A087207 -id:A087207 - cp's OEIS frontend

A351560 a(n) is a binary representation of the primes that divide sigma(n) [the sum of divisors of n function], shown in decimal.

0, 2, 1, 8, 3, 3, 1, 6, 32, 3, 3, 9, 9, 3, 3, 1024, 3, 34, 5, 11, 1, 3, 3, 7, 1024, 11, 5, 9, 7, 3, 1, 10, 3, 3, 3, 40, 129, 7, 9, 7, 11, 3, 17, 11, 35, 3, 3, 1025, 130, 1026, 3, 9, 3, 7, 3, 7, 5, 7, 7, 11, 1025, 3, 33, 1073741824, 11, 3, 65, 11, 3, 3, 3, 38, 2049, 131, 1025, 13, 3, 11, 5, 1027, 16, 11, 11, 9, 3, 19

Offset: 1

Antti Karttunen, Feb 19 2022

nonn

This is not additive sequence, but "oritive": For all coprime x, y (with gcd(x,y)=1), a(x*y) = a(x) OR a(y), where OR is bitwise-or (A003986). Compare also with A080398.

Cf. A000203, A003986, A007947, A048675, A080398, A087207, A351559 [= n AND a(n)].

{0}~Join~Array[Total[2^(PrimePi[#] - 1) & /@ FactorInteger[DivisorSigma[1, #]][[All, 1]]] &, 85, 2] (* Michael De Vlieger, Feb 20 2022 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A351560(n) = A048675(A007947(sigma(n)));

a(n) = A087207(A000203(n)) = A048675(A080398(n)) = A048675(A007947(A000203(n))).

A361111 The binary expansion of a(n) specifies which primes divide A360519(n).

Original entry on oeis.org

0, 3, 5, 12, 10, 3, 5, 20, 18, 3, 9, 24, 18, 6, 5, 17, 48, 34, 3, 9, 40, 36, 7, 65, 72, 10, 3, 33, 96, 66, 11, 129, 132, 6, 3, 17, 80, 68, 5, 257, 258, 130, 129, 33, 34, 6, 13, 513, 514, 1026, 1025, 9, 14, 2050, 2049, 65, 66, 4098, 4097, 5, 260, 264, 11, 7

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 03 2023

			A360519(6) = 12, which is divisible by 2, 3, but not 5, 7, 11, ... So we write down 1, 1, 0, 0, 0, .... Thus a(6) has binary expansion ...00011, and so a(6) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
N. J. A. Sloane, Table showing A360519(1)-A360519(13), also the smallest missing number (smn, A361109 and A361110), binary vectors showing which terms are divisible by the primes 2, 3, 5, 7, 11; and phi, a decimal representation of those binary vectors (A361111). This sequence forms the bottom row of the table.

Cf. A087207, A360519.

PARI
```
See Links section.
```

a(n) = A087207(A360519(n)). - Rémy Sigrist, Mar 03 2023

More terms from Rémy Sigrist, Mar 03 2023

A276010 a(0) = 0, for n >= 1, a(n) = A275736(n) OR a(A257684(n)), where OR is given by A003986.

Original entry on oeis.org

0, 1, 2, 3, 1, 1, 4, 5, 6, 7, 5, 5, 2, 3, 2, 3, 3, 3, 1, 1, 3, 3, 1, 1, 8, 9, 10, 11, 9, 9, 12, 13, 14, 15, 13, 13, 10, 11, 10, 11, 11, 11, 9, 9, 11, 11, 9, 9, 4, 5, 6, 7, 5, 5, 4, 5, 6, 7, 5, 5, 6, 7, 6, 7, 7, 7, 5, 5, 7, 7, 5, 5, 2, 3, 2, 3, 3, 3, 6, 7, 6, 7, 7, 7, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 3, 3, 1, 1, 5, 5, 7, 7, 5, 5, 3

Offset: 0

Antti Karttunen, Aug 17 2016

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A003986, A087207, A257684, A275734.

