A064800 -id:A064800 - cp's OEIS frontend

A317743 Valleys in A064800: terms which are smaller than their neighbors.

14, 18, 20, 27, 30, 32, 35, 38, 42, 44, 51, 54, 57, 59, 62, 67, 68, 72, 74, 80, 87, 90, 93, 98, 102, 104, 110, 114, 123, 128, 131, 132, 135, 138, 140, 143, 147, 150, 152, 158, 163, 164, 171, 174, 179, 182, 187, 192, 194, 198, 200

Offset: 1

Zak Seidov, Aug 05 2018

nonn

From Zak Seidov, Sep 21 2018: (Start)
First second-order valley is a(514) = 516
with {521, 517, 516, 517, 520}.
First third-order valley is a(k=265827) = 265829
with {265833, 265831, 265830, 265829, 265831, 265832, 265834}.
Are there minima of higher order? (End)

			a(1) = 14 is the first valley (local minimum) in A064800 as A064800(13) = 14 is less than A064800(12) = 15 and A064800(14) = 16.

Cf. A064800, A319504.

Transpose[ Select[ Partition[(# + PrimeOmega[#]) & /@ Range[201], 3, 1], #[[1]] > #[[2]] < #[[3]] &]][[2]] (* Giovanni Resta, Aug 09 2018 *)

lista(nn) = {my(v = vector(nn, n, n + bigomega(n))); for (n=2, nn, if ((v[n] < v[n-1]) && (v[n] < v[n+1]), print1(v[n], ", ")););} \\ Michel Marcus, Sep 07 2018

A319504 Indices of valleys (i.e. local minima) in A064800.

Original entry on oeis.org

13, 17, 19, 25, 29, 31, 33, 37, 41, 43, 49, 53, 55, 57, 61, 65, 67, 71, 73, 79, 85, 89, 91, 97, 101, 103, 109, 113, 121, 127, 129, 131, 133, 137, 139, 141, 145, 149, 151, 157, 161, 163, 169, 173, 177, 181, 185, 191, 193, 197, 199, 201, 205, 209, 211, 217, 221, 223, 229, 233, 235, 239, 241, 244, 249, 251, 253, 257, 265, 269, 271, 277, 281, 283, 289, 293, 295, 298

Offset: 1

Zak Seidov, Sep 20 2018

nonn

			A064800(13) is the first minimum in A064800: A064800(12) = 15, A064800(13) = 14 and A064800(14) = 16 and hence a(1) = 13.

Cf. A064800, A317743.

lista(nn) = {my(v = vector(nn, n, n + bigomega(n))); for (n=2, nn, if ((v[n] < v[n-1]) && (v[n] < v[n+1]), print1(n, ", ")););} \\ Michel Marcus, Sep 21 2018

More terms from Michel Marcus, Sep 21 2018

A069345 n minus the number of its prime-factors: a(n) = n - A001222(n).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 5, 7, 8, 10, 9, 12, 12, 13, 12, 16, 15, 18, 17, 19, 20, 22, 20, 23, 24, 24, 25, 28, 27, 30, 27, 31, 32, 33, 32, 36, 36, 37, 36, 40, 39, 42, 41, 42, 44, 46, 43, 47, 47, 49, 49, 52, 50, 53, 52, 55, 56, 58, 56, 60, 60, 60, 58, 63, 63, 66, 65

Offset: 1

Reinhard Zumkeller, Mar 18 2002

Prime factors are counted with multiplicity. - Harvey P. Dale, Sep 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A001222, A064800, A069355.

[n eq 1 select 1 else n-(&+[p[2]: p in Factorization(n)]): n in [1..100]]; // Vincenzo Librandi, Apr 09 2018

Table[n-PrimeOmega[n],{n,70}] (* Harvey P. Dale, Sep 20 2011 *)

a(n) = n - bigomega(n); \\ Michel Marcus, Apr 08 2018

A229109 a(n) = n plus the number of its distinct prime factors.

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 8, 9, 10, 12, 12, 14, 14, 16, 17, 17, 18, 20, 20, 22, 23, 24, 24, 26, 26, 28, 28, 30, 30, 33, 32, 33, 35, 36, 37, 38, 38, 40, 41, 42, 42, 45, 44, 46, 47, 48, 48, 50, 50, 52, 53, 54, 54, 56, 57, 58, 59, 60, 60, 63, 62, 64, 65, 65, 67, 69, 68

Offset: 1

Reinhard Zumkeller, Sep 13 2013

nonn

			a(40) = 42, since 40 has two distinct prime divisors (2 and 5), and so 40 + 2 = 42.
a(41) = 42 also, since 41 is prime and therefore 41 + 1 = 42.
a(42) = 45, since 42 has three distinct prime divisors (2, 3, 7), and so 42 + 3 = 45.

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

Cf. A064800, A001221, A000961.

