A007917 -id:A007917 - cp's OEIS frontend

A083714 (greatest prime <= n) - (greatest prime factor of n).

0, 0, 0, 1, 0, 2, 0, 5, 4, 2, 0, 8, 0, 6, 8, 11, 0, 14, 0, 14, 12, 8, 0, 20, 18, 10, 20, 16, 0, 24, 0, 29, 20, 14, 24, 28, 0, 18, 24, 32, 0, 34, 0, 32, 38, 20, 0, 44, 40, 42, 30, 34, 0, 50, 42, 46, 34, 24, 0, 54, 0, 30, 54, 59, 48, 50, 0, 50, 44, 60, 0, 68, 0, 36, 68, 54, 62, 60, 0, 74

Offset: 1

Reinhard Zumkeller, May 04 2003

nonn

a(n) = A007917(n) - A006530(n);

n>1: a(n) = 0 iff n is prime.

Cf. A083718, A083715, A083716, A083717.

Join[{0},Table[NextPrime[n+1,-1]-FactorInteger[n][[-1,1]],{n,2,90}]] (* Harvey P. Dale, Nov 27 2011 *)

A083715 (greatest prime <= n) mod (greatest prime factor of n).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 6, 3, 1, 0, 2, 0, 4, 5, 8, 0, 2, 3, 10, 2, 2, 0, 4, 0, 1, 9, 14, 3, 1, 0, 18, 11, 2, 0, 6, 0, 10, 3, 20, 0, 2, 5, 2, 13, 8, 0, 2, 9, 4, 15, 24, 0, 4, 0, 30, 5, 1, 9, 6, 0, 16, 21, 4, 0, 2, 0, 36, 3, 16, 7, 8, 0, 4, 1, 38, 0, 6, 15, 40, 25, 6, 0, 4, 11, 20, 27, 42

Offset: 1

Reinhard Zumkeller, May 04 2003

nonn

a(n) = A007917(n) mod A006530(n);

n>1: a(n) = 0 iff n is prime.

Cf. A083716, A083717, A083714, A083718.

A083717 (Greatest prime <= n) * (greatest prime factor of n).

Original entry on oeis.org

1, 4, 9, 6, 25, 15, 49, 14, 21, 35, 121, 33, 169, 91, 65, 26, 289, 51, 361, 95, 133, 209, 529, 69, 115, 299, 69, 161, 841, 145, 961, 62, 341, 527, 217, 93, 1369, 703, 481, 185, 1681, 287, 1849, 473, 215, 989, 2209, 141, 329, 235, 799, 611, 2809, 159, 583, 371

Offset: 1

Reinhard Zumkeller, May 04 2003

nonn

a(n) = A007917(n)*A006530(n);

Terms of A001358; A001248 is a subsequence.

Cf. A083715, A083716, A083714 A083718.

A083718 (greatest prime <= n) + (greatest prime factor of n).

Original entry on oeis.org

2, 4, 6, 5, 10, 8, 14, 9, 10, 12, 22, 14, 26, 20, 18, 15, 34, 20, 38, 24, 26, 30, 46, 26, 28, 36, 26, 30, 58, 34, 62, 33, 42, 48, 38, 34, 74, 56, 50, 42, 82, 48, 86, 54, 48, 66, 94, 50, 54, 52, 64, 60, 106, 56, 64, 60, 72, 82, 118, 64, 122, 92, 68, 63, 74, 72, 134, 84, 90, 74

Offset: 1

Reinhard Zumkeller, May 04 2003

nonn

a(n) = A007917(n) + A006530(n).

Cf. A083714, A083715, A083716, A083717.

Join[{2},Table[If[PrimeQ[n],n,NextPrime[n,-1]]+FactorInteger[n][[-1,1]],{n,2,70}]] (* Harvey P. Dale, Aug 11 2016 *)

A090119 a(n) = nextprime(A090117(n)), the smallest prime following squares listed in A090117 and also the distance of a(n) from the preceding prime is 2*n.

