A001221 -id:A001221 - cp's OEIS frontend

A307409 a(n) = (A001222(n) - 1)*A001221(n).

0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 3, 0, 4, 0, 4, 2, 2, 0, 6, 1, 2, 2, 4, 0, 6, 0, 4, 2, 2, 2, 6, 0, 2, 2, 6, 0, 6, 0, 4, 4, 2, 0, 8, 1, 4, 2, 4, 0, 6, 2, 6, 2, 2, 0, 9, 0, 2, 4, 5, 2, 6, 0, 4, 2, 6, 0, 8, 0, 2, 4, 4, 2, 6, 0, 8, 3, 2, 0, 9, 2, 2, 2, 6, 0, 9, 2, 4, 2, 2, 2, 10, 0, 4, 4, 6, 0, 6, 0, 6

Offset: 1

Mats Granvik, Apr 07 2019

nonn

a(n) + 2 appears to differ from A000005 at n=1 and when n is a term of A320632. Verified up to n=3000.
If A320632 contains the numbers such that A001222(n) - A051903(n) > 1, then this sequence contains precisely the numbers p^k and p^k*q for distinct primes p and q. The comment follows, since d(p^k) = k+1 = (k-1)*1 + 2 and d(p^k*q) = 2k+2 = ((k+1)-1)*2 + 2. - Charlie Neder, May 14 2019
Positions of first appearances are A328965. - Gus Wiseman, Nov 05 2019
Regarding Neder's comment above, see also my comments in A322437. - Antti Karttunen, Feb 17 2021

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Two less than A307408.
A113901(n) is bigomega(n) * omega(n).
A328958(n) is sigma_0(n) - bigomega(n) * omega(n).
Cf. A000005, A001221, A001222, A060687, A070175, A071625, A113901, A124010, A303555, A320632, A323023, A328956, A328957, A328964, A328965, A322437.

a[n_] := (PrimeOmega[n] - 1)*PrimeNu[n];
aa = Table[a[n], {n, 1, 104}];

a(n) = (bigomega(n) - 1)*omega(n); \\ Michel Marcus, May 15 2019

a(n) = (A001222(n) - 1)*A001221(n).
a(n) = binomial(A001222(n) - 1, 1)*binomial(A001221(n), 1).
a(n) = A307408(n) - 2.

A146289 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001221(n)), giving the number of divisors of n with m distinct prime factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 3, 2, 1, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 3, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 4, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 3, 1, 1, 1, 3, 3

Offset: 1

Matthew Vandermast, Nov 11 2008

The formula used in obtaining the n-th row (see below) also gives the number of divisors of the k-th power of n.
Two numbers have identical rows in the table if and only if they have the same prime signature.
T(n,0)=1.

			Rows begin:
1;
1,1;
1,1;
1,2;
1,1;
1,2,1;
1,1;
1,3;
1,2;
1,2,1;
...
12 has 1 divisor with 0 distinct prime factors (1); 3 with 1 (2, 3 and 4); and 2 with 2 (6 and 12), for a total of 6. The 12th row of the table therefore reads (1, 3, 2). These are the positive coefficients of the polynomial equation 1 + 3k + 2k^2 = (1 + 2k)(1 + k), derived from the prime factorization of 12 (namely, 2^2*3^1).

G. C. Greubel, Table of n, a(n) for the first 500 rows, flattened
Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Distinct Prime Factors
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

Row sums equal A000005(n).
T(n, 1) = A001222(n) for n>1. T(n, A001221(n)) = A005361(n).
Row n of A007318 is identical to row A002110(n) of this table and also identical to the row for any squarefree number with n prime factors.
Cf. A146290. Also cf. A146291, A146292.

f:= proc(n)
   local F,G,f,t,k;
   F:= ifactors(n)[2];
   G:= mul(1+f[2]*t, f= F);
   seq(coeff(G,t,k),k=0..nops(F));
end proc:
seq(f(n),n=1..100); # Robert Israel, Feb 10 2015

Join[{{1}}, Table[nn = DivisorSigma[0, n];CoefficientList[Series[Product[1 + i x, {i, FactorInteger[n][[All, 2]]}], {x, 0,nn}], x], {n, 2, 100}]] // Grid (* Geoffrey Critzer, Feb 09 2015 *)

tabf(nn) = {for (n=1, nn, vd = divisors(n); vo = vector(#vd, k, omega(vd[k])); for (k=0, vecmax(vo), print1(#select(x->x==k, vo), ", ");); print(););} \\ Michel Marcus, Apr 22 2017

If the canonical factorization of n into prime powers is Product p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (1 + e(p) k).

A181591 a(n) = binomial(bigOmega(n),omega(n)), where omega = A001221 and bigOmega = A001222.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 6, 2, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 6, 1, 1, 1, 6, 1, 1, 1, 3, 3, 1, 1, 10, 2, 3, 1, 3, 1, 6, 1, 6, 1, 1, 1, 4, 1, 1, 3, 6, 1, 1, 1, 3, 1, 1, 1, 10, 1, 1, 3, 3, 1, 1, 1, 10, 4, 1, 1, 4, 1, 1, 1, 6, 1, 4, 1, 3, 1, 1, 1, 15, 1, 3, 3, 6

Offset: 1

Reinhard Zumkeller, Nov 01 2010

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A001221, A001222, A007318, A136277.

