A013632 -id:A013632 - cp's OEIS frontend

A013598 a(n) = nextprime(3^n)-3^n.

1, 2, 2, 2, 2, 8, 4, 16, 2, 4, 2, 20, 16, 8, 2, 2, 26, 34, 10, 56, 8, 56, 4, 32, 2, 14, 2, 16, 26, 130, 4, 16, 70, 70, 34, 22, 2, 50, 8, 82, 118, 70, 4, 52, 8, 46, 68, 52, 56, 16, 28, 34, 50, 26, 28, 20, 62, 4, 158, 64, 16, 34, 122, 2, 92, 64, 28, 230, 20

Offset: 0

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

Zak Seidov, Table of n, a(n) for n = 0..1000

Cf. A013604.

Maple
```
seq(nextprime(3^i)-3^i,i=0..100);
```

np[n_]:=Module[{c=3^n},NextPrime[c]-c]; Array[np,80,0] (* Harvey P. Dale, Jul 14 2014 *)

a(n) = A151800(3^n)-3^n = A013632(3^n). - R. J. Mathar, Nov 28 2016

Corrected by Harvey P. Dale, Jul 14 2014

A013599 a(n) = nextprime(5^n) - 5^n.

Original entry on oeis.org

1, 2, 4, 2, 6, 12, 4, 12, 22, 26, 4, 14, 58, 6, 12, 42, 24, 2, 12, 56, 48, 24, 18, 38, 58, 14, 12, 38, 34, 62, 28, 92, 214, 122, 102, 168, 136, 18, 48, 102, 108, 126, 18, 126, 76, 108, 22, 204, 52, 122, 96, 114, 94, 14, 52, 38, 58, 248, 64, 56, 16, 102, 106

Offset: 0

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

Zak Seidov, Table of n, a(n) for n = 0..500

Cf. A000351, A013632, A054321, A151800,

Maple
```
seq(nextprime(5^i)-5^i, i=0..100);
```

NextPrime[#]-#&/@(5^Range[0,70]) (* Harvey P. Dale, Sep 29 2011 *)

a(n) = nextprime(5^n+1) - 5^n; \\ Michel Marcus, Jun 14 2020

a(n) = A151800(5^n)-5^n = A054321(n)-5^n = A013632(5^n). - R. J. Mathar, Nov 28 2016

A013600 a(n) = nextprime(6^n)-6^n.

Original entry on oeis.org

1, 1, 1, 7, 1, 13, 7, 5, 11, 25, 5, 35, 35, 35, 5, 91, 47, 35, 5, 17, 11, 7, 103, 61, 7, 13, 23, 7, 25, 47, 7, 73, 5, 41, 133, 77, 101, 103, 193, 61, 47, 187, 71, 35, 215, 83, 121, 95, 37, 95, 145, 35, 77, 13, 7, 5, 25, 77, 47, 283, 235, 23, 137, 137, 47

Offset: 0

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 0..1000

Cf. A000400, A013632, A063766.

Maple
```
seq(nextprime(6^i)-6^i,i=0..100);
```

(NextPrime[#]-#)&/@(6^Range[0,70]) (* Harvey P. Dale, May 04 2024 *)

a(n) = nextprime(6^n)-6^n; \\ Michel Marcus, Jan 08 2020

a(n) = A063766(n)-6^n = A013632(6^n). - R. J. Mathar, Nov 28 2016

A284597 a(n) is the least number that begins a run of exactly n consecutive numbers with a nondecreasing number of divisors, or -1 if no such number exists.

