A014076 -id:A014076 - cp's OEIS frontend

A244623 Odd prime powers that are not primes.

1, 9, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 529, 625, 729, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14641, 15625, 16129, 16807, 17161, 18769, 19321, 19683

Offset: 1

Jani Melik, Jul 02 2014

nonn

Intersection of A061345 and A014076.

A014076 set minus A061346.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A025475.

Cf. A061345 (odd prime powers), A061346 (odd neither prime nor prime power), A062739 (odd powerful), A075109 (perfect powers), A136141.

Join[{1},Select[Range[1,20001,2],PrimePowerQ[#]&&(!PrimeQ[#])&]] (* Harvey P. Dale, Dec 11 2018 *)

isok(p) = ((p%2) && !isprime(p) && isprimepower(p)) || (p==1); \\ Michel Marcus, Jul 06 2021

def isA244623(n) :
   return(n % 2 == 1 and is_prime_power(n) == 1 and is_prime(n) == 0)
[n for n in (1..20000) if isA244623(n)]

a(n) = A079290(n) at least in the range n=3..94, and perhaps beyond. - R. J. Mathar, Aug 20 2014

Sum_{n>=1} 1/a(n) = 1/2 + Sum_{p prime} 1/(p*(p-1)) = 1/2 + A136141. - Amiram Eldar, Dec 21 2020

A067800 Nonprime numbers k such that phi(k) > k/2.

Original entry on oeis.org

1, 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99, 111, 115, 117, 119, 121, 123, 125, 129, 133, 135, 141, 143, 145, 147, 153, 155, 159, 161, 169, 171, 175, 177, 183, 185, 187, 189, 201, 203, 205, 207, 209, 213, 215

Offset: 1

Benoit Cloitre, Feb 07 2002

nonn

Sequence is similar to A014076(n) giving odd nonprimes. Only 3 terms = 105, 165, 195 are not in the sequence among 59 terms < 210.
Cototient(m) > totient(m) equivalent to 2*phi(m) < m; the missing terms mentioned here seem to form A036798. - Labos Elemer, May 08 2003
The number 9075 is not in this sequence, is in A014076 and is not in A036798, which means that the missing terms mentioned here do not form A036798 (cf. A118700). - R. J. Mathar, Aug 08 2007

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

Cf. A000010 (phi), A014076, A036798, A051953, A083254, A118700.

Select[Range[250],!PrimeQ[#]&&EulerPhi[#]>#/2&] (* Harvey P. Dale, Aug 29 2021 *)

isok(k) = !isprime(k) && eulerphi(k) > k/2; \\ Amiram Eldar, May 08 2025

A125713 Smallest odd prime p such that (n+1)^p - n^p is prime.

Original entry on oeis.org

3, 3, 3, 3, 5, 3, 7, 7, 3, 3, 3, 17, 3, 3, 43, 5, 3, 1607, 5, 19, 127, 229, 3, 3, 3, 13, 3, 3, 149, 3, 5, 3, 23, 3, 5, 83, 3, 3, 37, 7, 3, 3, 37, 5, 3, 5, 58543, 3, 3, 7, 29, 3, 479, 5, 3, 19, 5, 3, 4663, 54517, 17, 3, 3, 5, 7, 3, 3, 17, 11, 47, 61, 19, 23, 3, 5, 19, 7, 5, 7, 3, 3

Offset: 1

Alexander Adamchuk, Dec 01 2006, Feb 15 2007

Corresponding smallest primes of the form (n+1)^p - n^p, where p = a(n) is an odd prime, are listed in A121091(n+1) = {7, 19, 37, 61, 4651, 127, 1273609, 2685817, 271, 331, 397, 6431804812640900941, 547, 631, ...}. a(n) = A058013(n) for n = {4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, ...} = A047845(n) = (n-1)/2, where n runs through odd nonprimes (A014076), for n>1. a(97) = 7. a(99)..a(112) = {5, 43, 5, 13, 7, 5, 3, 6529, 59, 3, 5, 5, 113, 5}. a(114) = 139. a(117)..a(129) = {7, 13, 3, 5, 5, 7, 3, 5167, 3, 41, 59, 3, 3}. a(131) = 101. a(n) is currently unknown for n = {113, 115, 116, 130, 132, ...}.
a(96) = 1307, a(98) = 709.
a(137) is probably 196873 from a prime of this form discovered by Jean-Louis Charton in December 2009 and reported to Henri Lifchitz's PRP Top. - Robert Price, Feb 17 2012
a(138) through a(150) are 113, >32401, 3, 7, 3, 8839, 5, 7, 13, 3, 5, 271, 13. - Robert Price, Feb 17 2012
a(137) = 196873 confirmed by Fischer link; a(139) > 260000. - Ray Chandler, Feb 26 2017

Robert Price and Ray Chandler, Table of n, a(n) for n = 1..138 (first 136 terms from Robert Price)
Richard Fischer, Generalized primes of the form (B+1)^N - B^N.