Cf. A088529, A136565 (equal for n = 2..23).

a[n_] := Binomial[PrimeOmega[n], PrimeNu[n]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 29 2013 *)

a(n) = binomial(bigomega(n), omega(n)); \\ Michel Marcus, Jul 10 2022

a(n) = A007318(A001222(n),A001221(n)).
a(n) = A088529(n) = A136565(n) for 2 <= n < 24.
a(n) = A136277(A156552(n)). - Antti Karttunen, May 29 2017

A074816 a(n) = 3^A001221(n) = 3^omega(n).

Original entry on oeis.org

1, 3, 3, 3, 3, 9, 3, 3, 3, 9, 3, 9, 3, 9, 9, 3, 3, 9, 3, 9, 9, 9, 3, 9, 3, 9, 3, 9, 3, 27, 3, 3, 9, 9, 9, 9, 3, 9, 9, 9, 3, 27, 3, 9, 9, 9, 3, 9, 3, 9, 9, 9, 3, 9, 9, 9, 9, 9, 3, 27, 3, 9, 9, 3, 9, 27, 3, 9, 9, 27, 3, 9, 3, 9, 9, 9, 9, 27, 3, 9, 3, 9, 3, 27, 9, 9, 9, 9, 3, 27, 9, 9, 9, 9, 9, 9, 3, 9, 9, 9

Offset: 1

Benoit Cloitre, Sep 08 2002

Old name was: a(n) = sum(d|n, tau(d)*mu(d)^2 ).
Terms are powers of 3.
The inverse Mobius transform of A074823, as the Dirichlet g.f. is product_{primes p} (1+2*p^(-s)) and the Dirichlet g.f. of A074816 is product_{primes p} (1+2*p^(-s))/(1-p^(-s)). - R. J. Mathar, Feb 09 2011
If n is squarefree, then a(n) = #{(x, y) : x, y positive integers, lcm (x, y) = n}. See Crandall & Pomerance. - Michel Marcus, Mar 23 2016

Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 2.3 p. 108.

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

Cf. A001221, A034444, A069212, A124508.

A074816[n_]:=3^PrimeNu[n]; (* Enrique Pérez Herrero, Jun 28 2010 *)

a(n) = 3^omega(n); \\ Michel Marcus, Mar 23 2016

for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Feb 16 2022

a(n) = 3^m if n is divisible by m distinct primes. i.e., a(n)=3 if n is in A000961; a(n)=9 if n is in A007774 ...
a(n) = 3^A001221(n) = 3^omega(n). Multiplicative with a(p^e)=3. - Vladeta Jovovic, Sep 09 2002.
a(n) = tau_3(rad(n)) = A007425(A007947(n)). - Enrique Pérez Herrero, Jun 24 2010
a(n) = abs(Sum_{d|n} A000005(d^3)*mu(d)). - Enrique Pérez Herrero, Jun 28 2010
a(n) = Sum_{d|n, gcd(d, n/d) = 1} 2^omega(d) (The total number of unitary divisors of the unitary divisors of n). - Amiram Eldar, May 29 2020, Dec 27 2024
a(n) = Sum_{d1|n, d2|n} mu(d1*d2)^2. - Wesley Ivan Hurt, Feb 04 2022
Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Feb 16 2022

Simpler definition at the suggestion of Michel Marcus. - N. J. A. Sloane, Mar 25 2016

A319899 Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 23, 26, 31, 33, 35, 39, 41, 51, 53, 55, 58, 59, 65, 67, 69, 74, 77, 83, 85, 86, 87, 91, 93, 94, 95, 97, 103, 109, 111, 119, 122, 123, 127, 129, 131, 142, 146, 155, 157, 158, 161, 165, 169, 177, 178, 179, 183, 185, 187, 191, 201, 202

Offset: 1

Gus Wiseman, Dec 17 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of square multiset multisystems, meaning the number of edges is equal to the number of distinct vertices.

			The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
  11: {{3}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  26: {{},{1,2}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  39: {{1},{1,2}}
  41: {{6}}
  51: {{1},{4}}
  53: {{1,1,1,1}}
  55: {{2},{3}}
  58: {{},{1,3}}
  59: {{7}}
  65: {{2},{1,2}}
  67: {{8}}
  69: {{1},{2,2}}
  74: {{},{1,1,2}}

Cf. A003963, A057151, A064573, A120732, A319616, A319877, A320325, A322527, A322530.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeOmega[#]==PrimeNu[Times@@primeMS[#]]&]

A212168 Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n)).

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140, 141, 142, 143

Offset: 1

Matthew Vandermast, May 22 2012

nonn

A225230(a(n)) > 1; A050326(a(n)) > 1. - Reinhard Zumkeller, May 03 2013

Subsequence of A130092. - Ivan N. Ianakiev, Sep 17 2019

			10 = 2^1*5^1 has 2 distinct prime factors, hence 2 positive exponents in its prime factorization (although the 1s are often left implicit). 2 is larger than the maximal exponent in 10's prime factorization, which is 1. Therefore, 10 belongs to the sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages)

Complement of A212165. See also A212164, A212166-A212167.