Original entry on oeis.org

46, 5, 43, 1, 1613, 241, 17011, 12853, 234613, 376741, 78312721, 125938261, 4019167441, 16586155153, 35237422882, 1296230533473, 42301168491121, 61118966262061

Offset: 1

Fred Schneider, Mar 29 2017

The words "begins" and "exactly" in the definition are crucial. The initial values of tau (number of divisors function, A000005) can be partitioned into nondecreasing runs as follows: {1, 2, 2, 3}, {2, 4}, {2, 4}, {3, 4}, {2, 6}, {2, 4, 4, 5}, {2, 6}, {2, 6}, {4, 4}, {2, 8}, {3, 4, 4, 6}, {2, 8}, {2, 6}, {4, 4, 4, 9}, {2, 4, 4, 8}, {2, 8}, {2, 6, 6}, {4}, {2, 10}, ... From this we can see that a(1) = 46 (the first singleton), a(2)=5 (the first pair), a(3)=43 (the first triple), a(4)=1, etc. - Bill McEachen and Giovanni Resta, Apr 26 2017. (see also A303577 and A303578 - N. J. A. Sloane, Apr 29 2018)
Initial values computed with a brute force C++ program.
It seems very likely that one can always find a(n) and that we never need to take a(n) = -1. But this is at present only a conjecture. - N. J. A. Sloane, May 04 2017
Conjecture follows from Dickson's conjecture (see link). - Robert Israel, Mar 30 2020
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
The analog sequence for sigma (sum of divisors) instead of tau (number of divisors) is A285893 (see also A028965). - M. F. Hasler, May 06 2017
a(n) > 3.37*10^14 for n > 18. - Robert Gerbicz, May 14 2017

			241 = 241^1 => 2 divisors
242 = 2^1 * 11^2 => 6 divisors
243 = 3^5 => 6 divisors
244 = 2^2 * 61^1 => 6 divisors
245 = 5^1 * 7^2 => 6 divisors
246 = 2^1 * 3^1 * 41^1 => 8 divisors
247 = 13^1 * 19^1 => 4 divisors
So, 247 breaks the chain. 241 is the lowest number that is the beginning of exactly 6 consecutive numbers with a nondecreasing number of divisors. So it is the 6th term in the sequence.
Note also that a(5) is not 242, even though tau evaluated at 242, 243,..., 246 gives 5 nondecreasing values, because here we deal with full runs and 242 belongs to the run of 6 values starting at 241.

Robert Israel, Dickson's conjecture implies all a(n)>0

Cf. A000005, A000203, A006558, A013632, A028965, A075046, A285893.

See also A286287, A286288, A286289, A303577, A303578.

Function[s, {46}~Join~Map[Function[r, Select[s, Last@ # == r &][[1, 1]]], Range[2, Max[s[[All, -1]] ] ]]]@ Map[{#[[1, 1]], Length@ # + 1} &, DeleteCases[SplitBy[#, #[[-1]] >= 0 &], k_ /; k[[1, -1]] < 0]] &@ MapIndexed[{First@ #2, #1} &, Differences@ Array[DivisorSigma[0, #] &, 10^6]] (* Michael De Vlieger, May 06 2017 *)

genit()={for(n=1,20,q=0;ibgn=1;for(m=ibgn,9E99,mark1=q;q=numdiv(m);if(mark1==0,summ=0;dun=0;mark2=m);if(q>=mark1,summ+=1,dun=1);if(dun>0&&summ==n,print(n," ",mark2);break);if(dun>0&&summ!=n,q=0;m-=1)));} \\ Bill McEachen, Apr 25 2017

A284597=vector(19);apply(scan(N,s=1,t=numdiv(s))=for(k=s+1,N,t>(t=numdiv(k))||next;k-s>#A284597||A284597[k-s]||printf(" a(%d)=%d,",k-s,s)||A284597[k-s]=s;s=k);done,[10^6]) \\ Finds a(1..10) in ~ 1 sec, but would take 100 times longer to get one more term with scan(10^8). You may extend the search using scan(END,START). - M. F. Hasler, May 06 2017

from sympy import divisor_count
def A284597(n):
    count, starti, s, i = 0,1,0,1
    while True:
        d = divisor_count(i)
        if d < s:
            if count == n:
                return starti
            starti = i
            count = 0
        s = d
        i += 1
        count += 1 # Chai Wah Wu, May 04 2017

a(1), a(2), a(4) corrected by Bill McEachen and Giovanni Resta, Apr 26 2017

a(17)-a(18) from Robert Gerbicz, May 14 2017

A378370 Distance between n and the least prime power >= n, allowing 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 27 2024

nonn

Prime powers allowing 1 are listed by A000961.