Cf. also A000120, A060502, A275736, A275728, A275808.

a(0) = 0, for n >= 1, a(n) = A275736(n) OR a(A257684(n)), where OR is given by A003986.
a(n) = A087207(A275734(n)).
Other identities. For all n >= 1:
A000120(a(n)) = A060502(n).

A324193 a(1) = 0; for n > 1, a(n) = Product_{d|n, d>1, dA297167(d)).

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 6, 3, 10, 1, 54, 1, 14, 15, 30, 1, 90, 1, 150, 21, 22, 1, 1350, 5, 26, 15, 294, 1, 2250, 1, 210, 33, 34, 35, 6750, 1, 38, 39, 5250, 1, 6174, 1, 726, 375, 46, 1, 66150, 7, 350, 51, 1014, 1, 3150, 55, 16170, 57, 58, 1, 1181250, 1, 62, 735, 2310, 65, 23958, 1, 1734, 69, 17150, 1, 1653750, 1, 74, 525, 2166, 77, 39546, 1, 404250, 105

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

An auxiliary sequence for defining A300827, which is the restricted growth sequence transform of this sequence. A324202 is a similar sequence, but is not limited to the proper divisors of n, and in contrast to this, also finds the least prime signature representative (A046523) of the product formed.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences computed from indices in prime factorization

Cf. A001221, A001222, A007913, A061395, A087207, A297167, A300827 (rgs-transform), A324120, A324180.

Cf. also A324202.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324193(n) = { my(m=1); if(n<=2, n-1, fordiv(n, d, if((d>1)&(dA297167(d)))); (m)); };

a(1) = 0; for n > 1, a(n) = Product_{d|n, d>1, dA297167(d)).
For all n > 0:
A001222(a(n)) = A000005(n)-2.
A001221(A007913(a(n))) = A324120(n).
A087207(A007913(a(n))) = A324180(n).

A330919 Lexicographically earliest sequence of distinct squarefree numbers such that for any n > 0, either a(n)/a(n+1) or a(n+1)/a(n) is a prime number.

Original entry on oeis.org

1, 2, 6, 3, 15, 5, 10, 30, 210, 42, 14, 7, 21, 105, 35, 70, 770, 110, 22, 11, 33, 66, 330, 165, 55, 385, 77, 154, 462, 231, 1155, 2310, 30030, 2730, 390, 78, 26, 13, 39, 195, 65, 130, 910, 182, 91, 273, 546, 6006, 858, 286, 143, 429, 2145, 715, 1430, 4290

Offset: 1

Rémy Sigrist, May 02 2020

In other words, consecutive terms differ exactly by one prime factor.
This sequence has strong connections with A163252:
- here consecutive terms differ by one prime factor, there by one binary digit,
- for any n > 0, A163252(n-1) encodes in binary form the prime numbers appearing in a(n).
Odd indexed terms have an even number of prime factors and vice versa.
For any prime number p: as there are only finitely many squarefree numbers with greatest prime factor < p, eventually the sequence contains a multiple of p.

			The first terms, alongside their prime factors, are:
  n   a(n)  prime factors
  --  ----  -------------
   1     1
   2     2  2
   3     6  2, 3
   4     3     3
   5    15     3, 5
   6     5        5
   7    10  2,    5
   8    30  2, 3, 5
   9   210  2, 3, 5, 7
  10    42  2, 3,    7

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A330919

Cf. A019565, A087207, A163252.

PARI
```
See Links section.
```

a(n) = A019565(A163252(n-1)).

A087207(a(n)) = A163252(n-1).

A334878 For any n > 0 with prime factorization Product_{k > 0} prime(k)^e_k (where prime(k) denotes the k-th prime number), let b_k = 1 + max_{k > 0} e_k; a(n) = Sum_{k > 0} e_k * b_k^(k-1).