Cf. A058013 (smallest prime p such that (n+1)^p - n^p is prime).
Cf. A065913 (smallest prime of form (n+1)^k - n^k).
Cf. A121091 (smallest nexus prime of the form n^p - (n-1)^p, where p is odd prime).
Cf. A047845, A014076.
Cf. A062585 (numbers n such that k^n - (k-1)^n is prime, where k is 19).
Cf. A000043, A057468, A059801, A059802, A062572-A062666.

A163395 a(n) = (n-th even nonprime)^(n-th even number).

Original entry on oeis.org

1, 16, 1296, 262144, 100000000, 61917364224, 56693912375296, 72057594037927936, 121439531096594251776, 262144000000000000000000, 705429498686404044207947776, 2315513501476187716057433112576

Offset: 1

Juri-Stepan Gerasimov, Jul 26 2009

nonn

Here n-th even nonprime = A163300(n), n-th even number = A005843(n), (A014076 U A163300 = A141468).

			a(1) = 0^0 = 1, a(2) = 4^2 = 16, a(3) = 6^4 = 1296.

G. C. Greubel, Table of n, a(n) for n = 1..100

Cf. A005843, A014076, A141468, A163300.

A163300 := proc(n) if n = 1 then 0; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a; end if; end do; end if; end proc: A005843 := proc(n) 2*n ; end: A163395 := proc(n) A163300(n)^A005843(n-1) ; end: seq(A163395(n),n=1..13) ; # R. J. Mathar, Oct 14 2009

Join[{1}, Table[(2*(n - 1 + Mod[(n + 1), n]))^(2 n - 2), {n, 2, 10}]] (* G. C. Greubel, Dec 21 2016 *)
With[{nn=30},Range[2,nn,2]^Range[0,nn-2,2]] (* Harvey P. Dale, Nov 09 2017 *)

a(n) = A163300(n)^A005843(n).

Extended by R. J. Mathar, Oct 14 2009

A175761 Odd nonprimes such that the arithmetic mean of all prime factors is not an integer.

Original entry on oeis.org

1, 45, 63, 75, 99, 117, 135, 147, 153, 165, 171, 175, 207, 245, 255, 261, 273, 279, 315, 325, 333, 345, 351, 363, 369, 375, 385, 387, 399, 405, 423, 435, 455, 459, 475, 477, 495, 507, 531, 539, 549, 561, 567, 595, 603, 615, 639, 651, 657, 665, 675, 705, 711, 715, 735

Offset: 1

Juri-Stepan Gerasimov, Aug 29 2010

The presence of 1 (which has no prime factors) is for compatibility with A175352.

			a(6) = 117 because 117 = 3*3*13 and (3 + 3 + 13)/3 is not an integer.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A141468, A175352.

fQ[n_] := Block[{fi = Flatten[ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, len = Length@ fi; len != 1 && ! IntegerQ[ Plus @@ fi/len]]; Join[{1},Select[1 + 2 Range@ 356, fQ]] (* Robert G. Wilson v, Aug 31 2010 *)

Equals: Intersection of A175352 and A014076.

Corrected (315, 345 inserted, 355 removed) by R. J. Mathar, Aug 30 2010

A270003 Least prime p such that n = p + q - r for some primes q and r with q > p.

Original entry on oeis.org

3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2

Offset: 1

Clark Kimberling, Apr 26 2016

p = 3 when n is an odd nonprime and p = 2 otherwise, so that 3 appears in positions given by A014076.

			n   p   q   r
1   3   5   7
2   2   3   3
3   2   3   2
4   2   5   3
5   2   5   2
6   2   7   3
7   2   7   2

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A000040, A270753, A271353.

t = Join[{{1, {3, 5, 7}}, {2, {2, 3, 3}}}, Table[If[PrimeQ[n], {n, {2, n, 2}}, p = If[EvenQ[2 + NextPrime[n, 1] - n], 3, 2]; NestWhile[# + 1 &, 1, ! PrimeQ[r = (p + (q = NextPrime[n, #])) - n] &]; {n, {p, q, r}}], {n, 3, 300}]];
Map[#[[2]][[1]] &, t] (* p, A270003 *)
Map[#[[2]][[2]] &, t] (* q, A270753 *)
Map[#[[2]][[3]] &, t] (* r, A271353 *)
(* Peter J. C. Moses, Apr 26 2016 *)

a(n)=if(n%2 && !isprime(n), 3, 2) \\ Charles R Greathouse IV, Apr 29 2016

A162022 Smallest prime factor of n-th odd composite integers A071904.

