A238949 -id:A238949 - cp's OEIS frontend

A383959 The number of prime powers p^e having the property that e is a unitary divisor of the p-adic valuation of n.

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

Offset: 1

Amiram Eldar, May 16 2025

First differs from A238949 at n = 64.	
First differs from A383960 at n = 256.		
Also, the number of prime powers p^e having the property that e is a squarefree divisor of the p-adic valuation of n.

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

Cf. A001221, A034444, A077761, A085548, A238949, A278908, A361255, A383863, A383960.

f[p_, e_] := 2^PrimeNu[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> 1 << omega(x), factor(n)[, 2]));

Additive with a(p^e) = A034444(e) = 2^A001221(e).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B - C + D), where B is Mertens's constant (A077761), C = Sum_{p prime} 1/p^2 (A085548), and D = Sum_{p prime, e>=2} (1-1/p)*A034444(e)/p^e = 0.92341081050532387352... .

A110475 Number of symbols '*' and '^' to write the canonical prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Sep 08 2005

nonn

It is conjectured that 1,2,3,4,5,6,7,9,11 are the only positive integers which cannot be represented as the sum of two elements of indices n such that a(n) = 1. - Jonathan Vos Post, Sep 11 2005

a(n) = 2 iff n is a sphenic number (A007304) or n is a prime p times a prime power q^e with e > 1 and q not equal to p. a(n) = 3 iff n has exactly four distinct prime factors (A046386); or n is the product of two prime powers (p^e)*(q^f) with e > 1, f > 1 and p not equal to q; or n is a semiprime s times a prime power r^g with g > 1 and r relatively prime to s. For a(n) > 3, Reinhard Zumkeller's description is a simpler description than the above compound descriptions. - Jonathan Vos Post, Sep 11 2005

			a(208029250) = a(2*5^3*11^2*13*23^2) = 4 '*' + 3 '^' = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.

Cf. A050252, A001358, A025475, A000040, A238949.
Cf. A007304, A046386, A008578.
Cf. A001221, A056170, A118914, A077761, A085548.

a110475 1 = 0
a110475 n = length us - 1 + 2 * length vs where
            (us, vs) = span (== 1) $ a118914_row n
-- Reinhard Zumkeller, Mar 23 2014

A110475[n_] := 2*Length[#] - 1 - Count[#, 1] & [FactorInteger[n][[All, 2]]];
Array[A110475, 100] (* Paolo Xausa, Mar 10 2025 *)

a(n) = A001221(n) - 1 + A056170(n) for n > 1.
a(n) = 0 iff n=1 or n is prime: a(A008578(n)) = 0.
a(n) = 1 iff n is a semiprime or a prime power p^e with e > 1.
From Amiram Eldar, Sep 27 2024: (Start)
a(n) = A238949(n) - 1 for n >= 2.
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C - 1), where B is Mertens's constant (A077761) and C = Sum_{p prime} 1/p^2 (A085548). (End)

A238956 Degree of divisor lattice in graded colexicographic order.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 2, 3, 4, 4, 4, 2, 3, 4, 4, 5, 5, 5, 2, 3, 4, 4, 4, 5, 6, 5, 6, 6, 6, 2, 3, 4, 4, 4, 5, 5, 6, 5, 6, 7, 6, 7, 7, 7, 2, 3, 4, 4, 4, 4, 5, 5, 6, 6, 5, 6, 6, 7, 8, 6, 7, 8, 7, 8, 8, 8, 2, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 9, 7, 8, 9, 8, 9, 9, 9

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  2, 2;
  2, 3, 3;
  2, 3, 4, 4, 4;
  2, 3, 4, 4, 5, 5, 5;
  2, 3, 4, 4, 4, 5, 6, 5, 6, 6, 6;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283 [math.NT], 2014.

Cf. A238949 in graded colexicographic order.

Cf. A036035, A238969.

C(sig)={sum(i=1, #sig, if(sig[i]>1, 2, 1))}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Apr 01 2020

T(n,k) = A238949(A036035(n,k)).

Offset changed and terms a(50) and beyond from Andrew Howroyd, Apr 01 2020

A238969 Degree of divisor lattice in divisor lattice in canonical order.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 2, 3, 4, 4, 4, 2, 3, 4, 4, 5, 5, 5, 2, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 2, 3, 4, 4, 4, 5, 5, 4, 5, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 2, 3, 4, 4, 4, 5, 5, 4, 5, 6, 6, 6, 5, 6, 6, 7, 7, 7, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  2, 2;
  2, 3, 3;
  2, 3, 4, 4, 4;
  2, 3, 4, 4, 5, 5, 5;
  2, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283 [math.NT], 2014.

