A109819 -id:A109819 - cp's OEIS frontend

A284262 a(n) = where A284259 for the first time obtains value n (positions of its records).

1, 2, 6, 105, 5005, 85085, 1616615, 37182145, 6685349671, 247357937827, 10141675450907, 436092044389001, 20496326086283047, 9156001667401012567, 558516101711461766587, 37420578814667938361329, 2656861095841423623654359, 193950859996423924526768207, 15322117939717490037614688353, 1271735788996551673122019133299

Offset: 0

Antti Karttunen, Mar 24 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..335

Cf. A001221, A001222, A002110, A003961, A242378, A284259 (a left inverse), A284263.

Cf. also A109819.

A[n_]:= If[n<1, 0, Block[{k=1}, While[Prime[n + k  - 1] > Prime[k]^2, k++]; k - 1]]; a[n_]:=If[n<2, n + 1, Product[Prime[i], {i, A[n] + 1, A[n] + n}]]; Table[a[n], {n, 0, 51}] (* Indranil Ghosh, Mar 24 2017 *)

A(n) = { my(k=1); if(0==n, 0, while(prime(n+k-1) > (prime(k)^2), k = k+1); (k-1)); };
a(n) = prod(i=A(n) + 1, A(n) + n, prime(i));
for(n=0, 51, print1(a(n),", ")) \\ Indranil Ghosh, after Antti Karttunen, Mar 24 2017

from sympy import prime
from operator import mul
from functools import reduce
def A(n):
    if n<1: return 0
    k=1
    while prime(n + k - 1)>prime(k)**2:k+=1
    return k - 1
def a(n): return n + 1 if n<2 else reduce(mul, [prime(i) for i in range(A(n) + 1, A(n) + n + 1)])
print([a(n) for n in range(21)]) # Indranil Ghosh, Mar 24 2017

(define (A284262 n) (A242378bi (A284263 n) (A002110 n))) ;; Where A242378bi(k,n) applies prime shift A003961(n) k times. See A242378.

For n > 1, a(n) = Product_{i = A284263(n)+1 .. A284263(n)+n} prime(i); a(0) = 1, a(1) = 2.
a(n) = A242378(A284263(n), A002110(n)) [shift the prime factorization of the n-th primorial A284263(n) steps towards larger primes].
Other identities. For all n >= 0:
A001221(a(n)) = A001222(a(n))  = n.
A284259(a(n)) = n.

A109818 Sum of primes between n and n^2.

Original entry on oeis.org

0, 5, 15, 36, 95, 150, 318, 484, 774, 1043, 1576, 2099, 2886, 3790, 4620, 6040, 7941, 9465, 11541, 13810, 16763, 19982, 23515, 26840, 32253, 37461, 42368, 48394, 55737, 62668, 70112, 80029, 89512, 100678, 111427, 124051, 135954, 148630, 166354

Offset: 1

Rick L. Shepherd, Jul 02 2005

			a(3) = 15 because 3, 5 and 7 are the A073882(3) = 3 primes in the interval from 3 to 3^2 inclusive and 3 + 5 + 7 = 15.

Cf. A109819 (product of same primes), A073882 (number of primes between n and n^2).

Join[{0},Table[Sum[Prime[i],{i,If[PrimeQ[n],PrimePi[n],PrimePi[n]+1],PrimePi[n^2]}],{n,2,39}]] (* James C. McMahon, Apr 02 2024 *)

for(n=1,50,print1(sum(k=n,n^2,if(isprime(k),k)),","))

A376740 Numbers that have at least one two-digit prime factor.

Original entry on oeis.org

11, 13, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 102, 104, 106, 110, 111, 114, 115, 116, 117

Offset: 1

Kishin Ikemoto, Oct 03 2024

Subsequence of A068191, first differing at A068191(55) = 101 which is not a term here.
Numbers k such that gcd(k,10978895066407230594062391177770267) > 1. - Chai Wah Wu, Nov 18 2024 [The big number is A109819(10) - Alois P. Heinz, Nov 18 2024]
The asymptotic density of this sequence is A051953(A109819(10))/A109819(10) = 1329644281346285477858013527/2807455661493975149742813527 = 0.473611... . - Amiram Eldar, Nov 19 2024

			201 = 3*67 is in this sequence because it has one two-digit prime factor.
202 = 2*101 is not, because neither of them is two-digit.

Kishin Ikemoto, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, order 10978895066407230594062391177770267.

Cf. A051953, A068191, A084891, A109819.

q:= convert(select(isprime, [seq(i,i=11 .. 99, 2)]),`*`):
filter:= n -> igcd(n,q) > 1:
select(filter, [$1..200]); # Robert Israel, Nov 18 2024

A376740Q[k_] := GCD[k, 10978895066407230594062391177770267] > 1;
Select[Range[200], A376740Q] (* Paolo Xausa, Jun 24 2025 *)

is(k) = {forprime(p = 11, 97, if(!(k % p), return(1))); 0;} \\ Amiram Eldar, Nov 19 2024

def ok(n): return any(n%p == 0 for p in [11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97])
print([k for k in range(1, 118) if ok(k)]) # Michael S. Branicky, Oct 15 2024

a(n + A051953(A109819(10))) = a(n) + A109819(10). - Amiram Eldar, Nov 19 2024

cp's OEIS Frontend

A284262 a(n) = where A284259 for the first time obtains value n (positions of its records).

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A109818 Sum of primes between n and n^2.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

A376740 Numbers that have at least one two-digit prime factor.

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Mathematica

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