Haskell
```
a229109 n  = a001221 n + n
```

Table[n + PrimeNu[n], {n, 80}] (* Harvey P. Dale, Jun 22 2015 *)

a(n) = n + omega(n); \\ Michel Marcus, Jan 25 2022

a(n) = n + A001221(n).

a(n) = n + 1 if and only if n is prime or a power of a prime (that is, n > 1 is in A000961). - Alonso del Arte, Nov 17 2016

A069348 Primes of the form k + Omega(k), where Omega(k) is the number of prime factors of k, A001222(k).

Original entry on oeis.org

3, 11, 17, 23, 31, 37, 41, 47, 53, 59, 67, 71, 73, 79, 89, 97, 101, 113, 127, 131, 139, 151, 157, 163, 167, 173, 179, 181, 191, 193, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 277, 293, 307, 311, 313, 317, 331, 337

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

The representation is ambiguous, see A069349.

			a(10) = 59 = A064800(57) = 57 + 2 (as 57 = 3*19).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A069346, A069357.

Select[Table[n + PrimeOmega[n], {n, 1, 400}], PrimeQ] // Union (* Jean-François Alcover, Jun 08 2013 *)

is(n)=for(k=1,log(n+.5)\log(2),if(bigomega(n-k)==k,return(isprime(n)))); 0 \\ Charles R Greathouse IV, Dec 10 2013

Corrected by T. D. Noe, Nov 01 2006

A069357 Numbers of form 2^i*3^j + (i+j) with i, j >= 0.

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 15, 20, 21, 28, 30, 37, 40, 53, 58, 70, 77, 85, 102, 113, 135, 150, 167, 199, 222, 248, 264, 295, 330, 392, 439, 492, 521, 584, 655, 735, 777, 872, 979, 1034, 1161, 1304, 1465, 1546, 1737, 1952, 2059, 2194, 2314, 2601, 2924, 3083, 3466, 3897

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

Cf. A055600, A069358, A064800, A022328, A022329.

Distinct values of A003586(k) + A069352(k). [Corrected by Georg Fischer, Dec 11 2022, further clarification by Sean A. Irvine, Apr 28 2024]

Missing a(1)=1 inserted and duplicate values removed by Sean A. Irvine, Apr 28 2024

A069349 Number of ways to represent primes as A069348(n)=k+A001222(k).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

			a(9)=3 as: A069348(9)=53=A064800(48)=A064800(50)=A064800(51)
53=2*2*2*2*3+5 and 53=2*5*5+3 and 51=3*17+2.

A337455 Numbers of the form m + bigomega(m) with m a positive integer.

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 12, 14, 15, 16, 17, 18, 20, 21, 23, 24, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 44, 45, 47, 48, 51, 53, 54, 55, 57, 58, 59, 60, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 84, 85, 87, 88, 89, 90, 92, 93

Offset: 1

Nathan J. McDougall, Aug 27 2020

nonn

If a(n) = m + A001222(m) then (a(n) - m) <= log(a(n))/log(2).

It appears that a(n)/n may converge to a constant around ~ 1.49.

			a(7) = 10 + A001222(10) = 10 + 2 = 12

Petr Kucheriaviy, On numbers not representable as n + ω(n), arXiv preprint (2022). arXiv:2203.12006 [math.NT]

Cf. A001222 (bigomega), A064800, A358973.
Numbers of the form k^n+n where k is prime are subsequences: A006127 (k=2), A104743 (k=3), A104745 (k=5), A226199 (k=7), A226737 (k=11).
Subsequences include A008864, A101340, and A160649 (excluding the first term).

m = 100; Select[Union @ Table[n + PrimeOmega[n], {n, 1, m}], # <= m &] (* Amiram Eldar, Aug 28 2020 *)

upto(limit)=Set(select(t->t<=limit, apply(m->m+bigomega(m), [1..limit]))) \\ Andrew Howroyd, Aug 27 2020

list(lim)=my(v=List()); forfactored(n=1, lim\1-1, my(t=n[1]+bigomega(n)); if(t<=lim, listput(v, t))); Set(v) \\ Charles R Greathouse IV, Dec 07 2022

Kucheriaviy proves that a(n) << n log log n and conjectures that a(n) ≍ n, that is, these numbers have positive lower density. - Charles R Greathouse IV, Dec 07 2022

cp's OEIS Frontend

A317743 Valleys in A064800: terms which are smaller than their neighbors.

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A319504 Indices of valleys (i.e. local minima) in A064800.

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A069345 n minus the number of its prime-factors: a(n) = n - A001222(n).

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A229109 a(n) = n plus the number of its distinct prime factors.

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A069348 Primes of the form k + Omega(k), where Omega(k) is the number of prime factors of k, A001222(k).

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A069357 Numbers of form 2^i*3^j + (i+j) with i, j >= 0.

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A069349 Number of ways to represent primes as A069348(n)=k+A001222(k).

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A337455 Numbers of the form m + bigomega(m) with m a positive integer.

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