Sequences obtained by adding n to each term are placed in parentheses below.
For prime instead of prime power we have A007920 (A007918), strict A013632.
For perfect power we have A074984 (A377468), opposite A069584 (A081676).
For squarefree we have A081221 (A067535).
The restriction to the prime numbers is A377281 (A345531).
The strict version is A377282 = a(n) + 1.
For non prime power instead of prime power we have A378371 (A378372).
The opposite version is A378457, strict A276781.
A000015 gives the least prime power >= n, opposite A031218.
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.
Prime-powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A053707, A065514, A065890, A343249, A376596, A376597, A377051, A377054, A377289, A377781.

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

a(n) = A000015(n) - n.

a(n) = A377282(n - 1) - 1 for n > 1.

A378367 Greatest non prime power <= n, allowing 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 6, 6, 6, 10, 10, 12, 12, 14, 15, 15, 15, 18, 18, 20, 21, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 30, 33, 34, 35, 36, 36, 38, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 48, 50, 51, 52, 52, 54, 55, 56, 57, 58, 58, 60, 60, 62, 63, 63, 65, 66, 66

Offset: 1

Gus Wiseman, Nov 29 2024

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, ...

			The greatest non prime power <= 7 is 6, so a(7) = 6.

Sequences obtained by subtracting each term from n are placed in parentheses below.
For prime we have A007917 (A064722).
For nonprime we have A179278 (A010051 almost).
For perfect power we have A081676 (A069584).
For squarefree we have A070321.
For nonsquarefree we have A378033.
For non perfect power we have A378363.
The opposite is A378372, subtracting n A378371.
For prime power we have A031218 (A276781 - 1).
Subtracting from n gives (A378366).
A000015 gives the least prime power >= n (A378370).
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 (A013632), weak version A007918 (A007920).
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A065514, A113646, A345531 (A377281), A377051, A377054, A377282, A377468 (A074984), A378358, A378457.
Cf. A356068.

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

a(n) = n - A378366(n).

a(n) = A361102(A356068(n)). - Ridouane Oudra, Aug 22 2025

A013601 a(n) = nextprime(7^n)-7^n.

Original entry on oeis.org

1, 4, 4, 4, 10, 4, 10, 4, 16, 4, 18, 10, 16, 40, 4, 66, 6, 24, 48, 24, 16, 52, 102, 4, 46, 60, 10, 24, 76, 10, 114, 18, 90, 40, 24, 36, 6, 72, 22, 24, 232, 10, 54, 60, 216, 160, 174, 34, 48, 24, 382, 88, 48, 10, 124, 10, 58, 34, 132, 214, 46, 22, 40, 136

Offset: 0

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 0..1000

Cf. A013632, A063767.

Maple
```
seq(nextprime(7^i)-7^i,i=0..100);
```

a(n) = nextprime(7^n+1) - 7^n; \\ Michel Marcus, Aug 13 2019

a(n) = A063767(n)-7^n = A013632(7^n). - R. J. Mathar, Nov 28 2016

A203074 a(0)=1; for n > 0, a(n) = next prime after 2^(n-1).

Original entry on oeis.org

1, 2, 3, 5, 11, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659

Offset: 0

Frank M Jackson and N. J. A. Sloane, Dec 28 2011

nonn

Equals {1} union A014210. Unlike A014210, every positive integer can be written in one or more ways as a sum of terms of this sequence. See A203075, A203076.

a(n)*2^(n-1) = A133814(n-1) for n > 1 and a(n)*2^(n-1) for n > O is a subsequence of primitive practical numbers (A267124). - Frank M Jackson, Dec 29 2024

			a(5) = 17, since this is the next prime after 2^(5-1) = 2^4 = 16.