Original entry on oeis.org

5, 11, 29, 367, 149, 631, 127, 1949, 541, 907, 3251, 1693, 2503, 10427, 5779, 10831, 10007, 22229, 30631, 25301, 121123, 76207, 93047, 157627, 212557, 35729, 119027, 1121509, 190979, 672439, 693943, 1004027, 259099, 1646101, 675713, 1207841

Offset: 1

Labos Elemer, Jan 09 2004

nonn

			a(7) = 127 because 127-113 = 14 = 2*7 and 121 = 11^2 is between {127,113} closest primes to 121 a suitable square number. Also 127 is the smallest prime with this property.

Cf. A090116, A090117, A090118, A007917, A007918, A000720, A000040, A053001, A007491, A000290.

pre[x_] := Prime[PrimePi[x]]; nex[x_] := Prime[PrimePi[x]+1]; de[x_] := Prime[PrimePi[x]+1]-Prime[PrimePi[x]]; de[1] = 0; t=Table[de[w^2], {w, 1, 50000}]; mt=Table[Min[Flatten[Position[t, 2*j]]], {j, 1, 100}]; Table[nex[Part[mt, j]^2], {j, 1, Length[mt]}]

a(n) = nextprime(A090117(n)) = nextprime(A090116(n)^2).

a(n) = A007918(A090117(n)) = prime(1+pi(A090117(n))).

Name corrected by Jason Yuen, Jun 23 2025

A104089 Largest prime <= 4^n.

Original entry on oeis.org

3, 13, 61, 251, 1021, 4093, 16381, 65521, 262139, 1048573, 4194301, 16777213, 67108859, 268435399, 1073741789, 4294967291, 17179869143, 68719476731, 274877906899, 1099511627689, 4398046511093, 17592186044399, 70368744177643

Offset: 1

Cino Hilliard, Mar 03 2005

Cf. A014234, A104082, A104088, A104090-A104094, A003618, A104096.

Cf. A000040, A000302, A007917.

NextPrime[4^Range[30], -1] (* Paolo Xausa, Oct 28 2024 *)

g(n,b) = for(x=0,n,print1(precprime(b^x)","))

a(n) = A007917(A000302(n)). - Paolo Xausa, Oct 28 2024

A175090 Composites c with result 0 under iterations of {r mod (max prime p <= r)} starting at r = c.

Original entry on oeis.org

9, 10, 15, 16, 21, 22, 25, 26, 28, 33, 34, 36, 39, 40, 45, 46, 49, 50, 52, 55, 56, 58, 63, 64, 66, 69, 70, 75, 76, 78, 81, 82, 85, 86, 88, 91, 92, 94, 96, 99, 100, 105, 106, 111, 112, 115, 116, 118, 120, 122, 123, 124, 126, 129, 130, 133, 134, 136, 141, 142

Offset: 1

Jaroslav Krizek, Jan 28 2010

nonn

Intersection of A002808 and A175089.

Composites c such that A121559(c) = 0. - Michel Marcus, Aug 22 2014

			Iteration procedure for a(3) = 15: 15 mod 13 = 2, 2 mod 2 = 0.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007917 and A064722 (both for the iterations).

g:= proc(x) if isprime(x) then 0 else x mod prevprime(x) fi end proc:
f:= proc(x) local y; y:= x; while y > 1 do y:= g(y) od; y = 0 end proc:
select(not(isprime) and f, [$4..200]); # Robert Israel, Feb 09 2015

Composites := Select[Range[2, 200], ! PrimeQ[#] &]; Select[Composites, PrimeQ[# - NextPrime[#, -1]] &] (* Carlos Eduardo Olivieri, Feb 09 2015 *)

Missing term 55 inserted, more terms added, Michel Marcus, Aug 22 2014

A273282 Largest prime not exceeding the geometric mean of all prime divisors of n counted with multiplicity.

