A036234 -id:A036234 - cp's OEIS frontend

A158976 a(n) = sum of numbers k <= n such that not all proper divisors of k are divisors of n.

0, 0, 0, 0, 4, 0, 10, 6, 18, 23, 37, 10, 49, 45, 54, 66, 94, 75, 112, 90, 123, 149, 175, 120, 199, 220, 241, 251, 305, 236, 335, 307, 358, 396, 409, 385, 505, 501, 534, 499, 622, 568, 664, 630, 632, 749, 799, 688, 847, 857, 937, 959, 1049, 985, 1078, 1039, 1205

Offset: 1

Jaroslav Krizek, Apr 01 2009

nonn

For primes p, a(p) = A000217(p) - A158662(p) = A000217(p) - A014284(A036234(p)).

			For n = 7 we have the following proper divisors for k <= n: {1}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1}. Only 4 and 6 have proper divisors that are not divisors of 7, viz. 2 and 2, 3. Hence a(7) = 4 + 6 = 10.

Cf. A000040, A000217, A158662, A014284, A036234, A158974.

[ IsEmpty(S) select 0 else &+S where S is [ k: k in [1..n] | exists(t){ d: d in Divisors(k) | d ne k and d notin Divisors(n) } ]: n in [1..57] ];

Edited and extended by Klaus Brockhaus, Apr 06 2009

A159070 Count of numbers k in the range 1 < k <= n such that set of proper divisors of k is a subset of the set of proper divisors of n.

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 4, 6, 5, 6, 5, 10, 6, 8, 8, 9, 7, 11, 8, 12, 10, 10, 9, 16, 10, 11, 11, 13, 10, 17, 11, 15, 13, 13, 13, 19, 12, 14, 14, 19, 13, 19, 14, 18, 19, 16, 15, 24, 16, 19, 17, 19, 16, 22, 18, 23, 18, 18, 17, 28, 18, 20, 23, 23, 20, 24, 19, 23, 21, 26, 20, 31, 21, 23, 25, 25

Offset: 1

Jaroslav Krizek, Apr 04 2009

nonn

Here proper divisors include 1 but not the argument (k or n, respectively) in the divisor set, as defined in A032741.

			a(8) = 6 admits the following 6 k: 2 {1}, 3 {1}, 4 {1, 2}, 5 {1}, 7 {1}, 8 {1, 2, 4} with subsets of the proper divisors {1, 2, 4} for n = 8.

Cf. A158973, A000040, A036234, A000720.

a(n) = A158973(n) - 1.

If p = prime, element of A000040, a(p) = A158973(p) - 1 = A036234(p) - 1 = A000720(p).

Edited and extended by R. J. Mathar, Apr 06 2009

A159073 Sum of the k in the range 1

Original entry on oeis.org

0, 2, 5, 9, 10, 20, 17, 29, 26, 31, 28, 67, 41, 59, 65, 69, 58, 95, 77, 119, 107, 103, 100, 179, 125, 130, 136, 154, 129, 228, 160, 220, 202, 198, 220, 280, 197, 239, 245, 320, 238, 334, 281, 359, 402, 331, 328, 487, 377, 417, 388, 418, 381, 499, 461, 556, 447, 443, 440

Offset: 1

Jaroslav Krizek, Apr 04 2009

nonn

Here proper divisors include 1 but not the argument (k or n, respectively) in the divisor set, as defined in A032741.

Terms of the sum are counted in A159070.

			a(8) = 29 is the sum of the following six k: 2 {1}, 3 {1}, 4 {1, 2}, 5 {1}, 7 {1}, 8 {1, 2, 4} with subsets of the proper divisors {1, 2, 4} of n = 8. 2 + 3 + 4 + 5 + 7 + 8 = 29.

Cf.: A158975, A000040, A014284, A036234.

a(n) = A158975(n) - 1.

If p = prime, element of A000040, a(p) = A158662(p) - 1 = A014284(A036234(p)) - 1.

Edited and extended by R. J. Mathar, Apr 06 2009

A328879 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j) + 1), where pi = A000720.

