where d (number of divisors) = A000005, omega = A001221, bigomega = A001222.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, -1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, -1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, -1, 1, 0, 0, -1, 1, 2, 1, 0, 0, 0, 0, -1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, -1, 1, 0, -1

Offset: 1

Gus Wiseman, Nov 02 2019

a(n) = sigma_0(n) - omega(n) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222. - The original name of the sequence.

			a(144) = sigma_0(144) - omega(144) * nu(144) = 15 - 6 * 2 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from exponents in factorization of n.

Positions of first appearances are A328962.
Zeros are A328956.
Nonzeros are A328957.
omega(n) * nu(n) is A113901(n).
(omega(n) - 1) * nu(n) is A307409(n).
sigma_0(n) - 2 - (omega(n) - 1) * nu(n) is A328959(n).
Cf. A000005, A001221, A001222, A112798, A124010, A323023, A328960, A328961, A328963, A328964.

Table[DivisorSigma[0,n]-PrimeOmega[n]*PrimeNu[n],{n,100}]

A328958(n) = (numdiv(n)-(omega(n)*bigomega(n))); \\ Antti Karttunen, Jan 27 2025

a(n) = A000005(n) - A001222(n) * A001221(n) = A000005(n) - A113901(n).

More terms added and the function names in the definition replaced with standard OEIS ones - Antti Karttunen, Jan 27 2025

A339304 Irregular triangle read by rows T(n,k) in which row n has length the partition number A000041(n-1) and columns k give the number of divisors function A000005, 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 29 2020

T(n,k) is also the number of divisors of A336811(n,k).

Conjecture: the sum of row n equals A138137(n), the total number of parts in the last section of the set of partitions of n.

			Triangle begins:
  1;
  2;
  2, 1;
  3, 2, 1;
  2, 2, 2, 1, 1;
  4, 3, 2, 2, 2, 1, 1;
  2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1;
  4, 4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1;
  3, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of the triangle, flattened)

Number of divisors of A336811.
Row n has length A000041(n-1).
Every column gives A000005.
Row sums give A138137 (conjectured).
Cf. A135010, A138121, A138879, A187219.

A339304row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[0,n-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]];Array[A339304row,10] (* Paolo Xausa, Sep 01 2023 *)

a(m) = A000005(A336811(m)).

T(n,k) = A000005(A336811(n,k)).

A229276 Composite squarefree numbers n such that p-tau(n) divides n+sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

Original entry on oeis.org

6, 10, 15, 66, 145, 231, 435, 1221, 11571, 99093, 105502, 292434, 449854, 585429, 643858, 968014, 1372494, 1787091, 1939434, 4659114, 5524014, 5654334, 6250371, 6974007, 19495374, 19821714, 28488039, 34701369, 46183893, 81133734, 213352233, 230140869

Offset: 1

Paolo P. Lava, Sep 19 2013

nonn

Subsequence of A120944.

			Prime factors of 435 are 3, 5, 29 and sigma(435) = 720, tau(435) = 8.
435 + 720 = 1155 and 1155 / (3 - 8) = -231, 1155 / (5 - 8) = -385, 1155 / (29 - 8) = 55.

Kevin P. Thompson, Table of n, a(n) for n = 1..62

Cf. A000005, A000203, A228299-A228302, A229273-A229275.

with (numtheory); P:=proc(q) local a, b, c, i, ok, p, n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 or a[i][1]=tau(n) then ok:=0; break;
else if not type((n+sigma(n))/(a[i][1]-tau(n)), integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(2*10^6);

a(21)-a(33) from Giovanni Resta, Sep 20 2013

First term deleted by Paolo P. Lava, Sep 23 2013

A253139 a(n) = lcm_{d|n} tau(d), where tau(d) represents the number of divisors of d (A000005(d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 12, 6, 4, 2, 12, 2, 4, 4, 60, 2, 12, 2, 12, 4, 4, 2, 24, 6, 4, 12, 12, 2, 8, 2, 60, 4, 4, 4, 36, 2, 4, 4, 24, 2, 8, 2, 12, 12, 4, 2, 120, 6, 12, 4, 12, 2, 24, 4, 24, 4, 4, 2, 24, 2, 4, 12, 420, 4, 8, 2, 12, 4, 8, 2, 72, 2, 4, 12, 12, 4, 8

Offset: 1

Matthew Vandermast, Dec 27 2014

A divisibility sequence (cf. Ward link and second formula).

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			The divisors of 20 are 1, 2, 4, 5, 10 and 20, which have 1, 2, 3, 2, 4 and 6 divisors respectively. The least common multiple of 1, 2, 3, 2, 4 and 6 is 12; therefore, a(20) = 12.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Morgan Ward, A note on divisibility sequences, Bull. Amer. Math. Soc., 45 (1939), 334-336.

