A120430 -id:A120430 - cp's OEIS frontend

A376308 Run-compression of the sequence of first differences of prime-powers.

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

Offset: 1

Gus Wiseman, Sep 20 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, run-compression removes 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 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, ...
The run-compression is A376308 (this sequence).

For primes instead of prime-powers we have A037201, halved A373947.
For squarefree numbers instead of prime-powers we have A376305.
For run-lengths instead of compression we have A376309.
For run-sums instead of compression 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, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A024619 and A361102 list non-prime-powers, differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A007916, A025475, A034296, A053289, A076259, A110969, A120430, A124767, A174965, A374251.

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

A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 17, 43, 70, 1077, 6635, 12369, 43578, 105102700

Offset: 1

Gus Wiseman, Sep 04 2024

The corresponding prime-powers A246655(a(n)) are given by A006549.

From A006549, it is not known whether this sequence is infinite.

			The fifth prime-power is 7 and the sixth is 8, so 5 is in the sequence.

For nonprime numbers (A002808) we have A375926, differences A373403.
Positions of 1's in A057820.
First differences are A373671.
For nonsquarefree numbers we have A375709, differences A373409.
For non-prime-powers we have A375713.
For non-perfect-powers we have A375740.
For squarefree numbers we have A375927, differences A373127.
Prime-powers:
- terms: A000961, complement A024619.
- differences: A057820.
- runs: A373675, A373673, A373674, A174965
- anti-runs: A373576, A120430, A006549, A373671
Non-prime-powers:
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969 (A373669, sorted A373670).
- anti-runs: A373679, A373575, A255346, A373672
A000040 lists all of the primes, differences A001223.
A025528 counts prime-powers up to n.
Cf. A007916, A014963, A027833, A046933, A053289, A093555, A375714.

Join@@Position[Differences[Select[Range[100],PrimePowerQ]],1]

Numbers k such that A246655(k+1) - A246655(k) = 1.

The inclusive version is a(n) + 1 shifted.

a(14) from Amiram Eldar, Sep 24 2024

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]]]

A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

Original entry on oeis.org

1, 5, 10, 13, 19, 25, 199, 35, 118, 48, 28195587, 61, 3745011205066703, 80, 6635, 312, 1079, 207, 3249254387600868788, 179, 43580, 216, 21151968922, 615, 762951923, 403, 1962, 466, 12371, 245, 1480223716, 783, 494890212533313, 1110, 2064590, 1235, 375744164943287809536

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

For odd n either a(n) or a(n)+1 is in A024622 (unless a(n) = 0), corresponding to cases where the smaller or the larger term in the pair of consecutive prime powers, respectively, is a power of 2. - Pontus von Brömssen, Sep 27 2024

			a(4) = 13, because the first occurrence of 4 in A057820 is at index 13. The corresponding first pair of consecutive prime powers with difference 4 is (19, 23), and a(4) = A025528(23) = 13.
a(61) = A024622(96), because the first pair of consecutive prime powers with difference 61 is (2^96, 2^96+61), and A025528(2^96+61) = A024622(96).

Pontus von Brömssen, Table of n, a(n) for n = 1..60

For compression instead of first appearances we have A376308.
For run-lengths instead of first appearances we have A376309.
For run-sums instead of first appearances we have A376310.
For squarefree numbers instead of prime-powers we have A376311.
The sorted version is A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A024619 and A361102 list non-prime-powers, first differences A375708.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A116861 counts partitions by compressed sum, by compressed length A116608.
Cf. A001597, A006549, A007916, A024622, A025475, A025528, A037201, A038664, A053289, A110969, A120430, A174965, A373400, A375706.

mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&]];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

A057820(a(n)) = n whenever a(n) > 0. - Pontus von Brömssen, Sep 24 2024

Definition modified by Pontus von Brömssen, Sep 26 2024

More terms from Pontus von Brömssen, Sep 27 2024

A375738 Minimum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

Original entry on oeis.org

2, 3, 6, 7, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 83, 84, 85, 86, 87, 88

Offset: 1

Gus Wiseman, Sep 10 2024

nonn

Non-perfect-powers (A007916) are numbers with no proper integer roots.

An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.

			The initial anti-runs are the following, whose minima are a(n):
  (2)
  (3,5)
  (6)
  (7,10)
  (11)
  (12)
  (13)
  (14)
  (15,17)
  (18)
  (19)
  (20)
  (21)
  (22)
  (23)
  (24,26,28)

For composite numbers we have A005381, runs A008864 (except first term).
For prime-powers we have A120430, runs A373673 (except first term).
For squarefree numbers we have A373408, runs A072284.
For nonsquarefree numbers we have A373410, runs A053806.
For non-prime-powers we have A373575, runs A373676.
For anti-runs of non-perfect-powers:
- length: A375736
- first: A375738 (this)
- last: A375739
- sum: A375737
For runs of non-perfect-powers:
- length: A375702
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
Cf. A007674, A045542, A046933, A061399, A216765, A251092, A373403, A373679, A375708, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Min/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most

A375928 Positions of adjacent non-prime-powers (exclusive) differing by more than 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 11, 12, 13, 14, 18, 21, 22, 25, 26, 29, 34, 35, 37, 39, 42, 43, 48, 49, 50, 55, 62, 65, 66, 69, 70, 73, 80, 83, 84, 86, 91, 92, 101, 102, 107, 112, 115, 116, 119, 124, 125, 134, 135, 138, 139, 150, 161, 164, 165, 168, 173, 174, 175, 182

Offset: 1

Gus Wiseman, Sep 12 2024

nonn

			The non-prime-powers (exclusive) are 1, 6, 10, 12, 14, 15, 18, 20, ... which increase by more than 1 after positions 1, 2, 3, 4, 6, 7, ...

For prime-powers inclusive (A000961) we have A376163, differences A373672.
For nonprime numbers (A002808) we have A014689, differences A046933.
First differences are A110969.
The complement is A375713.
For non-perfect-powers we have A375714, complement A375740.
The complement for prime-powers (exclusive) is A375734, differences A373671.
The complement for nonprime numbers is A375926, differences A373403.
A000040 lists the prime numbers, differences A001223.
A000961 lists prime-powers (inclusive), differences A057820.
A007916 lists non-perfect-powers, differences A375706.
A024619 lists non-prime-powers (inclusive), differences A375735.
A246655 lists prime-powers (exclusive), differences A174965.
A361102 lists non-prime-powers (exclusive), differences A375708.
Cf. A006549, A053289, A073783, A093555, A120430, A176246, A251092.

ce=Select[Range[100],!PrimePowerQ[#]&];
Select[Range[Length[ce]-1],!ce[[#+1]]==ce[[#]]+1&]

The inclusive version is a(n+1) - 1.

A377043 The n-th perfect-power A001597(n) minus the n-th power of a prime A000961(n).

Original entry on oeis.org

0, 2, 5, 5, 11, 18, 19, 23, 25, 36, 48, 64, 81, 98, 100, 101, 115, 138, 164, 179, 184, 200, 209, 240, 271, 284, 300, 336, 374, 413, 439, 450, 495, 542, 587, 632, 683, 738, 793, 852, 887, 903, 964, 1029, 1097, 1165, 1194, 1230, 1295, 1370, 1443, 1518, 1561

Offset: 1

Gus Wiseman, Oct 25 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root.

Excluding 1 from the powers of primes gives A377044.
A000015 gives the least prime-power >= n.
A031218 gives the greatest prime-power <= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A025475 lists numbers that are both a perfect-power and a prime-power.
A080101 counts prime-powers between primes (exclusive).
A106543 lists numbers that are neither a perfect-power nor a prime-power.
A131605 lists perfect-powers that are not prime-powers.
A246655 lists the prime-powers, complement A361102 (differences A375708).
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Cf. A023055, A045542, A052410, A053707, A069623, A110969, A216765, A377051.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
per=Select[Range[1000],perpowQ];
per-NestList[NestWhile[#+1&,#+1,!PrimePowerQ[#]&]&,1,Length[per]-1]

from sympy import mobius, primepi, integer_nthroot
def A377043(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-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    def g(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return bisection(f,n,n)-bisection(g,n,n) # Chai Wah Wu, Oct 27 2024

a(n) = A001597(n) - A000961(n).

A377044 The n-th perfect-power A001597(n) minus the n-th prime-power A246655(n).

Original entry on oeis.org

-1, 1, 4, 4, 9, 17, 18, 21, 23, 33, 47, 62, 77, 96, 98, 99, 113, 137, 159, 175, 182, 196, 207, 236, 265, 282, 297, 333, 370, 411, 433, 448, 493, 536, 579, 628, 681, 734, 791, 848, 879, 899, 962, 1028, 1094, 1159, 1192, 1220, 1293, 1364, 1437, 1514, 1559, 1591

Offset: 1

Gus Wiseman, Oct 25 2024

sign

Perfect-powers (A001597) are numbers with a proper integer root.

Including 1 with the prime-powers gives A377043.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, A093555, A376596.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A025475 lists numbers that are both a perfect-power and a prime-power.
A031218 gives the greatest prime-power <= n.
A080101 counts prime-powers between primes (exclusive).
A106543 lists numbers that are neither a perfect-power nor a prime-power.
A131605 lists perfect-powers that are not prime-powers.
A246655 lists the prime-powers, complement A361102, A375708.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Cf. A023055, A045542, A052410, A053707, A069623, A110969, A216765, A376560, A376561, A377051.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
per=Select[Range[1000],perpowQ];
per-NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,2,Length[per]-1]

from sympy import mobius, primepi, integer_nthroot
def A377044(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-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    def g(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return bisection(f,n,n)-bisection(g,n,n) # Chai Wah Wu, Oct 27 2024

a(n) = A001597(n) - A246655(n).

A376163 Positions of adjacent non-prime-powers (inclusive, so 1 is a prime-power) differing by 1.

Original entry on oeis.org

4, 7, 8, 14, 15, 16, 18, 19, 22, 23, 26, 27, 29, 30, 31, 32, 35, 37, 39, 40, 43, 44, 45, 46, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 66, 67, 70, 71, 73, 74, 75, 76, 77, 78, 80, 81, 84, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 102, 103, 104, 105

Offset: 1

Gus Wiseman, Sep 13 2024

nonn

			The non-prime-powers (inclusive) are 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ... which increase by 1 after positions 4, 7, 8, ...

For prime-powers inclusive (A000961) we have A375734, differences A373671.
For nonprime numbers (A002808) we have A375926, differences A373403.
For prime-powers exclusive (A246655) we have A375734(n+1) + 1.
First differences are A373672.
The exclusive version is a(n) - 1 = A375713.
Positions of 1's in A375735.
For non-perfect-powers we have A375740.
Prime-powers inclusive:
- terms: A000961
- differences: A057820
- runs: A373675, A373673, A373674, A174965
- antiruns: A373576, A120430, A006549, A373671
Non-prime-powers inclusive:
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969
- antiruns: A373679, A373575, A255346, A373672
A000040 lists all of the primes, differences A001223.
A007916 lists non-perfect-powers, differences A375706.
Cf. A046933, A053289, A073783, A093555, A176246, A251092, A375714.

ce=Select[Range[2,100],!PrimePowerQ[#]&];
Select[Range[Length[ce]-1],ce[[#+1]]==ce[[#]]+1&]

cp's OEIS Frontend

A376308 Run-compression of the sequence of first differences of prime-powers.

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A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.

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

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

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A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

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A375738 Minimum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

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A375928 Positions of adjacent non-prime-powers (exclusive) differing by more than 1.

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A377043 The n-th perfect-power A001597(n) minus the n-th power of a prime A000961(n).

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