A075028 -id:A075028 - cp's OEIS frontend

A285010 a(n+1) = a(n) + p, where p is the largest prime less than a(n); a(1) = 3.

3, 5, 8, 15, 28, 51, 98, 195, 388, 771, 1540, 3071, 6138, 12271, 24540, 49073, 98142, 196271, 392518, 785021, 1570038, 3140045, 6280086, 12560153, 25120300, 50240589, 100481176, 200962343, 401924670, 803849309, 1607698612, 3215397195, 6430794388, 12861588761, 25723177510

Offset: 1

David James Sycamore, Apr 07 2017

After a(1) the sequence alternates between odd and even numbers (obviously).
There is at least 1 prime between p and 2p (Bertrand), and since there is no prime between p and a(n) there must be at least one prime between a(n) and a(n) + p. Hence the sequence continues indefinitely, and each added prime is added once only.
The underlying sequence of added primes is: 2, 3, 7, 13, 23, 47, 97, ...; namely A075028 or A068524 but without their initial terms (1, 2 respectively).
Four primes occur in the first 24 terms, a(1) = 3, a(2) = 5, a(18) = 196271, and a(24) = 12560153, suggesting a higher density of primes here than in related sequence A123196, in which only three primes arise in the first 500 terms. It would be interesting to examine this further, once more terms become available.

			a(1) = 3, the first odd prime. The greatest prime less than 3 is 2, so a(2) = 3 + 2 = 5. Greatest prime less than 5 is 3 so a(3) = 5 + 3 = 8. Likewise a(4) = 8 + 7 = 15; etc.

Giovanni Resta, Table of n, a(n) for n = 1..3000

Cf. A123196, A075058, A068524.

a[1]=3; a[n_] := a[n] = a[n-1] + NextPrime[a[n-1], -1]; Array[a, 35] (* Giovanni Resta, Apr 10 2017 *)
NestList[#+NextPrime[#,-1]&,3,40] (* Harvey P. Dale, Aug 22 2020 *)

lista(nn) = { print1(a=3, ", "); for (n=2, nn, a += precprime(a-1); print1(a, ", ");); } \\ Michel Marcus, Apr 08 2017

a(n) ~ c * 2^n, where c = 0.748642996358317338.... - Bill McEachen, May 09 2024

More terms from Michel Marcus, Apr 08 2017

A227444 Numbers n such that d(n-8) < d(n-7) < ... < d(n), where d(n) = A000005(n) is the number of divisors of n.

Original entry on oeis.org

1584614385, 27604567617, 32094058449, 33284383029, 35585046850, 38045133489, 38045133490, 40156586901, 41126483649, 41126483650, 47777659029, 51564159909, 53303194065, 53690225745, 55555595955, 55576199109, 61048056225

Offset: 1

Alex Ratushnyak, Jul 11 2013

nonn

Cf. A000005, A075028.

A364704 Smallest initial number k of n consecutive numbers satisfying sigma(k) < sigma(k+1) < ... < sigma(k+n-1).

Original entry on oeis.org

1, 1, 1, 1, 36721681, 36721681

Offset: 1

Seiichi Manyama, Aug 03 2023

a(7) > 5*10^10. (Calculated using the b-file in A028965.)

Cf. A000203, A075028, A285893, A364680.

Cf. A028965, A053224, A364659, A364662.

PARI
```
b(n) = my(k=n); while(sigma(k)
				
```

cp's OEIS Frontend

A285010 a(n+1) = a(n) + p, where p is the largest prime less than a(n); a(1) = 3.

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A227444 Numbers n such that d(n-8) < d(n-7) < ... < d(n), where d(n) = A000005(n) is the number of divisors of n.

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A364704 Smallest initial number k of n consecutive numbers satisfying sigma(k) < sigma(k+1) < ... < sigma(k+n-1).

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PARI