A013632 -id:A013632 - cp's OEIS frontend

A075046 a(n) = the smallest number k such that the number of divisors of the n numbers from k through k+n-1 are in nondescending order.

1, 1, 1, 1, 241, 241, 12853, 12853, 234613, 376741, 78312721, 125938261, 4019167441, 16586155153, 35237422882, 1296230533473, 42301168491121, 61118966262061

Offset: 1

Amarnath Murthy, Sep 03 2002

tau(k) <= tau(k+1) <= ... <= tau(k+n-1).
a(16) > 10^12. - Donovan Johnson, Oct 13 2009
a(17) > 10^13. - Giovanni Resta, Apr 12 2017
a(19) > 2.64*10^15. - Jud McCranie, Mar 27 2019
If a(n) > 1, then A013632(a(n)) >= n. Might be useful to help speed up brute force search. - Chai Wah Wu, May 04 2017

			a(5) = 241 = a(6) as tau(241) = 2 < tau(242) = tau(243) = tau(244) = tau(245) = 6 < tau(246).

Cf. A075028, A075029, A075031, A045983, A006073, A075044, A055927, A075047.

k = 1; Do[ While[t = Table[ DivisorSigma[0, i], {i, k, k + n - 1}]; t != Sort[t], k++ ]; Print[k], {n, 1, 11}]

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003
a(11) from Robert G. Wilson v, Sep 07 2003
a(12)-a(15) from Donovan Johnson, Oct 13 2009
a(16) from Fred Schneider, Mar 29 2017
a(17)-a(18) from Jud McCranie, Mar 27 2019

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).

A013634 a(n) = nextprime(n) + n.

Original entry on oeis.org

2, 3, 5, 8, 9, 12, 13, 18, 19, 20, 21, 24, 25, 30, 31, 32, 33, 36, 37, 42, 43, 44, 45, 52, 53, 54, 55, 56, 57, 60, 61, 68, 69, 70, 71, 72, 73, 78, 79, 80, 81, 84, 85, 90, 91, 92, 93, 100, 101, 102, 103, 104, 105, 112, 113, 114, 115, 116, 117, 120, 121, 128, 129, 130

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A013632, A151800.

[NextPrime(n) + n: n in [0..80]]; // Vincenzo Librandi, Dec 27 2018

Maple
```
[ seq(nextprime(i)+i,i=0..100) ];
```

Table[n+NextPrime[n],{n,0,120}] (* Harvey P. Dale, May 03 2013 *)
Array[NextPrime[#] + # &, 80, 0] (* Vincenzo Librandi, Dec 27 2018 *)

a(n) = A151800(n) + n.

A060846 Smallest prime > the n-th nontrivial power of a prime.

Original entry on oeis.org

5, 11, 11, 17, 29, 29, 37, 53, 67, 83, 127, 127, 131, 173, 251, 257, 293, 347, 367, 521, 541, 631, 733, 853, 967, 1031, 1361, 1373, 1693, 1861, 2053, 2203, 2203, 2213, 2411, 2819, 3137, 3491, 3727, 4099, 4493, 4919, 5051, 5333, 6247, 6563, 6863, 6899, 7927

Offset: 1

Labos Elemer, May 03 2001

nonn

			78125=5^7 is followed by 78137.

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

Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711, A060845, A068435, A246547.

NextPrime[Select[Range[10^4], !PrimeQ[#] && PrimePowerQ[#] &]] (* Amiram Eldar, Oct 04 2024 *)

ispp(x) = !isprime(x) && isprimepower(x);
lista(nn) = apply(x->nextprime(x), select(x->ispp(x), [1..nn])); \\ Michel Marcus, Aug 24 2019

from sympy import primepi, integer_nthroot, nextprime
def A060846(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    return nextprime(bisection(f,n,n)) # Chai Wah Wu, Sep 15 2024

a(n) = nextprime(A025475(n+1)) = A007918(A025475(n+1)) = Min{p| p>A025475(n+1)}. [corrected by Michel Marcus, Aug 24 2019]

A072918 a(n) = p(n) - sigma(n), where p(n) is the least prime greater than sigma(n).