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 5, 3, 3, 5, 3, 3, 7, 3, 5, 3, 7, 3, 5, 3, 3, 3, 5, 3, 7, 11, 3, 5, 3, 7, 3, 3, 11, 5, 3, 3, 5, 3, 7, 3, 13, 3, 5, 3, 3, 5, 11, 3, 3, 3, 7, 5, 3, 11, 3, 5, 7, 3, 13, 3, 3, 5, 3, 3, 5, 13, 3, 11, 3, 7, 3, 5, 3, 3, 5, 3, 3, 7, 17, 3, 5, 3, 13, 7, 3, 5, 3, 3, 11, 3, 17, 5, 3, 7, 3

Offset: 1

Zak Seidov, Jun 25 2009

nonn

Records are for n's such that A071904(n) = squares of primes.

a(n) = A020639(A071904(n)). [Reinhard Zumkeller, Oct 10 2011]

			A071904(1)=9, hence a(1)=3, A071904(4)=25, hence a(4)=5.

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

Cf. A014076, A071904.

nn=501;With[{ci=Complement[Range[9,nn,2],Prime[Range[PrimePi[nn]]]]}, FactorInteger[ #][[1,1]]&/@ci] (* Harvey P. Dale, Nov 30 2012 *)

from sympy import primepi, primefactors
def A162022(n):
    if n == 1: return 3
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return min(primefactors(m)) # Chai Wah Wu, Jul 31 2024

Corrected example a(4)=5 Francesco Antoni (francesco_antoni(AT)yahoo.com), Aug 04 2010

A251558 a(n) = smallest odd number not in {A098550(1), A098550(2), ..., A098550(n)} which is neither a prime nor a term of A251542.

Original entry on oeis.org

9, 9, 9, 9, 15, 15, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 27, 27, 33, 33, 33, 33, 33, 45, 45, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 57, 57, 57, 57, 57, 57, 57, 69, 69, 75, 75, 75, 75, 75, 75, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 105, 105, 105

Offset: 1

N. J. A. Sloane, Dec 23 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015 and J. Int. Seq. 18 (2015) 15.6.7.

Cf. A098550, A251542, A251546, A251557, A251559.

Cf. A014076.

import Data.List (delete); import Data.List.Ordered (minus)
a251558 n = a251558_list !! (n-1)
a251558_list = 9 : 9 : 9 : f 2 3 [4..] (tail a014076_list) where
   f u v ws zs = g ws where
     g (x:xs) = if gcd x u > 1 && gcd x v == 1
                   then y : f v x (delete x ws) ys else g xs
                where ys@(y:_) = zs `minus` [x]
-- Reinhard Zumkeller, Mar 11 2015

terms = 70; max = 2 terms;
f[lst_] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]];
A098550 = Nest[f, {1, 2, 3}, max-3];
sel = Select[Transpose[{Range[max], A098550}], PrimeQ[#[[2]]]&][[All,1]]+2;
A251542 = A098550[[sel]]/A098550[[sel-2]] ;
a[n_] := For[k = 1, k <= max, k = k+2, If[CompositeQ[k] && FreeQ[A098550[[1 ;; n]], k] && FreeQ[A251542, k], Return[k]]];
Table[a[n], {n, 1, terms}] (* Jean-François Alcover, Dec 06 2018, after Robert G. Wilson v in A098550 *)

A354370 Successive pairs of terms (i, j) such that (i + j) is a prime number and at least i is a prime number. This is the lexicographically earliest infinite sequence of distinct terms > 1 with this property.