A250270 gives range of values. A141586 lists numbers n such that a(n) divides n.

Cf. A003418, A253141.

Table[LCM@@DivisorSigma[0,Divisors[n]],{n,100}] (* Harvey P. Dale, Sep 01 2017 *)
lcm[n_] := lcm[n] = LCM @@ Range[n]; a[1] = 1; a[n_] := Times @@ (lcm [Last[#] + 1] & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

a(n) = my(d = divisors(n)); lcm(vector(#d, k, numdiv(d[k]))); \\ Michel Marcus, Jan 23 2015

If n = Product_ prime(i)^e(i), then a(n) = Product_ A003418(e(i)+1).

a(n) = Product_{d|n} A253141(d).

A262511 Numbers k for which there is exactly one solution to x - d(x) = k, where d(k) is the number of divisors of k (A000005). Positions of ones in A060990.

Original entry on oeis.org

2, 3, 4, 5, 9, 10, 12, 14, 15, 16, 18, 21, 23, 26, 30, 31, 32, 41, 42, 44, 45, 47, 53, 54, 59, 60, 61, 71, 72, 73, 76, 77, 80, 82, 83, 84, 86, 89, 90, 92, 93, 94, 95, 97, 99, 101, 104, 105, 106, 110, 115, 119, 121, 122, 127, 135, 139, 146, 148, 149, 151, 154, 158, 161, 169, 171, 173, 176, 177, 183, 186, 188, 189, 190, 191, 192, 194, 195, 199, 200, 202

Offset: 1

Antti Karttunen, Sep 25 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000005, A049820, A060990.
Cf. A262512 (gives the corresponding x).
Cf. A262510 (a subsequence).
Subsequence of A236562.

allocatemem(123456789);
uplim = 14414400 + 504; \\ = A002182(49) + A002183(49).
v060990 = vector(uplim);
for(n=3, uplim, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
uplim2 = 14414400;
n=0; k=1; while(n <= uplim2, if(1==A060990(n), write("b262511_big.txt", k, " ", n); k++); n++;);

;; With Antti Karttunen's IntSeq-library.
(define A262511 (ZERO-POS 1 1 (COMPOSE -1+ A060990)))

Other identities. For all n >= 1:

a(n) = A049820(A262512(n)).

A320779 Inverse Euler transform of the number of divisors function A000005.

Original entry on oeis.org

1, 1, 0, 0, -1, 1, -1, 0, 1, -1, 0, 1, -1, -1, 2, 1, -2, -2, 2, 3, -4, 0, 3, -3, 3, -2, -2, 2, 1, 7, -15, 0, 17, -11, -1, 0, 9, -4, -18, 26, -10, -10, 24, -17, -15, 21, 27, -42, -37, 69, 43, -113, -11, 149, -98, -24, 67, -57, 24, -53, 213, -243, -193, 704

Offset: 1

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n).

Alois P. Heinz, Table of n, a(n) for n = 1..5000
OEIS Wiki, Euler transform

Cf. A000005.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-SumOfDivisors(n, 0))):
seq(a(n), n = 1..64); # Peter Luschny, Nov 21 2022

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Table[DivisorSigma[0,n],{n,100}]]

from functools import lru_cache
from sympy import mobius, divisors, divisor_count
def A320779(n):
    @lru_cache(maxsize=None)
    def b(n): return divisor_count(n)
    @lru_cache(maxsize=None)
    def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
    return sum(mobius(d)*c(n//d) for d in divisors(n,generator=True))//n # Chai Wah Wu, Jul 15 2024

A365399 Length of the longest subsequence of 1, ..., n on which the number of divisors function tau A000005 is nondecreasing.

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Sep 03 2023

nonn

The sequence was inspired by A365339.

			The terms of the subsequences of A000005 are marked by '*'. They start:
  1*, 2,  2 , 3,  2, 4,  2, 4,  ... -> a(1) = 1
  1*, 2*, 2 , 3,  2, 4,  2, 4,  ... -> a(2) = 2
  1*, 2*, 2*, 3,  2, 4,  2, 4,  ... -> a(3) = 3
  1*, 2*, 2*, 3*, 2, 4,  2, 4,  ... -> a(4) = 4
  1*, 2*, 2*, 3*, 2, 4,  2, 4,  ... -> a(5) = 4
  1*, 2*, 2*, 3*, 2, 4*, 2, 4,  ... -> a(6) = 5
  1*, 2*, 2*, 3*, 2, 4*, 2, 4,  ... -> a(7) = 5
  1*, 2*, 2*, 3*, 2, 4*, 2, 4*, ... -> a(8) = 6
Example: a(2000000) = 450033.

Peter Luschny, Table of n, a(n) for n = 1..10000
Plot2, A365399 vs A365339.

Cf. A000005, A365339, A365398.