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 3, 2, 4, 1, 1, 1, 3, 5, 5, 6, 1, 2, 3, 1, 5, 1, 5, 1, 6, 1, 1, 3, 1, 1, 5, 4, 5, 5, 5, 6, 3, 1, 3, 7, 1, 1, 3, 5, 1, 1, 5, 3, 2, 4, 1, 3, 5, 7, 1, 7, 3, 7, 1, 5, 5, 1, 3, 4, 5, 5, 3, 1, 1, 5, 1, 2, 5, 13, 3, 9, 1, 5, 3, 5, 6, 1, 5, 3, 1, 5, 7, 1, 7, 5, 1, 5, 3, 5, 7, 5, 3, 2, 1, 6

Offset: 1

Joseph L. Pe, Aug 11 2002

			phi(4) = 7 and the least prime > 7 is 11; hence a(4) = 11 - 7 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A013632, A202111.

ps[n_]:=Module[{sn=DivisorSigma[1,n]}, NextPrime[sn]-sn]; ps/@Range[100] (* Harvey P. Dale, Feb 02 2011 *)

A072918(n) = (nextprime(1+sigma(n)) - sigma(n)); \\ Antti Karttunen, Nov 07 2017

a(n) = A013632(A000203(n)). - Antti Karttunen, Nov 07 2017

A084695 Triangle read by rows in which row n lists the n smallest positive numbers k such that k + n is a prime.

Original entry on oeis.org

1, 1, 3, 2, 4, 8, 1, 3, 7, 9, 2, 6, 8, 12, 14, 1, 5, 7, 11, 13, 17, 4, 6, 10, 12, 16, 22, 24, 3, 5, 9, 11, 15, 21, 23, 29, 2, 4, 8, 10, 14, 20, 22, 28, 32, 1, 3, 7, 9, 13, 19, 21, 27, 31, 33, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 1, 5, 7, 11, 17, 19, 25, 29, 31, 35, 41, 47

Offset: 1

Amarnath Murthy and Jason Earls, Jul 12 2003

			Triangle begins:
  1;
  1,  3;
  2,  4,  8;
  1,  3,  7,  9;
  2,  6,  8, 12, 14;
  1,  5,  7, 11, 13, 17;
  4,  6, 10, 12, 16, 22, 24;

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

First column gives A013632, last gives A084747.

Cf. A000040, A000720.

[NthPrime(#PrimesUpTo(n) +k) -n: k in [1..n], n in [1..16]]; // G. C. Greubel, May 12 2023

nn=30;Flatten[With[{prs=Prime[Range[nn]]},Table[Take[prs,{PrimePi[n]+1, PrimePi[n]+n}]-n,{n,Floor[nn/2]}]]] (* Harvey P. Dale, Dec 07 2012 *)
Table[Prime[PrimePi[n] +k] -n, {n,16}, {k,n}]//Flatten (* G. C. Greubel, May 12 2023 *)

def A084695(n,k): return nth_prime(prime_pi(n) + k) - n
flatten([[A084695(n,k) for k in range(1,n+1)] for n in range(1,17)]) # G. C. Greubel, May 12 2023

T(n, k) = prime(PrimePi(n) + k) - n. - G. C. Greubel, May 12 2023

A117217 Common prime gap associated with the primes A122535.

Original entry on oeis.org

2, 6, 6, 6, 12, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 12, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 12, 6, 6, 6, 6, 6, 12, 6, 6, 6, 6, 12, 12, 6, 12, 6, 6, 6, 6, 6, 6, 6, 6, 12, 6, 6, 6, 6, 6, 6, 6, 6, 12, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Lekraj Beedassy, Mar 04 2006

nonn

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A006562, A013632, A122535. - Zak Seidov, Feb 12 2013

a(n)=A013632(A122535(n)). - R. J. Mathar, Apr 11 2008

a(n)=A006562(n)-A122535(n). - Zak Seidov, Feb 12 2013

Corrected and extended by R. J. Mathar, Apr 11 2008

A159978 a(n) = (smallest prime > Fibonacci(n)) - Fibonacci(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 2, 3, 4, 8, 5, 6, 2, 3, 4, 4, 7, 20, 14, 3, 2, 4, 13, 4, 10, 11, 16, 14, 23, 4, 4, 25, 10, 14, 35, 6, 24, 3, 2, 6, 7, 12, 20, 9, 48, 10, 5, 28, 18, 23, 14, 14, 11, 16, 10, 21, 4, 62, 13, 38, 12, 7, 16, 12, 19, 36, 28, 143, 32, 58, 29, 96, 100, 33, 2, 30, 27, 12, 62, 25

Offset: 1

Enoch Haga, Apr 28 2009

			a(6)=3 because the 6th Fibonacci term is 8 and the distance to nextprime(6) is 3 (11-8=3).

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000045, A013632, A159977.

A159978 := proc(n) local f; f := combinat[fibonacci](n) ; nextprime(f)-f ; end: seq(A159978(n),n=1..100) ; # R. J. Mathar, Apr 29 2009

Table[f = Fibonacci[n]; NextPrime[f] - f, {n, 200}] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)

a(n) = my(f=fibonacci(n)); nextprime(f+1) - f; \\ Michel Marcus, Sep 22 2022

10 'FiboB 20 A=1:print A; 30 B=1:print B; 40 C=A+B:print C;:T=T+1:print "<";nxtprm(C)-C;">"; 50 D=B+C:print D;:print "<";nxtprm(D)-D;">"; 60 A=C:B=D:if T>22 then stop:else 40

a(n) = A013632(A000045(n)). - R. J. Mathar, Apr 29 2009

Extended by R. J. Mathar, Apr 29 2009

A171228 n^(p-n) where p is smallest prime > n.

Original entry on oeis.org

0, 1, 2, 9, 4, 25, 6, 2401, 512, 81, 10, 121, 12, 28561, 2744, 225, 16, 289, 18, 130321, 8000, 441, 22, 148035889, 7962624, 390625, 17576, 729, 28, 841, 30, 887503681, 33554432, 1185921, 39304, 1225, 36, 1874161, 54872, 1521, 40, 1681, 42, 3418801

Offset: 0

Juri-Stepan Gerasimov, Dec 05 2009

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A013632, A013636.

Table[n^(NextPrime[n]-n),{n,0,50}] (* Harvey P. Dale, Jun 20 2013 *)

Definition corrected by Andrew Weimholt, Dec 08 2009

A353072 Numbers k such that nextprime(k)-k is a positive square.

Original entry on oeis.org

1, 2, 4, 6, 7, 10, 12, 13, 16, 18, 19, 22, 25, 28, 30, 33, 36, 37, 40, 42, 43, 46, 49, 52, 55, 58, 60, 63, 66, 67, 70, 72, 75, 78, 79, 82, 85, 88, 93, 96, 97, 100, 102, 103, 106, 108, 109, 112, 118, 123, 126, 127, 130, 133, 136, 138, 140, 145, 148, 150, 153

Offset: 1

Tanya Khovanova, Apr 21 2022

nonn

Numbers p-1, where p is prime is a subsequence (see A006093).

			The next prime after 7 is 11, and 11-7 = 4 a square, so 7 is in this sequence.
The next prime after 118 is 127, 127-118 = 9 is a square, so 118 is a term.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A006093, A013632, A029710.

Select[Range[2000], NextPrime[#] - # > 0 && IntegerQ[Sqrt[NextPrime[#] - #]] &]
npsQ[n_]:=With[{c=NextPrime[n]-n},c>0&&IntegerQ[Sqrt[c]]]; Select[Range[200],npsQ] (* Harvey P. Dale, May 05 2023 *)

upto(n) = {my(res = List(1), q = 2, u = nextprime(n + 1)); forprime(p = 3, u, forstep(i = sqrtint(p - q), 1, -1, listput(res, p-i^2) ); q = p ); res } \\ David A. Corneth, Apr 22 2022

isok(k) = issquare(nextprime(k+1)-k); \\ Michel Marcus, Apr 22 2022

cp's OEIS Frontend

A075046 a(n) = the smallest number k such that the number of divisors of the n numbers from k through k+n-1 are in nondescending order.

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

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A013634 a(n) = nextprime(n) + n.

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Magma

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A060846 Smallest prime > the n-th nontrivial power of a prime.

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PARI

Python

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A072918 a(n) = p(n) - sigma(n), where p(n) is the least prime greater than sigma(n).

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PARI

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A084695 Triangle read by rows in which row n lists the n smallest positive numbers k such that k + n is a prime.

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Magma

Mathematica

SageMath

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A117217 Common prime gap associated with the primes A122535.

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A159978 a(n) = (smallest prime > Fibonacci(n)) - Fibonacci(n).

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