Original entry on oeis.org

2, 3, 5, 6, 7, 4, 11, 8, 13, 10, 17, 12, 19, 18, 23, 14, 29, 24, 31, 16, 37, 22, 41, 20, 43, 28, 47, 26, 53, 30, 59, 38, 61, 36, 67, 34, 71, 32, 73, 40, 79, 48, 83, 44, 89, 42, 97, 52, 101, 50, 103, 46, 107, 56, 109, 54, 113, 60, 127, 64, 131, 62, 137, 74, 139, 58, 149, 78, 151, 72, 157, 66, 163, 70

Offset: 1

Eric Angelini and Carole Dubois, May 24 2022

nonn

The terms 1, 9, 15, 21, 25, 27, 33, 35, 39, 45, ... will never appear in the sequence; they form A014076, the "Odd nonprimes". Two prime terms can form a pair (2 and 3 for instance) but the first term must always be a prime [the pair (5, 6) is ok].

			The earliest pairs with their prime sum: (2, 3) = 5, (5, 6) = 11, (7, 4) = 11, (11, 8) = 19, (13, 10) = 23, (17, 12) = 29, (19, 18) = 37, (23, 14) = 37, etc.

Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..10^4, showing records in red and local minima in blue, highlighting fixed points in gold and composite powers of 2 in magenta.

Cf. A354367, A354368, A354369 (same idea), A014076.

nn = 120; c[] = 0; a[1] = 2; c[2] = 1; u = 3; Do[If[EvenQ[i], k = u; While[Nand[c[k] == 0, PrimeQ[# + k]] &[a[i - 1]], k++], k = u; While[Nand[c[k] == 0, PrimeQ[k]], k++]]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[Or[c[u] > 0, And[OddQ[u], CompositeQ[u]]], u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, May 24 2022 *)

from sympy import isprime
from itertools import islice
def agen(): # generator of terms
    i, j, v, aset = 2, 3, 4, set()
    while True:
        aset.update((i, j)); yield from (i, j)
        i = j = v
        while i in aset or not isprime(i): i += 1
        while j == i or j in aset or not isprime(i+j): j += 1
        while v in aset: v += 1
print(list(islice(agen(), 74))) # Michael S. Branicky, Jun 24 2022

A065824 Smallest solution m to (n+1)phi(m) = nsigma(m), or -1 if no solution exists.

Original entry on oeis.org

3, 5, 7, 323, 11, 13, 899, 17, 19, 1763, 23, 5249, 3239, 29, 31, 979801, 5459, 37, 10763, 41, 43, 9179, 47, 9701, 10403, 53, 12319, 5646547, 59, 61, 24569, 19109, 67, 19043, 71, 73, 22499, 50819, 79, 41309, 83, 32639, 46979, 89, 34579, 39059, 125969

Offset: 1

Labos Elemer, Nov 23 2001

If p = a(n) is a prime solution, then (n+1)*(p-1) = n*(p+1) and p = 2n+1, so position for p if it is in fact a minimal solution is at n = (p-1)/2. E.g. 29 appears at 14th position shown by A005097. On the other hand large and (seemingly always composite) solutions arise at indices shown essentially by A047845. Also, differences between the sites of two consecutive small prime solutions appears to be d/2, half the difference between consecutive primes (A001223).

Donovan Johnson, Table of n, a(n) for n = 1..456

Cf. A000010, A000203, A062699, A065818, A065819, A065822, A065823.

See also A005097, A047845, A014076, A001223.

max = 10^7; a[n_] := For[m = 3, m <= max, m++, If[(n+1)*EulerPhi[m] == n*DivisorSigma[1, m], Print[m]; Return[m]]] /. Null -> -1; Array[a, 50] (* Jean-François Alcover, Oct 08 2016 *)

from itertools import count
from math import prod
from sympy import factorint
def A065824(n):
    for m in count(1):
        f = factorint(m)
        if (n+1)*m*prod((p-1)**2 for p in f)==n*prod(p**(e+2)-p for p,e in f.items()):
            return m # Chai Wah Wu, Aug 12 2024

(n+1)*A000010(a(n)) = n*A000203(a(n)), smallest x=a(n) solutions.

cp's OEIS Frontend

A244623 Odd prime powers that are not primes.

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A067800 Nonprime numbers k such that phi(k) > k/2.

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A125713 Smallest odd prime p such that (n+1)^p - n^p is prime.

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A163395 a(n) = (n-th even nonprime)^(n-th even number).

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A175761 Odd nonprimes such that the arithmetic mean of all prime factors is not an integer.

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A270003 Least prime p such that n = p + q - r for some primes q and r with q > p.

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A162022 Smallest prime factor of n-th odd composite integers A071904.

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A251558 a(n) = smallest odd number not in {A098550(1), A098550(2), ..., A098550(n)} which is neither a prime nor a term of A251542.

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A065824 Smallest solution m to (n+1)phi(m) = nsigma(m), or -1 if no solution exists.