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 3, 3, 2, 17, 2, 19, 2, 3, 3, 23, 2, 5, 5, 3, 3, 29, 3, 31, 2, 5, 5, 5, 2, 37, 5, 5, 2, 41, 3, 43, 3, 3, 5, 47, 2, 7, 3, 7, 3, 53, 2, 7, 2, 7, 7, 59, 2, 61, 7, 3, 2, 7, 3, 67, 3, 7, 3, 71, 2, 73, 7, 3, 3, 7, 3, 79, 2, 3, 7

Offset: 2

Giuseppe Coppoletta, May 19 2016

nonn

a(n) = n iff n is prime.

a(n) <= A079866(n) with equality iff A079866(n) is prime.

			a(46) = 5 because 5 is the greatest prime not bigger than sqrt(2*23).
For n = 5^3 * 11 * 89, a(n)=7 and A273283(n)=11 because A001222(n)=5 and 7 < n^(1/5) < 11.

Giuseppe Coppoletta, Table of n, a(n) for n = 2..10000

Cf. A273283, A273284, A273288, A079866, A001222.

a[n_] := NextPrime[ Floor[n^ (1/PrimeOmega[n])] + 1, -1]; a /@ Range[2, 100] (* Giovanni Resta, May 25 2016 *)

a(n) = precprime(sqrtnint(n, bigomega(n))); \\ Michel Marcus, May 24 2016

[previous_prime(floor(n^(1/sloane.A001222(n)))+1) for n in (2..100)]

For n>=2, a(n) = A007917(A079866(n)).

A366994 The largest divisor of n that is not a term of A322448.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 32, 65, 66, 67, 68

Offset: 1

Amiram Eldar, Oct 31 2023

First differs from A365683 at n = 64.
The largest divisor of n whose prime factorization has exponents that are all either 1 or primes.
The number of these divisors is A366991(n) and their sum is A366992(n).

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

Cf. A007917, A322448, A365683, A366989, A366991, A366992.

f[p_, e_] := p^If[e == 1, 1, NextPrime[e+1, -1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1], f[i, 1]^precprime(f[i, 2])));}

Multiplicative with a(p) = p and a(p^e) = p^A007917(e) for e >= 2.
a(n) <= n, with equality if and only if n is not in A322448.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} f(1/p) = 0.48535795387619596052..., where f(x) = (1 - x) * (1 + Sum_{k>=1} x^(2*k-s(k))), s(k) = A007917(k) for k >= 2, and s(1) = 1.

A378366 Difference between n and the greatest non prime power <= n (allowing 1).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

Sequences obtained by subtracting each term from n are placed in parentheses below.
For nonprime we almost have A010051 (A179278).
For prime we have A064722 (A007917).
For perfect power we have A069584 (A081676).
For squarefree we have (A070321).
For prime power we have A378457 = A276781-1 (A031218).
For nonsquarefree we have (A378033).
For non perfect power we almost have A075802 (A378363).
Subtracting from n gives (A378367).
The opposite is A378371, adding n A378372.
A000015 gives the least prime power >= n (cf. A378370 = A377282 - 1).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A151800 gives the least prime > n, weak version A007918.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A007920, A013632, A065514, A074984, A113646, A345531, A377051, A377054, A377281, A377289, A378357.

Table[n-NestWhile[#-1&,n,PrimePowerQ[#]&],{n,100}]

a(n) = n - A378367(n).

cp's OEIS Frontend

A083714 (greatest prime <= n) - (greatest prime factor of n).

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A083715 (greatest prime <= n) mod (greatest prime factor of n).

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A083717 (Greatest prime <= n) * (greatest prime factor of n).

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A083718 (greatest prime <= n) + (greatest prime factor of n).

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A090119 a(n) = nextprime(A090117(n)), the smallest prime following squares listed in A090117 and also the distance of a(n) from the preceding prime is 2*n.

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A104089 Largest prime <= 4^n.

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A175090 Composites c with result 0 under iterations of {r mod (max prime p <= r)} starting at r = c.

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A273282 Largest prime not exceeding the geometric mean of all prime divisors of n counted with multiplicity.

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A366994 The largest divisor of n that is not a term of A322448.

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