M. F. Hasler & Bill McEachen, Table of n, a(n) for n = 0..1300 (missing lines n = 1159..1165 from Bill McEachen)
Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Cf. A013632, A013597, A014210, A104080, A203075, A203076.

[1] cat [NextPrime(2^(n-1)): n in [1..40]]; // Vincenzo Librandi, Feb 23 2018

nextprime[n_Integer] := (k=n+1;While[!PrimeQ[k], k++];k); aprime[m_Integer] := (If[m==0, 1, nextprime[2^(m-1)]]); Table[aprime[l], {l,0,100}]
nxt[{n_,a_}]:={n+1,NextPrime[2^n]}; NestList[nxt,{0,1},40][[All,2]] (* Harvey P. Dale, Oct 10 2017 *)

a(n)=if(n,nextprime(2^n/2+1),1) \\ Charles R Greathouse IV

A203074(n)=nextprime(2^(n-1)+1)-!n  \\ M. F. Hasler, Mar 15 2012

A203074(n) = 2^(n-1) + A013597(n-1), for n > 0. - M. F. Hasler, Mar 15 2012

a(n) = A104080(n-1) for n > 2. - Georg Fischer, Oct 23 2018

A378457 Difference between n and the greatest prime power <= n, allowing 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Prime powers allowing 1 are listed by A000961.

			The greatest prime power <= 6 is 5, so a(6) = 1.

Sequences obtained by subtracting each term from n are placed in parentheses below.
For nonprime we have A010051 (almost) (A179278).
Subtracting from n gives (A031218).
For prime we have A064722 (A007917).
For perfect power we have A069584 (A081676).
For squarefree we have (A070321).
Adding one gives A276781.
For nonsquarefree we have (A378033).
For non perfect power we have (A378363).
For non prime power we have A378366 (A378367).
The opposite is A378370 = A377282-1.
A000015 gives the least prime power >= n.
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. A001597, A007920, A013632, A065514, A074984, A377051, A377054, A377281, A377289, A377468, A378357, A378371.

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

a(n) = n - A031218(n).

a(n) = A276781(n) - 1.

A127796 a(n) = nextprime(9^n) - 9^n.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 16, 2, 26, 10, 8, 4, 2, 2, 26, 4, 70, 34, 2, 8, 118, 4, 8, 68, 56, 28, 50, 28, 62, 158, 16, 122, 92, 28, 20, 110, 140, 70, 28, 44, 20, 124, 316, 38, 8, 44, 136, 58, 110, 2, 148, 170, 116, 170, 40, 2, 182, 10, 46, 254, 56, 14, 8, 2, 190, 148, 382, 10, 56, 10

Offset: 0

Artur Jasinski, Jan 29 2007

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 0..1000

Cf. A013597, A013598, A013602, A013599, A013600, A013601, A127796, A033873, A127797, A127798, A127799.

Cf. A013632, A001019.

<< NumberTheory`NumberTheoryFunctions` a = {}; Do[k = NextPrime[9^x] - 9^x; AppendTo[a, k], {x, 0, 100}]; a

a(n) = A013632(A001019(n)). - Michel Marcus, Nov 18 2019

cp's OEIS Frontend

A013598 a(n) = nextprime(3^n)-3^n.

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A013599 a(n) = nextprime(5^n) - 5^n.

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A013600 a(n) = nextprime(6^n)-6^n.

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A284597 a(n) is the least number that begins a run of exactly n consecutive numbers with a nondecreasing number of divisors, or -1 if no such number exists.

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A378370 Distance between n and the least prime power >= n, allowing 1.

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A378367 Greatest non prime power <= n, allowing 1.

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A013601 a(n) = nextprime(7^n)-7^n.

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A203074 a(0)=1; for n > 0, a(n) = next prime after 2^(n-1).

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