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 3, 6, 5, 16, 5, 32, 9, 6, 4, 64, 7, 128, 11, 10, 17, 256, 7, 18, 33, 12, 29, 512, 7, 1024, 5, 18, 65, 12, 8, 2048, 129, 34, 19, 4096, 11, 8192, 83, 15, 257, 16384, 9, 54, 19, 66, 245, 32768, 13, 20, 67, 130, 513, 65536, 14, 131072, 1025

Offset: 1

Rémy Sigrist, May 14 2020

In other words, a(n) encodes the prime factorization of n in base 1 + A051903(n).

Every nonnegative integer appears finitely many times in this sequence.

			For n = 84:
- 84 = 7 * 3 * 2^2 = prime(4) * prime(2) * prime(1)^2,
- b_84 = 1 + 2 = 3,
- so a(84) = 1*3^(4-1) + 1*3^(2-1) + 2*3^(1-1) = 32.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A051903, A087207.

a(n) = { if (n==1, 0, my (f=factor(n), b=1+vecmax(f[,2]~)); sum(k=1, #f~, f[k,2]*b^(primepi(f[k,1])-1))) }

a(2^e) = e for any e >= 0.
a(prime(k)) = 2^(k-1) for any k > 0.
a(prime(k)^e) = e*(e+1)^(k-1) for any k > 0 and e >= 0.
a(n) = A087207(n) for any squarefree number n.

A361322 The binary expansion of a(n) specifies which primes divide A361321(n).

Original entry on oeis.org

0, 3, 5, 12, 10, 18, 17, 9, 40, 34, 6, 20, 24, 11, 33, 36, 68, 65, 129, 130, 66, 72, 13, 7, 258, 264, 136, 132, 21, 257, 288, 48, 19, 513, 516, 14, 35, 1025, 1028, 260, 259, 2049, 2052, 22, 514, 520, 25, 4097, 4098, 1026, 1032, 41, 8193, 8194, 2050, 2056, 73

Offset: 1

Scott R. Shannon, Rémy Sigrist and N. J. A. Sloane, Mar 09 2023

Conjecture: The sequence is a permutation of A057716.

Rémy Sigrist, PARI program

Cf. A057716, A087207, A361321.

PARI
```
\\ See Links section.
```

a(n) = A087207(A360519(n)).

A361643 The binary expansion of a(n) specifies which primes divide A359804(n).

Original entry on oeis.org

0, 1, 2, 4, 1, 3, 5, 8, 2, 1, 6, 9, 16, 3, 5, 10, 17, 4, 3, 9, 7, 18, 12, 1, 3, 5, 11, 17, 6, 8, 33, 2, 5, 9, 3, 20, 10, 1, 7, 13, 19, 32, 36, 65, 34, 6, 129, 24, 3, 5, 11, 17, 68, 66, 257, 7, 40, 18, 33, 132, 3, 9, 5, 130, 14, 513, 21, 258, 9, 260, 3, 72, 7

Offset: 1

Rémy Sigrist, Mar 19 2023

			The first terms, in decimal and in binary, alongside A359804(n) and its divisibility by small prime numbers, are:
  n   a(n)  bin(a(n))  A359804(n)  Divisibility by:
                                      7  5  3  2
  --  ----  ---------  ----------     -  -  -  -
   1     0          0           1
   2     1          1           2              X
   3     2         10           3           X
   4     4        100           5        X
   5     1          1           4              X
   6     3         11           6           X  X
   7     5        101          10        X     X
   8     8       1000           7     X
   9     2         10           9           X
  10     1          1           8              X

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program

Cf. A087207, A359804.

PARI
```
See Links section.
```

a(n) = A087207(A359804(n)).

A332411 If n = Product (p_j^k_j) then a(n) = Sum (n^(pi(p_j) - 1)), where pi = A000720.

