A110969 -id:A110969 - cp's OEIS frontend

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

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

A375710 Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 less than the next nonsquarefree number.

Original entry on oeis.org

5, 6, 9, 19, 20, 21, 33, 34, 36, 49, 57, 58, 62, 63, 66, 76, 77, 88, 89, 91, 96, 97, 103, 104, 113, 114, 118, 119, 130, 131, 132, 136, 142, 149, 150, 161, 162, 174, 175, 187, 188, 189, 190, 201, 202, 206, 215, 217, 218, 225, 226, 231, 232, 245, 246, 249, 253

Offset: 1

Gus Wiseman, Sep 09 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 2 after the fifth and sixth terms.

Positions of 2's in A078147.
For prime numbers we have A029707.
For nonprime numbers we appear to have A014689.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A373573, A375927.

Join@@Position[Differences[Select[Range[1000], !SquareFreeQ[#]&]],2]

Complement of A375709 U A375711 U A375712.

A375711 Numbers k such that A013929(k+1) - A013929(k) = 3. In other words, the k-th nonsquarefree number is 3 less than the next nonsquarefree number.

Original entry on oeis.org

3, 16, 23, 27, 31, 44, 46, 51, 55, 60, 68, 74, 79, 86, 95, 101, 105, 107, 112, 116, 121, 126, 129, 146, 147, 152, 159, 164, 167, 172, 177, 182, 185, 191, 195, 199, 204, 209, 220, 223, 229, 234, 237, 242, 244, 257, 262, 270, 275, 285, 286, 291, 299, 305, 312

Offset: 1

Gus Wiseman, Sep 09 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 3 after the third term.

Positions of 3's in A078147.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A014689, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A375927.

Join@@Position[Differences[Select[Range[1000],!SquareFreeQ[#]&]],3]

Complement of A375709 U A375710 U A375712.

A375712 Numbers k such that A013929(k+1) - A013929(k) = 4. In other words, the k-th nonsquarefree number is 4 less than the next nonsquarefree number.

Original entry on oeis.org

1, 4, 7, 11, 12, 13, 14, 22, 25, 26, 29, 32, 35, 39, 40, 41, 42, 50, 53, 54, 61, 64, 70, 71, 72, 75, 78, 81, 82, 83, 84, 87, 90, 98, 99, 102, 109, 110, 117, 120, 123, 124, 127, 135, 139, 140, 144, 151, 154, 155, 156, 157, 160, 163, 168, 169, 170, 173, 176, 179

Offset: 1

Gus Wiseman, Sep 09 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 4 after the first, fourth, and seventh terms.

For prime numbers we have A029709.
Positions of 4's in A078147.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A014689, A029707, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A375927.

Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&]],4]

Complement of A375709 U A375710 U A375711.

A373574 Numbers k such that the k-th maximal antirun of nonsquarefree numbers has length different from all prior maximal antiruns. Sorted positions of first appearances in A373409.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 18, 52, 678

Offset: 1

Gus Wiseman, Jun 10 2024

The unsorted version is A373573.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
Is this sequence finite? Are there only 9 terms?

			The maximal antiruns of nonsquarefree numbers begin:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99
The a(n)-th rows are:
     4    8
     9   12   16   18   20   24
    28   32   36   40   44
    49
    64   68   72   75
    81   84   88   90   92   96   98
   148  150  152
   477  480  484  486  488  490  492  495
  6345 6348 6350 6352 6354 6356 6358 6360 6363

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For squarefree runs we have the triple (1,3,5), firsts of A120992.
For prime runs we have the triple (1,2,3), firsts of A175632.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For squarefree antiruns: A373200, firsts of A373127, unsorted A373128.
For composite runs we have A373400, firsts of A176246, unsorted A073051.
For prime antiruns we have A373402, firsts of A027833, unsorted A373401.
For composite antiruns we have the triple (1,2,7), firsts of A373403.
Sorted positions of first appearances in A373409.
The unsorted version is A373573.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A007674, A025157, A049094, A061399, A068781, A072284, A077643, A110969, A251092, A294242, A373410, A373412.

t=Length/@Split[Select[Range[100000],!SquareFreeQ[#]&],#1+1!=#2&];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

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.

A375930 Numbers k such that A005117(k+1) - A005117(k) > 1. In other words, the k-th squarefree number is more than 1 less than the next.

Original entry on oeis.org

3, 6, 8, 11, 12, 13, 16, 17, 20, 23, 26, 29, 31, 32, 33, 34, 37, 39, 42, 45, 47, 50, 52, 55, 56, 57, 60, 61, 64, 67, 70, 73, 75, 77, 78, 81, 83, 86, 89, 91, 92, 93, 95, 98, 99, 100, 103, 104, 106, 109, 112, 115, 117, 120, 121, 122, 125, 127, 130, 133, 136, 139

Offset: 1

Gus Wiseman, Sep 12 2024

nonn

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2-1)) = 1 - A065469 = 0.46928817... . - Amiram Eldar, Sep 15 2024

			The squarefree numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ... which first increase by more than one after positions 3, 6, 8, 11, ...

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

For nonprime numbers: A014689, complement A375926, differences A373403.
For composite numbers: A065890 shifted, complement A375929.
Positions of terms > 1 in A076259.
First differences are A120992, complement A373127.
The complement is A375927.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
Cf. A007674, A061399, A065469, A068781, A072284, A110969, A294242, A375709.

Join@@Position[Differences[Select[Range[100],SquareFreeQ[#]&]],_?(#>1&)]

lista(kmax) = {my(is1 = 1, is2, c = 1); for(k = 2, kmax, is2 = issquarefree(k); if(is2, c++); if(is1 && !is2, print1(c, ", ")); is1 = is2);} \\ Amiram Eldar, Sep 15 2024

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

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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A375710 Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 less than the next nonsquarefree number.

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A375711 Numbers k such that A013929(k+1) - A013929(k) = 3. In other words, the k-th nonsquarefree number is 3 less than the next nonsquarefree number.

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A375712 Numbers k such that A013929(k+1) - A013929(k) = 4. In other words, the k-th nonsquarefree number is 4 less than the next nonsquarefree number.

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A373574 Numbers k such that the k-th maximal antirun of nonsquarefree numbers has length different from all prior maximal antiruns. Sorted positions of first appearances in A373409.

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

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A375930 Numbers k such that A005117(k+1) - A005117(k) > 1. In other words, the k-th squarefree number is more than 1 less than the next.

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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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