# Computes the first N terms of the sequence using function tau from A000005.
function LLS_list(seq, N)
    lst = zeros(Int64, N)
    dyn = zeros(Int64, N)
    for n in 1:N
        p = seq(n)
        nxt = dyn[p] + 1
        while p <= N && dyn[p] < nxt
            dyn[p] = nxt
            p += 1
        end
        lst[n] = dyn[n]
    end
    return lst
end
A365399List(N) = LLS_list(tau, N)
println(A365399List(69))

from bisect import bisect
from sympy import divisor_count
def A365399(n):
    plist, qlist, c = tuple(divisor_count(i) for i in range(1,n+1)), [0]*(n+1), 0
    for i in range(n):
        qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i
        c = max(c,a)
    return c # Chai Wah Wu, Sep 04 2023

a(n+1) - a(n) <= 1.

A067888 Numbers k such that tau(k+1) = tau(k-1) where tau(k) = A000005(k).

Original entry on oeis.org

4, 6, 7, 9, 12, 18, 19, 30, 34, 41, 42, 51, 55, 56, 60, 72, 86, 92, 94, 102, 103, 108, 124, 129, 137, 138, 142, 144, 150, 153, 160, 180, 183, 184, 185, 186, 192, 198, 199, 202, 204, 214, 216, 218, 220, 228, 231, 236, 240, 243, 244, 247, 248, 249, 266, 270, 282

Offset: 1

Benoit Cloitre, Mar 02 2002

If (p,p+2) are twin primes, then the composite number p+1 is in this sequence. The primes occurring in this sequence are listed in A067889. See A055574 for the analog with sigma instead of tau. - M. F. Hasler, Aug 06 2015

Giovanni Resta, Table of n, a(n) for n = 1..10000

Equals A062832 + 1. - Michel Marcus, Feb 11 2018

Cf. A000005, A055574, A067889.

Select[Range[300], Equal @@ DivisorSigma[0, # + {-1, 1}] &] (* Amiram Eldar, Jan 23 2025 *)

is_A067888(n)=n>1&&numdiv(n-1)==numdiv(n+1) \\ M. F. Hasler, Aug 06 2015

A088880 Number of different values of A000005(m) when A056239(m) is equal to n.

Original entry on oeis.org

1, 1, 2, 2, 5, 4, 8, 6, 12, 10, 16, 13, 25, 18, 28, 25, 40, 32, 51, 40, 62, 51, 76, 62, 99, 77, 112, 92, 138, 109, 165, 130, 189, 153, 220, 178, 267, 208, 292, 240, 347, 274, 397, 315, 445, 361, 512, 407, 591, 464, 647, 524, 746, 588, 830, 664, 928, 746, 1034

Offset: 0

Naohiro Nomoto, Nov 28 2003

Number of distinct values of Product_{k=1..n} (m(k,P)+1) where m(k,P) is multiplicity of part k in partition P, as P ranges over all partitions of n. - Vladeta Jovovic, May 24 2008

Alois P. Heinz, Table of n, a(n) for n = 0..222

Cf. A088314, A215366.

multipl := proc(P,p)
        local a;
        a := 0 ;
        for el in P do
                if el = p then
                        a := a+1 ;
                end if;
        end do;
        a ;
end proc:
A088880 := proc(n)
        local pro,pa,m,p;
        pro := {} ;
        for pa in combinat[partition](n) do
                m := 1 ;
                for p from 1 to n do
                        m := m*(1+multipl(pa,p)) ;
                end do:
                pro := pro union {m} ;
        end do:
        nops(pro) ;
end proc: # R. J. Mathar, Sep 27 2011
# second Maple program
b:= proc(n, i) option remember; `if`(n=0 or i<2, {n+1},
       {seq(map(p->p*(j+1), b(n-i*j, i-1))[], j=0..n/i)})
    end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 09 2012

b[n_, i_] := b[n, i] = If[n==0 || i<2, {n+1}, Table[b[n-i*j, i-1]*(j+1), {j, 0, n/i}] // Flatten // Union]; a[n_] := Length[b[n, n]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

cp's OEIS Frontend

A328956 Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

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A328958 a(n) = d(n) - (omega(n) * bigomega(n)), where d (number of divisors) = A000005, omega = A001221, bigomega = A001222.

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A339304 Irregular triangle read by rows T(n,k) in which row n has length the partition number A000041(n-1) and columns k give the number of divisors function A000005, 1 <= k <= n.

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A229276 Composite squarefree numbers n such that p-tau(n) divides n+sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

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A253139 a(n) = lcm_{d|n} tau(d), where tau(d) represents the number of divisors of d (A000005(d)).

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A262511 Numbers k for which there is exactly one solution to x - d(x) = k, where d(k) is the number of divisors of k (A000005). Positions of ones in A060990.

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A320779 Inverse Euler transform of the number of divisors function A000005.

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