Cf. A238956 in canonical order.

Cf. A063008, A238949.

C(sig)={sum(i=1, #sig, if(sig[i]>1, 2, 1))}
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 26 2020

T(n,k) = A238949(A063008(n,k)). - Andrew Howroyd, Mar 26 2020

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 26 2020

A383960 The number of prime powers p^e having the property that e is an infinitary divisor of the p-adic valuation of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 16 2025

First differs from A238949 at n = 64.

First differs from A383959 at n = 256.

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

Cf. A037445, A085548, A238949, A383760, A383865, A383959.

f[p_, e_] := 2^DigitCount[e, 2, 1]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 0; ff[p_, e_] := d[e]; a[n_] := Plus @@ ff @@@ FactorInteger[n]; Array[a, 100]

d(n) = vecprod(apply(x -> 1 << hammingweight(x), factor(n)[, 2]));
a(n) = vecsum(apply(x -> d(x), factor(n)[, 2]));

Additive with a(p^e) = A037445(e).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B - C + D), where B is Mertens's constant (A077761), C = Sum_{p prime} 1/p^2 (A085548), and D = Sum_{p prime, e>=2} (1-1/p)*A037445(e)/p^e = 0.92752481299257205938... .

A348477 Drop all 1 but the first 1 in A035306.

Original entry on oeis.org

1, 2, 3, 2, 2, 5, 2, 3, 7, 2, 3, 3, 2, 2, 5, 11, 2, 2, 3, 13, 2, 7, 3, 5, 2, 4, 17, 2, 3, 2, 19, 2, 2, 5, 3, 7, 2, 11, 23, 2, 3, 3, 5, 2, 2, 13, 3, 3, 2, 2, 7, 29, 2, 3, 5, 31, 2, 5, 3, 11, 2, 17, 5, 7, 2, 2, 3, 2, 37, 2, 19, 3, 13, 2, 3, 5, 41, 2, 3, 7, 43, 2, 2, 11, 3, 2, 5, 2, 23, 47, 2, 4, 3, 7, 2, 2, 5, 2, 3, 17, 2, 2, 13, 53, 2, 3, 3, 5, 11, 2, 3, 7, 3, 19, 2, 29

Offset: 1

Seiichi Manyama, Oct 20 2021

List of prime divisors of n and their exponents, ignoring the exponent 1. - Michael De Vlieger, Oct 20 2021

			   n   prime factorization  triangle
   1 = 1.                 ->  1;
   2 = 2.                 ->  2;
   3 = 3.                 ->  3;
   4 = 2^2.               ->  2, 2;
   5 = 5.                 ->  5;
   6 = 2*3.               ->  2, 3;
   7 = 7.                 ->  7;
   8 = 2^3.               ->  2, 3;
   9 = 3^2.               ->  3, 2;
  10 = 2*5.               ->  2, 5;
  11 = 11.                -> 11;
  12 = 2^2*3.             ->  2, 2, 3;
  13 = 13.                -> 13;
  14 = 2*7                ->  2, 7;
  15 = 3*5.               ->  3, 5;
  16 = 2^4.               ->  2, 4;

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.

Column 1 is A020639.
Row lengths are A238949(n) for n > 1.
Cf. A027746, A035306.

Array[DeleteCases[Flatten@ FactorInteger[#], 1] &, 58] /. {} -> {1} // Flatten (* Michael De Vlieger, Oct 20 2021 *)

tabf(nn) = if(nn==1, print1(1, ", "), my(f=factor(nn)); for(i=1, #f~, for(j=1, 2, if((k=f[i, j])>j-1, print1(k, ", ")))));

require 'prime'
def A348477(n)
  ary = (2..n).map{|i| i.prime_division}.flatten
  ary.delete(1)
  [1] + ary
end
p A348477(60)

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A383959 The number of prime powers p^e having the property that e is a unitary divisor of the p-adic valuation of n.

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A110475 Number of symbols '*' and '^' to write the canonical prime factorization of n.

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A238956 Degree of divisor lattice in graded colexicographic order.

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A238969 Degree of divisor lattice in divisor lattice in canonical order.

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A383960 The number of prime powers p^e having the property that e is an infinitary divisor of the p-adic valuation of n.

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A348477 Drop all 1 but the first 1 in A035306.

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