A082686 -id:A082686 - cp's OEIS frontend

A375205 PrimePi(greatest prime < sqrt(Q)) - PrimePi(greatest prime factor(Q) < sqrt(Q)), with Q = A082686(n).

0, 0, 1, 1, 0, 1, 0, 2, 1, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 3, 2, 3, 1, 2, 3, 0, 2, 1, 3, 2, 3, 1, 0, 4, 2, 4, 4, 3, 1, 2, 0, 4, 2, 3, 4, 1, 4, 3, 2, 4, 0, 1, 3, 4, 4, 2, 0, 4, 1, 3, 2, 4, 3, 4, 0, 1, 4, 3, 2, 5, 4, 2, 1, 3, 5, 4, 5, 3

Offset: 1

Bill McEachen, Oct 15 2024

nonn

New records appear to be in consecutive numerical order, suggesting every integer should be seen in this infinite sequence. Considering a(n)=0, empirically a power fit Y=k*x^c correlates well with the "x-th" occurrence. For example, the 491st 0 value is at n=99808.

			A082686(8)=51, with square root = 7.14... so the greatest prime < 7.14 is 7, while the greatest prime factor of 51 < 7.14 is 3. The prime count from 3 to 7 is 2, so a(8)=2.
A082686(999)=2883 with square root = 53.69... so the greatest prime < 53.69 is 53, while the greatest prime factor of 2883 < 53.69 is 31. The prime count from 31 to 53 is 5, so a(999)=5.

Bill McEachen, Table of n, a(n) for n = 1..10000

Cf. A082686.

nmax=85;q={};m=15;Until[Length[q]==nmax,If[ !PrimeQ[m]&&EvenQ[DivisorSigma[0, m]],AppendTo[q,m]];m=m+2];Table[PrimePi[NextPrime[Sqrt[q[[n]]],-1]]-PrimePi[Select[First/@FactorInteger[q[[n]]],#James C. McMahon, Dec 06 2024 *)

A364286 Composite numbers k for which A324644(k)/A324198(k) = 2.

Original entry on oeis.org

33, 51, 69, 91, 99, 135, 141, 145, 153, 159, 187, 207, 213, 217, 285, 295, 303, 321, 339, 391, 411, 423, 427, 435, 445, 477, 507, 519, 573, 637, 639, 679, 681, 699, 771, 783, 799, 843, 855, 861, 885, 895, 901, 909, 933, 951, 963, 1017, 1041, 1057, 1059, 1071, 1081, 1083, 1147, 1149, 1173, 1185, 1195, 1203, 1207

Offset: 1

Antti Karttunen, Jul 17 2023

nonn

See comments in A351458.

All terms are odd. Of the 63 initial terms of A349169, only term 13923 occurs also in this sequence. The first common term with A332458 is 161257. - Antti Karttunen, Mar 10 2024

Cf. A000203, A276086, A324198, A324644, A332458, A349169, A351458, A371082 (subsequence).

Subsequence of A082686.

f[x_] := Block[{m, i, n = x, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m]; Select[Select[Range[1350], CompositeQ], GCD[#2, #3]/GCD[#1, #3] == 2 & @@ {#, DivisorSigma[1, #], f[#]} &] (* Michael De Vlieger, Mar 10 2024 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA364286(n) = if(isprime(n), 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u))); \\ Antti Karttunen, Mar 10 2024

A226025 Odd composite numbers that are not squares of primes.

Original entry on oeis.org

15, 21, 27, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 123, 125, 129, 133, 135, 141, 143, 145, 147, 153, 155, 159, 161, 165, 171, 175, 177, 183, 185, 187, 189, 195, 201, 203, 205, 207, 209, 213, 215, 217

Offset: 1

Arkadiusz Wesolowski, Jun 07 2013

nonn

Numbers that are in A071904 (odd composite numbers) but not in A001248 (squares of primes).
First differs from its subsequence A082686 in a(16)=81 which is not in A082686. More precisely, A226025 \ A082686 = A062532 \ {1} = A014076^2 \ {1}. - M. F. Hasler, Oct 20 2013
Odd numbers that are greater than the square of their least prime factor - Odimar Fabeny, Sep 08 2014

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Subsequence of A071904. Cf. A226603.

a226025 n = a226025_list !! (n-1)
a226025_list = filter ((/= 2) . a100995) a071904_list
-- Reinhard Zumkeller, Jun 15 2013

[n: n in [3..217 by 2] | not IsPrime(n) and not IsSquare(n) or IsSquare(n) and not IsPrime(Floor(n^(1/2)))];

select(n -> not(isprime(n)) and (not(issqr(n)) or not(isprime(sqrt(n)))), [seq(2*i+1,i=1..1000)]); # Robert Israel, Sep 08 2014

Select[Range[3, 217, 2], ! PrimeQ[#] && ! PrimeQ@Sqrt[#] &]
r = Prime@Range[2, 6]^2; Complement[Select[Range[3, Last[r] - 2, 2], ! PrimeQ[#] &], Most[r]]
Select[Range[3,251,2],NoneTrue[{#,Sqrt[#]},PrimeQ]&] (* Harvey P. Dale, Sep 06 2021 *)

is_A226025(n)={bittest(n,0)&&!isprime(n,0)&&!(issquare(n)&&isprime(sqrtint(n)))&&n>1} \\ - M. F. Hasler, Oct 20 2013

A226025 = { odd x>1 | A100995(x) = 0 or A100995(x) > 2 }. - M. F. Hasler, Oct 20 2013

A158942 Nonsquares coprime to 10.

Original entry on oeis.org

3, 7, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 147, 149, 151, 153, 157, 159, 161, 163

Offset: 1

Eric Desbiaux, Mar 31 2009

Odd primes + odd nonprime integers that have an odd numbers of proper divisors A082686, are the result of a suppression of integers satisfying: 2n (A005843); n^2 (A000290); n^2+n (A002378). Of these, we can suppress the multiples of 5 (A008587).
Decimal expansion of 1/10^(n^2+n) + 1/10^(n^2) + 1/10^(5*n) + 1/10^(2*n) gives a 0 for these integers.
2n + n(n+1) + n^2 = 2n^2 + 3n = A014106.
2n^2 + 3n + 5n = 2n^2 + 8n = 2n(n+4) = A067728 8(8+n) is a perfect square.

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

Intersection of A000037 and A045572.

Cf. A082868, A002378, A000290, A005843, A008587, A014106, A067728.

Select[Range@ 163, ! IntegerQ@ Sqrt@ # && CoprimeQ[#, 10] &] (* Michael De Vlieger, Dec 11 2015 *)

isok(n) = (n % 2) && (n % 5) && (isprime(n) || (numdiv(n) % 2 == 0)); \\ Michel Marcus, Aug 27 2013

is(n)=gcd(n,10)==1 && !issquare(n) \\ Charles R Greathouse IV, Sep 05 2013

New name from Charles R Greathouse IV, Sep 05 2013

cp's OEIS Frontend

A375205 PrimePi(greatest prime < sqrt(Q)) - PrimePi(greatest prime factor(Q) < sqrt(Q)), with Q = A082686(n).

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A364286 Composite numbers k for which A324644(k)/A324198(k) = 2.

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A226025 Odd composite numbers that are not squares of primes.

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A158942 Nonsquares coprime to 10.

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