Original entry on oeis.org

0, 1, 3, 1, 25, 7, 343, 1, 9, 101, 14641, 13, 371293, 2745, 240, 1, 24137569, 19, 893871739, 401, 9282, 234257, 78310985281, 25, 625, 11881377, 27, 21953, 14507145975869, 931, 819628286980801, 1, 1185954, 1544804417, 44100, 37, 177917621779460413, 114415582593, 90224238, 1601

Offset: 1

Ilya Gutkovskiy, Feb 11 2020

nonn

			a(21) = a(3 * 7) = a(prime(2) * prime(4)) = 21^1 + 21^3 = 9282;
9282 in base 21 (reverse order of digits with leading zero) = 0101.
                                                               | |
                                                               2 4

Index entries for sequences computed from indices in prime factorization

Cf. A000079 (without a(0) gives the positions of 1's), A000244 (without a(0) gives the fixed points), A000720, A087207, A090883, A276379 (a(n) written in base n), A308814.

a:= n-> add(n^(numtheory[pi](i[1])-1), i=ifactors(n)[2]):
seq(a(n), n=1..42);  # Alois P. Heinz, Feb 11 2020

a[n_] := Plus @@ (n^(PrimePi[#[[1]]] - 1) & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 40}]
Table[SeriesCoefficient[Sum[n^(k - 1) x^Prime[k]/(1 - x^Prime[k]), {k, 1, n}], {x, 0, n}], {n, 1, 40}]

a(n) = my(f=factor(n)); sum(k=1, #f~, n^(primepi(f[k,1])-1)); \\ Michel Marcus, Feb 11 2020

a(n) = [x^n] Sum_{k>=1} n^(k - 1) * x^prime(k) / (1 - x^prime(k)).

A345298 a(n) = Sum_{p|n, p prime} tau(p #).

Original entry on oeis.org

0, 2, 4, 2, 8, 6, 16, 2, 4, 10, 32, 6, 64, 18, 12, 2, 128, 6, 256, 10, 20, 34, 512, 6, 8, 66, 4, 18, 1024, 14, 2048, 2, 36, 130, 24, 6, 4096, 258, 68, 10, 8192, 22, 16384, 34, 12, 514, 32768, 6, 16, 10, 132, 66, 65536, 6, 40, 18, 260, 1026, 131072, 14, 262144, 2050, 20, 2, 72

Offset: 1

Wesley Ivan Hurt, Jun 13 2021

nonn

If p is prime, a(p) = Sum_{p|p} tau(p #) = tau(p) * tau(prevprime(p)) * ... * tau(2) = 2 * 2 * ... * 2 ( pi(p) times ) = 2^pi(p).

			a(14) = Sum_{p|14} tau(p #) = tau(2 #) + tau(7 #) = 2^pi(2) + 2^pi(7) = 2^1 + 2^4 = 18.

Equals twice A087207.

Cf. A000005 (tau), A002110, A345284.

Table[Sum[DivisorSigma[0, Product[i^(PrimePi[i] - PrimePi[i - 1]), {i, k}]](PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]

G.f.: Sum_{k>=1} 2^k * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Aug 18 2021

cp's OEIS Frontend

A351560 a(n) is a binary representation of the primes that divide sigma(n) [the sum of divisors of n function], shown in decimal.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A361111 The binary expansion of a(n) specifies which primes divide A360519(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A276010 a(0) = 0, for n >= 1, a(n) = A275736(n) OR a(A257684(n)), where OR is given by A003986.

Views

Author

Keywords

Links

Crossrefs

Formula

A324193 a(1) = 0; for n > 1, a(n) = Product_{d|n, d>1, dA297167(d)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A330919 Lexicographically earliest sequence of distinct squarefree numbers such that for any n > 0, either a(n)/a(n+1) or a(n+1)/a(n) is a prime number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A334878 For any n > 0 with prime factorization Product_{k > 0} prime(k)^e_k (where prime(k) denotes the k-th prime number), let b_k = 1 + max_{k > 0} e_k; a(n) = Sum_{k > 0} e_k * b_k^(k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A361322 The binary expansion of a(n) specifies which primes divide A361321(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A361643 The binary expansion of a(n) specifies which primes divide A359804(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs