A057820 -id:A057820 - cp's OEIS frontend

A375926 Numbers k such that A018252(k+1) = A018252(k) + 1. In other words, the k-th nonprime number is 1 less than the next.

4, 5, 8, 9, 12, 13, 15, 16, 17, 18, 21, 22, 23, 24, 26, 27, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 44, 45, 46, 47, 49, 50, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 73, 74, 77, 78, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Gus Wiseman, Sep 11 2024

nonn

			The nonprime numbers are 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... which increase by 1 after term 4, term 5, term 8, etc.

The complement appears to be A014689, except the first term.
Positions of 1's in A065310 (see also A054546, A073783).
First differences are A373403 (except first).
The version for non-prime-powers is A375713, differences A373672.
The version for prime-powers is A375734, differences A373671.
The version for non-perfect-powers is A375740.
The version for composite numbers is A375929.
A000040 lists the prime numbers, differences A001223.
A018252 lists the nonprimes, exclusive A002808.
A046933 counts composite numbers between primes.
Cf. A000961, A006549, A057820, A176246, A246655, A251092, A375708.

Join@@Position[Differences[Select[Range[100],!PrimeQ[#]&]],1]

from sympy import primepi
def A375926(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 n+bisection(lambda y:primepi(x+1+y))-1
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

Original entry on oeis.org

3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, 205, 206, 215, 243, 244, 303, 324, 326, 401, 465, 483, 604, 800, 879, 938, 1032, 1054, 1076, 1233, 1280, 1720, 1889, 1890, 1905, 1939, 1959, 1961, 2256, 2289, 2409, 2879, 3149

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

			The prime-powers exclusive (A246655) are:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, ...
with first differences (A057820 except first term) :
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 3, ...
with first differences (A376596 except first term):
  0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, ...
with first appearances (A376654):
  1, 3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, ...

For first differences we have A376340.
These are the sorted positions of first appearances in A376596 except first term.
The inclusive version is a(n) + 1 = A376653(n), except first term.
For squarefree instead of prime-power we have A376655.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A053707, A057820, A064113, A069623, A093555, A174965, A333254, A336416, A361102.

q=Differences[Select[Range[1000],PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A377781 First differences of A065514(n) = greatest number < prime(n) that is 1 or a prime-power.

Original entry on oeis.org

1, 2, 1, 4, 2, 5, 1, 2, 8, 2, 3, 5, 4, 2, 6, 4, 6, 5, 3, 4, 2, 8, 2, 6, 8, 4, 2, 4, 2, 16, 3, 3, 6, 2, 10, 2, 6, 6, 6, 4, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 4, 13, 1, 6, 6, 2, 6, 4, 8, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 8, 6, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10

Offset: 1

Gus Wiseman, Nov 14 2024

nonn

Note 1 is a power of a prime but not a prime-power.

Differences of A065514, which is the restriction of A031218 (differences A377782).
The opposite is A377703 (restriction of A000015), differences of A345531.
The opposite for nonsquarefree is A377784, differences of A377783.
For nonsquarefree we have A378034, differences of A378032 (restriction of A378033).
The opposite for squarefree is A378037, differences of A112926 (restriction of A067535).
For squarefree we have A378038, differences of A112925 (restriction of A070321).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime-powers, differences A057820.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A361102 lists the non-powers of primes, differences A375708.
Prime-powers between primes:
- A053607 primes
- A080101 count (exclusive)
- A304521 by bits
- A366833 count
- A377057 positive
- A377286 zero
- A377287 one
- A377288 two
Cf. A001597, A007916, A008864, A053289, A120327, A343249, A375706, A377281, A377282, A377289.

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

A378371 Distance between n and the least non prime power >= n, allowing 1.

Original entry on oeis.org

0, 4, 3, 2, 1, 0, 3, 2, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 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 28 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, ...

			The least non prime power >= 4 is 6, so a(4) = 2.

Sequences obtained by adding n to each term are placed in parentheses below.
For prime we have A007920 (A151800), strict A013632.
For composite we have A010051 (A113646 except initial terms).
For perfect power we have A074984 (A377468)
For squarefree we have A081221 (A067535).
For nonsquarefree we have (A120327).
For non perfect power we have A378357 (A378358).
The opposite version is A378366 (A378367).
For prime power we have A378370, strict A377282 (A000015).
This sequence is A378371 (A378372).
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.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A053707, A065514, A276781 (A031218), A343249, A345531, A376596, A376597, A377051, A377054, A377289, A378457.

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

a(n) = A378372(n) - n.

A378372 Least non prime power >= n, allowing 1.

Original entry on oeis.org

1, 6, 6, 6, 6, 6, 10, 10, 10, 10, 12, 12, 14, 14, 15, 18, 18, 18, 20, 20, 21, 22, 24, 24, 26, 26, 28, 28, 30, 30, 33, 33, 33, 34, 35, 36, 38, 38, 39, 40, 42, 42, 44, 44, 45, 46, 48, 48, 50, 50, 51, 52, 54, 54, 55, 56, 57, 58, 60, 60, 62, 62, 63, 65, 65, 66, 68

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, ...

			The least non prime power >= 4 is 6, so a(4) = 6.

Sequences obtained by subtracting n from each term are placed in parentheses below.
For prime power we have A000015 (A378370).
For squarefree we have A067535 (A081221).
For composite we have A113646 (A010051).
For nonsquarefree we have A120327.
For prime we have A151800 (A007920), strict (A013632).
Run-lengths are 1 and A375708.
For perfect power we have A377468 (A074984).
For non-perfect power we have A378358 (A378357).
The opposite is A378367, distance A378366.
This sequence is A378372 (A378371).
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.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A007916, A031218 (A276781), A053707, A065514, A345531 (A377281), A377051, A377282, A377289, A378457.

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

a(n) = A378371(n) + n.

A070198 Smallest nonnegative number m such that m == i (mod i+1) for all 1 <= i <= n.

Original entry on oeis.org

0, 1, 5, 11, 59, 59, 419, 839, 2519, 2519, 27719, 27719, 360359, 360359, 360359, 720719, 12252239, 12252239, 232792559, 232792559, 232792559, 232792559, 5354228879, 5354228879, 26771144399, 26771144399, 80313433199, 80313433199

Offset: 0

Benoit Cloitre, May 06 2002

Also, smallest k such that, for 0 <= i < n, i+1 divides k-i.
Suggested by Chinese Remainder Theorem. This sequence can generate others: smallest b(n) such that b(n) == i (mod (i+2)), 1 <= i <= n, gives b(1)=1 and b(n) = a(n+1)-1 for n > 1; smallest c(n) such that c(n) == i (mod (i+3)), 1 <= i <= n, gives c(1)=1, c(2)=17 and c(n) = a(n+2) - 2 for n > 2; smallest d(n) such that c(n) == i (mod (i+4)), 1 <= i <= n, gives d(1)=1, d(2)=26, d(3)=206 and d(n) = a(n+3) - 3 for n > 3, etc.
A208768(n) occurs A057820(n) times. - Reinhard Zumkeller, Mar 01 2012
From Kival Ngaokrajang, Oct 10 2013: (Start)
A070198(n-1) is m such that max(Sum_{i=1..n} m (mod i)) = A000217(n-1).
Example for n = 3:
  m\i = 1  2  3  sum
  1     0  1  1   2
  2     0  0  2   2
  3     0  1  0   1
  4     0  0  1   1
  5     0  1  2   3 <--max remainder sum = 3 = A000217(2)
  6     0  0  0   0  first occurs at m = 5 = A070198(2)
(End)

			a(3) = 11 because 11 == 1 (mod 2), 11 == 2 (mod 3) and 11 == 3 (mod 4).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Chinese Remainder Theorem
Wikipedia, Chinese remainder theorem
Index entries for sequences related to lcm's

Cf. A053664, A072562.
Cf. A057825 (indices of primes). - R. J. Mathar, Jan 14 2009
Cf. A116151. - Zak Seidov, Mar 11 2014

a070198 n = a070198_list !! n
a070198_list = map (subtract 1) $ scanl lcm 1 [2..]
-- Reinhard Zumkeller, Mar 01 2012

[Exponent(SymmetricGroup(n))-1 : n in [1..30]]; /* Vincenzo Librandi, Oct 31 2014 - after Arkadiusz Wesolowski in A003418 */

seq(ilcm($1..n) - 1, n=1..100); # Robert Israel, Nov 03 2014

f[n_] := ChineseRemainder[ Range[0, n - 1], Range[n]]; Array[f, 28] (* or *)
f[n_] := LCM @@ Range@ n - 1; Array[f, 28] (* Robert G. Wilson v, Oct 30 2014 *)

from math import lcm
def A070198(n): return lcm(*range(1,n+2))-1 # Chai Wah Wu, May 02 2023

a(n) = lcm(1, 2, 3, ..., n+1) - 1 = A003418(n+1) - 1.

Edited by N. J. A. Sloane, Nov 18 2007, at the suggestion of Max Alekseyev

A366835 In the pair (A246655(n), A246655(n+1)), how many primes are there?

Original entry on oeis.org

2, 1, 1, 2, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2

Offset: 1

Paolo Xausa, Oct 25 2023

nonn

First 0 terms appear at n = 6, 14, 41, 359, 3589, corresponding to consecutive prime powers (8,9), (25,27), (121,125), (2187,2197) and (32761,32768), respectively (cf. A068315 and A068435).

There cannot be primes strictly between consecutive prime powers, so we get the same result considering the whole interval (not just the pair). - Gus Wiseman, Dec 25 2024

			a(1) = 2 because in the first prime power pair (2 and 3) there are two primes.
a(14) = 0 because in the 14th prime power pair (25 and 27) there are no primes.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, 1038 X 1038 raster of a(n), n = 1..1077444, read left to right in rows, then top to bottom, showing a(n) = 0 in white, a(n) = 1 in red, and a(n) = 2 in dark blue.

For perfect powers instead of prime powers we have A080769.
Positions of 1 are A379155, indices of A379157.
Positions of 0 are A379156, indices of A068315.
Positions of 2 are A379158, indices of A379541.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A031218 gives the greatest prime power <= n.
A065514 gives the greatest prime power < prime(n), difference A377289.
A080101 and A366833 count prime powers between primes, see A053607, A304521.
A246655 lists the prime powers, differences A057820.
Cf. A000961, A025474, A046933, A067871, A068435, A175106, A178700, A345531, A362965, A377281.

With[{upto=500},Map[Count[#,_?PrimeQ]&,Partition[Select[Range[upto],PrimePowerQ],2,1]]] (* Considers prime powers up to 500 *)

lista(nn) = my(v=[p| p <- [1..nn], isprimepower(p)]); vector(#v-1, k, isprime(v[k]) + isprime(v[k+1])); \\ Michel Marcus, Oct 26 2023

A376309 Run-lengths of the sequence of first differences of prime-powers.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with lengths A376309 (this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000

For runs of prime-powers increasing by one we have A174965.
For primes instead of prime-powers we have A333254.
For squarefree numbers instead of prime-powers we have A376306.
For compression instead of run-lengths we have A376308.
For run-sums instead of run-lengths we have A376310.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A025475, A037201, A053289, A110969, A120430, A251092, A373822, A375706.

Length/@Split[Differences[Select[Range[100],PrimePowerQ]]]

up_to = 20000;
A376309list(up_to) = { my(v=vector(up_to), ppp=2, pd=1, d, rl=0, k=2, i=0); while(i<#v, k++; if(isprimepower(k), d = k-ppp; ppp = k; if(d == pd, rl++, i++; v[i] = rl; rl = 1; pd = d))); (v); };
v376309 = A376309list(up_to);
A376309(n) = v376309[n]; \\ Antti Karttunen, Jan 18 2025

More terms from Antti Karttunen, Jan 18 2025

A376310 Run-sums of the sequence of first differences of prime-powers.

Original entry on oeis.org

3, 2, 2, 4, 3, 1, 2, 4, 8, 1, 5, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 12, 4, 2, 4, 6, 2, 10, 2, 4, 2, 24, 4, 2, 4, 6, 4, 8, 5, 1, 12, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 6, 2, 18, 4, 6, 8, 4, 8, 10, 2

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with sums A376310 (this sequence).

For primes instead of prime-powers we have A373822, halved A373823.
For squarefree numbers instead of prime-powers we have A376307.
For compression instead of run-sums we have A376308.
For run-lengths instead of run-sums we have A376309.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A007916, A025475, A037201, A053289, A054265, A110969, A120430, A174965, A251092, A375706.

Total/@Split[Differences[Select[Range[100],PrimePowerQ]]]

A376342 Positions of 1's in the run-compression (A376305) of the first differences (A076259) of the squarefree numbers (A005117).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 22, 24, 26, 28, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 51, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 124, 126, 128, 130

Offset: 1

Gus Wiseman, Sep 24 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, we can remove all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The sequence of squarefree numbers (A005117) is:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, ...
with run-compression (A376305):
  1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 2, 1, ...
with ones at (A376342):
  1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 22, 24, 26, 28, 30, 32, 34, 36, 38, ...

Before compressing we had A076259.
Positions of 1's in A376305.
The version for nonsquarefree numbers gives positions of ones in A376312.
For prime instead of squarefree numbers we have A376343.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
A116861 counts partitions by compressed sum, by compressed length A116608.
A274174 counts contiguous compositions, ranks A374249.
Cf. A007674, A037201, A053797, A053806, A061398, A072284, A120992, A373198, A376306, A376307, A376308, A376311.

Join@@Position[First /@ Split[Differences[Select[Range[100],SquareFreeQ]]],1]

cp's OEIS Frontend

A375926 Numbers k such that A018252(k+1) = A018252(k) + 1. In other words, the k-th nonprime number is 1 less than the next.

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A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

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A377781 First differences of A065514(n) = greatest number < prime(n) that is 1 or a prime-power.

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A378371 Distance between n and the least non prime power >= n, allowing 1.

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A378372 Least non prime power >= n, allowing 1.

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A070198 Smallest nonnegative number m such that m == i (mod i+1) for all 1 <= i <= n.

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A366835 In the pair (A246655(n), A246655(n+1)), how many primes are there?

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PARI

A376309 Run-lengths of the sequence of first differences of prime-powers.

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