Subsequence of A188654.

import Data.List (findIndices)
a212168 n = a212168_list !! (n-1)
a212168_list = map (+ 1) $ findIndices (> 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] < Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)
Select[Range[200],Max[FactorInteger[#][[All,2]]]Harvey P. Dale, Nov 21 2018 *)

is(n,f=factor(n))=my(e=f[,2]); #e && vecmax(e)<#e \\ Charles R Greathouse IV, Jan 09 2022

A261256 Let S_k denote the sequence of numbers j such that A001222(j) - A001221(j) = k. Then a(n) is the n-th term of S_n.

Original entry on oeis.org

4, 24, 72, 160, 432, 896, 2592, 5632, 12800, 26624, 61440, 124416, 278528, 622592, 1376256, 2949120, 5971968, 12058624, 25690112, 60817408, 130023424, 262144000, 528482304, 1107296256, 2264924160, 4586471424, 9395240960, 19864223744, 40265318400, 83751862272

Offset: 1

Carlos Eduardo Olivieri, Aug 12 2015

nonn

S_0 would correspond to the squarefree numbers (A005117), that is, numbers j such that A001222(j) = A001221(j). Note that S_0 is excluded from the scheme. - Michel Marcus, Sep 21 2015

			For n = 1, S_1 = {4, 9, 12, 18, 20, 25, ...}, so a(1) = S_1(1) = 4.
For n = 2, S_2 = {8, 24, 27, 36, 40, 54, ...}, so a(2) = S_2(2) = 24.
For n = 3, S_3 = {16, 48, 72, 80, 81, 108, ...}, so a(3) = S_3(3) = 72.
For n = 4, S_4 = {32, 96, 144, 160, 216, 224, ...}, so a(4) = S_4(4) = 160.
For n = 5, S_5 = {64, 192, 288, 320, 432, 448, ...}, so a(5) = S_5(5) = 432.

Charlie Neder, Table of n, a(n) for n = 1..500

Cf. A001221, A001222.
Cf. A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093.
Cf. A046660, A257851, A264959.

a261256 n = a257851 n (n - 1)  -- Reinhard Zumkeller, Nov 29 2015

OutSeq = {}; For[i = 1, i <= 16, i++, l = Select[Range[10^2*2^i], PrimeOmega[#] - PrimeNu[#] == i &]; AppendTo[OutSeq, l[[i]]]]; OutSeq

a(n) = {ik = 1; nbk = 0; while (nbk != n, ik++; if (bigomega(ik) == omega(ik) + n, nbk++);); ik;} \\ Michel Marcus, Oct 06 2015

a(n+1) > 2*a(n).
a(n) >= 2^prime(n) for n < 5.
a(n) = A257851(n,n-1). - Reinhard Zumkeller, Nov 29 2015
a(n) = b(n)*2^(n+1), where b(n) consists of the values of k/2^excess(k) over odd k, sorted in ascending order. In particular, a(n) <= prime(n)*2^(n+1), with equality only when n = 2. - Charlie Neder, Jan 31 2019

a(17)-a(21) from Jon E. Schoenfield, Sep 12 2015

More terms from Charlie Neder, Jan 31 2019

A293548 Expansion of Product_{k>=2} 1/(1 - x^k)^omega(k), where omega(k) is the number of distinct primes dividing k (A001221).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 4, 8, 9, 15, 16, 28, 29, 46, 54, 77, 90, 131, 150, 211, 251, 337, 401, 540, 637, 839, 1006, 1296, 1551, 1995, 2373, 3013, 3610, 4523, 5410, 6754, 8045, 9965, 11897, 14614, 17410, 21313, 25316, 30816, 36615, 44307, 52539, 63387, 74975, 90078

Offset: 0

Ilya Gutkovskiy, Oct 11 2017

nonn

Euler transform of A001221.

N. J. A. Sloane, Transforms

Cf. A001221, A006171, A293549.

nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^PrimeNu[k], {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d PrimeNu[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]

G.f.: Product_{k>=2} 1/(1 - x^k)^b(k), where b(k) = [x^k] Sum_{j>=1} x^prime(j)/(1 - x^prime(j)).

a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} a(n-k)*b(k), b(k) = Sum_{d|k} d*omega(d).

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

Original entry on oeis.org

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

cp's OEIS Frontend

A307409 a(n) = (A001222(n) - 1)*A001221(n).

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A146289 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001221(n)), giving the number of divisors of n with m distinct prime factors.

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A181591 a(n) = binomial(bigOmega(n),omega(n)), where omega = A001221 and bigOmega = A001222.

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A074816 a(n) = 3^A001221(n) = 3^omega(n).

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A319899 Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).

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A212168 Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n)).

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Haskell

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A261256 Let S_k denote the sequence of numbers j such that A001222(j) - A001221(j) = k. Then a(n) is the n-th term of S_n.

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