Original entry on oeis.org

1, 2, 3, 2, 4, 6, 5, 2, 3, 8, 6, 6, 7, 10, 12, 2, 8, 6, 9, 8, 15, 12, 10, 6, 4, 14, 3, 10, 11, 24, 12, 2, 18, 16, 20, 6, 13, 18, 21, 8, 14, 30, 15, 12, 12, 20, 16, 6, 5, 8, 24, 14, 17, 6, 24, 10, 27, 22, 18, 24, 19, 24, 15, 2, 28, 36, 20, 16, 30, 40, 21, 6, 22, 26, 12

Offset: 1

Ilya Gutkovskiy, Oct 29 2019

a(n) is the product of indices of distinct prime factors of n if 1 is considered as a prime (see A008578).

			a(36) = 6 because 36 = 2^2 * 3^2 = prime(1)^2 * prime(2)^2 and (1 + 1) * (2 + 1) = 6.

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A007947, A008578, A036234, A048250, A064553, A156061.

a[n_] := Times @@ ((PrimePi[#[[1]]] + 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 75}]

a(n)={my(f=factor(n)[,1]); prod(i=1, #f, 1 + primepi(f[i]))} \\ Andrew Howroyd, Oct 29 2019

A143078 Triangle read by rows: row n (n >= 2) has length pi(n) (see A000720) and the k-th term gives the exponent of prime(k) in the prime factorization of n.

Original entry on oeis.org

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

Offset: 2

Roger L. Bagula and Gary W. Adamson, Oct 14 2008

If we suppress the 0's at the ends of the rows we get A067255. The number of 0's suppressed is A036234(n)-A061395(n)-1. - Jacques ALARDET, Jan 11 2012

Otherwise said, the number of suppressed (= trailing) 0's in row n is A000720(n)-A061395(n). - M. F. Hasler, Mar 10 2013

			Triangle begins
{1},
{0, 1},
{2, 0},
{0, 0, 1},
{1, 1, 0}, (the 6th row, and 6 = prime(1)*prime(2))
{0, 0, 0, 1},
{3, 0, 0, 0},
{0, 2, 0, 0},
{1, 0, 1, 0},
...

Cf. A000720, A001222, A067255.

Clear[t, T, n, m, k]; t[n_, m_, k_] := If[PrimeQ[FactorInteger[ n][[m]][[1]]] && FactorInteger[n][[m]][[1]] == Prime[k], FactorInteger[n][[m]][[2]], 0]; T = Table[Apply[Plus, Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], { m, 1, Length[FactorInteger[n]]}]], {n, 1, 10}]; Flatten[%]

my(r(n)=vector(primepi(n),i,valuation(n,prime(i)))); concat(vector(20,n,r(n))) \\ [M. F. Hasler, Mar 10 2013]

t(n,m,k)=If[PrimeQ[FactorInteger[n][[m]][[1]]] && FactorInteger[n][[m]][[ 1]] == Prime[k], FactorInteger[n][[m]][[2]], 0]; T(n,m)=vector_sum overk of t(n,m,k).

Edited by N. J. A. Sloane, Jan 12 2012

More terms from M. F. Hasler, Mar 10 2013

A158975 a(n) = sum of numbers k <= n such that all proper divisors of k are divisors of n.

Original entry on oeis.org

1, 3, 6, 10, 11, 21, 18, 30, 27, 32, 29, 68, 42, 60, 66, 70, 59, 96, 78, 120, 108, 104, 101, 180, 126, 131, 137, 155, 130, 229, 161, 221, 203, 199, 221, 281, 198, 240, 246, 321, 239, 335, 282, 360, 403, 332, 329, 488, 378, 418, 389, 419, 382, 500, 462, 557, 448

Offset: 1

Jaroslav Krizek, Apr 01 2009

nonn

For primes p, a(p) = A158662(p) = A014284(A036234(p)).

			For n = 8 we have the following proper divisors of k <= n: {1}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1}, {1, 2, 4}. Only k = 6 has a proper divisor that is not a divisor of 8, viz. 3. Hence a(8) = 1 + 2 + 3 + 4 + 5 + 7 + 8 = 30.

Cf. A000040, A158662, A014284, A036234, 158973.

[ &+[ k: k in [1..n] | forall(t){ d: d in Divisors(k) | d eq k or d in Divisors(n) } ]: n in [1..57] ];

Edited and extended by Klaus Brockhaus, Apr 06 2009

A159072 Count of numbers k in the range 1<=k<= n such that set of proper divisors of k is not a subset of the set of the proper divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 4, 4, 6, 2, 7, 6, 7, 7, 10, 7, 11, 8, 11, 12, 14, 8, 15, 15, 16, 15, 19, 13, 20, 17, 20, 21, 22, 17, 25, 24, 25, 21, 28, 23, 29, 26, 26, 30, 32, 24, 33, 31, 34, 33, 37, 32, 37, 33, 39, 40, 42, 32, 43, 42, 40, 41, 45, 42, 48, 45, 48, 44, 51, 41, 52, 51, 50, 51, 54

Offset: 1

Jaroslav Krizek, Apr 04 2009

nonn

Here proper divisors include 1 but not the argument (k or n, respectively) in the divisor set, as defined in A032741.

We use the (nonstandard) terminology that the empty set (the proper divisors of 1) is not a subset of another set.

			a(8) = 2 counts k=6 with divisors set {1, 2, 3} (not subset of the divisors {1, 2, 4} of n = 8), and k=1 without proper divisors.

Cf.: A158974, A000040, A036234, A000720.

a(n)+A159070(n) = n. - R. J. Mathar, Apr 06 2009

Edited and extended by R. J. Mathar, Apr 06 2009

A347403 Step at which n is removed by the sieve of Eratosthenes or 0 if n is prime.

Original entry on oeis.org

1, 0, 0, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 4, 2, 3, 2, 0, 2, 0, 2, 3, 2, 4, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 5, 2, 3, 2, 0, 2, 4, 2, 3, 2, 0, 2, 0, 2, 3, 2, 4, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 5, 2, 0, 2, 3, 2, 0, 2, 4, 2, 3, 2, 0, 2, 5, 2, 3, 2, 4, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 4, 2, 3, 2, 5, 2, 6, 2, 3, 2, 4, 2, 0

Offset: 1

Nicola De Mitri, Aug 30 2021

nonn

Here, 1 is removed by the sieve at step 1 (a(1) = 1); all even numbers greater than 2 are removed at step 2 (a(2*n) = 2 for n>1); all multiples of 3 greater than 3, that have not been already removed (i.e., all odd multiples of 3), are removed at step 3; and so on (each time removing multiples of the next number not yet removed). Prime numbers are never removed and are assigned the default value of 0.

For all primes p, and k > 1: a(p^k) = PrimePi(p)+1 and a(i) < PrimePi(p)+1 for i < p^2.

Nicola De Mitri, Table of n, a(n) for n = 1..15000

Cf. A036234, A063918, A055396.

{1}~Join~Array[If[PrimeQ[#], 0, PrimePi@ FactorInteger[#][[-1, 1]]] &, 104, 2] (* Michael De Vlieger, Sep 01 2021 *)

UPPER = 1000
number_to_step = [float("NaN"), 1] + [0 for _ in range(2, UPPER+1)]
curstep = 1
for sieve_val in range(2, int(UPPER**.5) + 1):
    if number_to_step[sieve_val]:
        continue
    curstep += 1
    for j in range(2*sieve_val, UPPER + 1, sieve_val):
        if not number_to_step[j]:
            number_to_step[j] = curstep
def A347403(n):
    return number_to_step[n]

from sympy import isprime, primepi, primefactors
def a(n):
    return 0 if isprime(n) else primepi(min(primefactors(n), default=0)) + 1
print([a(n) for n in range(1, 128)]) # Michael S. Branicky, Aug 31 2021

a(n) = f(A063918(n)) where f(0) = 0 and f(m) = A036234(m) for m > 0.

a(n) = 0 if n is prime, else A000720(A020639(n))+1. - Michael S. Branicky, Aug 31 2021

A159074 Sum of the k in the range 1<=k<=n such that set of proper divisors of k is not a subset of the set of proper divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 11, 7, 19, 24, 38, 11, 50, 46, 55, 67, 95, 76, 113, 91, 124, 150, 176, 121, 200, 221, 242, 252, 306, 237

Offset: 1

Jaroslav Krizek, Apr 04 2009

nonn

The nomenclature of A159072 applies, where the terms in that sum are counted.

			a(8) = 7 adds k = 6, where {1, 2, 3} is not a subset of the divisor set {1, 2, 4} of n = 8, and k = 1, with an empty proper divisor set.

Cf.: A158976, A000217, A000040, A014284, A036234.

a(n) = A158976(n) + 1.
If p = prime, a(p) = A000217(p) - A158662(p) + 1 = A000217(p) - A014284[A036234(p)] + 1.
a(n)+A159073(n)=A000217(n). - R. J. Mathar, Apr 06 2009

Edited by R. J. Mathar, Apr 06 2009

A229945 Triangle read by rows in which row n lists the union of the primes <= n and the divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 5, 1, 2, 3, 5, 6, 1, 2, 3, 5, 7, 1, 2, 3, 4, 5, 7, 8, 1, 2, 3, 5, 7, 9, 1, 2, 3, 5, 7, 10, 1, 2, 3, 5, 7, 11, 1, 2, 3, 4, 5, 6, 7, 11, 12, 1, 2, 3, 5, 7, 11, 13, 1, 2, 3, 5, 7, 11, 13, 14, 1, 2, 3, 5, 7, 11, 13, 15, 1, 2, 3, 4, 5, 7, 8, 11, 13, 16

Offset: 1

Omar E. Pol, Nov 04 2013

Also row n lists the divisors of n and the primes < n that do not divide n, in increasing order.
Also row n lists the nonprime divisors of n and the primes <= n, in increasing order.
Note that if n is 1 or prime then row n lists the first A036234(n) terms of A008578.
The motivation for this sequence is A046022 which is also the union of the odd primes and the divisors of 4. Here the n-th row of triangle can be interpreted as the initial terms of the infinite sequence defined as the union of the prime numbers and the divisors of n.

			For n = 10, the divisors of 10 are 1, 2, 5, 10. The primes less than 10 that do not divide 10 are 3 and 7. So row 10 is 1, 2, 3, 5, 7, 10.
On the other hand, the primes <= n are 2, 3, 5, 7. The nonprime divisors of n are 1, 10. So row 10 is 1, 2, 3, 5, 7, 10.
Written as an irregular triangle the sequence begins:
1;
1, 2;
1, 2, 3;
1, 2, 3, 4;
1, 2, 3, 5;
1, 2, 3, 5, 6;
1, 2, 3, 5, 7;
1, 2, 3, 4, 5, 7, 8;
1, 2, 3, 5, 7, 9;
1, 2, 3, 5, 7, 10;
1, 2, 3, 5, 7, 11;
1, 2, 3, 4, 5, 6, 7, 11, 12;
1, 2, 3, 5, 7, 11, 13;
1, 2, 3, 5, 7, 11, 13, 14;
1, 2, 3, 5, 7, 11, 13, 15;
1, 2, 3, 4, 5, 7, 8, 11, 13, 16;
1, 2, 3, 5, 7, 11, 13, 17;
1, 2, 3, 5, 6, 7, 9, 11, 13, 17, 18;
1, 2, 3, 5, 7, 11, 13, 17, 19;
1, 2, 3, 4, 5, 7, 10, 11, 13, 17, 19, 20;
1, 2, 3, 5, 7, 11, 13, 17, 19, 21;
1, 2, 3, 5, 7, 11, 13, 17, 19, 22;
1, 2, 3, 5, 7, 11, 13, 17, 19, 23;
1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 17, 19, 23, 24;

Columns 1-3: A000012, A007395, A010701.
Right border gives A000027.
Cf. A000005, A000040, A000720, A008578, A027750, A036234, A046022.

cp's OEIS Frontend

A158976 a(n) = sum of numbers k <= n such that not all proper divisors of k are divisors of n.

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A159070 Count of numbers k in the range 1 < k <= n such that set of proper divisors of k is a subset of the set of proper divisors of n.

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A159073 Sum of the k in the range 1

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A328879 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j) + 1), where pi = A000720.

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A143078 Triangle read by rows: row n (n >= 2) has length pi(n) (see A000720) and the k-th term gives the exponent of prime(k) in the prime factorization of n.

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A158975 a(n) = sum of numbers k <= n such that all proper divisors of k are divisors of n.

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Magma

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A159072 Count of numbers k in the range 1<=k<= n such that set of proper divisors of k is not a subset of the set of the proper divisors of n.

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A347403 Step at which n is removed by the sieve of Eratosthenes or 0 if n is prime.

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