A053707 -id:A053707 - cp's OEIS frontend

A376519 Positions of terms not appearing for the first time in the first differences (A053289) of perfect-powers (A001597).

8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, 112, 114, 128, 136, 144, 145, 162, 180, 188, 198, 216, 226, 235, 246, 264, 265, 275, 285, 295, 305, 316, 317, 325, 328, 338, 350, 360, 367, 373, 385, 406, 416, 417, 419, 431, 443

Offset: 1

Gus Wiseman, Sep 28 2024

nonn

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with positions of latter appearances (A376519):
  8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, ...

These are the sorted positions of latter appearances in A053289 (union A023055).
The complement is A376268.
A053707 lists first differences of consecutive prime-powers.
A333254 lists run-lengths of differences between consecutive primes.
Other families of numbers and their first differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310.
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A025475, A045542, A046933, A052410, A069623, A174965, A216765, A303707, A305630, A305631, A375736.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
q=Differences[Select[Range[1000],perpowQ]];
Select[Range[Length[q]],MemberQ[Take[q,#-1],q[[#]]]&]

A378456 Number of composite numbers between consecutive nonprime prime powers (exclusive).

Original entry on oeis.org

1, 0, 4, 5, 1, 2, 12, 11, 12, 31, 3, 1, 32, 59, 11, 25, 46, 13, 125, 14, 80, 88, 94, 103, 52, 261, 35, 267, 147, 172, 120, 9, 9, 163, 355, 279, 313, 207, 329, 347, 376, 108, 257, 805, 283, 262, 25, 917, 242, 1081, 702, 365, 752, 389, 251, 535, 1679, 877, 447

Offset: 1

Gus Wiseman, Nov 30 2024

nonn

The inclusive version is a(n) + 2.

Nonprime prime powers (A246547) begin: 4, 8, 9, 16, 25, 27, 32, 49, ...

			The initial terms count the following composite numbers:
  {6}, {}, {10,12,14,15}, {18,20,21,22,24}, {26}, {28,30}, ...
The composite numbers for a(77) = 6 together with their prime indices are the following. We have also shown the nonprime prime powers before and after:
  32761: {42,42}
  32762: {1,1900}
  32763: {2,19,38}
  32764: {1,1,1028}
  32765: {3,847}
  32766: {1,2,14,31}
  32767: {4,11,36}
  32768: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

For prime instead of composite we have A067871.
For nonsquarefree numbers we have A378373, for primes A236575.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A002808 lists the composite numbers.
A031218 gives the greatest prime-power <= n.
A046933 counts composite numbers between primes.
A053707 gives first differences of nonprime prime powers.
A080101 = A366833 - 1 counts prime powers between primes.
A246655 lists the prime-powers not including 1, complement A361102.
A345531 gives the nearest prime power after prime(n) + 1, difference A377281.
Cf. A377286, A377287, A377288 (primes A053706).
Cf. A024619, A053607, A065890, A076259, A078147, A243348, A276781, A377057, A377282.

nn=1000;
v=Select[Range[nn],PrimePowerQ[#]&&!PrimeQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

A167186 Record gaps between nonprime prime powers.

Original entry on oeis.org

3, 4, 7, 9, 17, 40, 41, 74, 151, 307, 312, 408, 424, 912, 1032, 1217, 1872, 2518, 3713, 4920, 5208, 8400, 8520, 8892, 9297, 12840, 16008, 21840, 24360, 35880, 38808, 80760, 102168, 129480, 167160, 183960, 201072, 258720, 290760, 301242, 358848, 375468, 415920

Offset: 1

Michael B. Porter, Oct 29 2009, Oct 31 2009, Nov 03 2009

nonn

			17 is in the sequence since A025475(9) - A025475(8) = 49 - 32 = 17, and no previous gap is larger.
A025475(10) - A025475(9) = 64 - 49 = 15, but the previous gap is larger, so 15 is not in the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..200

List of nonprime prime powers: A025475.
Gaps between nonprime prime powers: A053707.
Record gaps between prime powers including primes: A121492.

Join[{3},DeleteDuplicates[Differences[Select[Range[10^6],PrimePowerQ[#] && !PrimeQ[ #]&]], GreaterEqual]] (* Harvey P. Dale, Feb 28 2023 *)

isA025475(n) = (omega(n) == 1 & !isprime(n)) || (n == 1)
d_max=0;n_prev=1;for(n=2,32e6,if(isA025475(n),d=n-n_prev;n_prev=n;if(d>d_max,print(d);d_max=d)))

A167188 Smaller prime power associated with record gap in A167186.

Original entry on oeis.org

1, 4, 9, 16, 32, 81, 128, 169, 361, 1024, 1369, 2401, 4489, 5329, 6889, 8192, 12769, 19683, 32768, 39601, 44521, 85849, 177241, 218089, 262144, 279841, 436921, 597529, 744769, 786769, 1142761, 1771561, 5340721, 6135529, 8826841, 10699441, 17447329, 18464209

Offset: 1

Michael B. Porter, Nov 01 2009

nonn

			32 is in the sequence since the gap between 32 and the next prime power, 49, is greater than any previous gap.

Donovan Johnson, Table of n, a(n) for n = 1..200

Larger prime power: A167189.
Record gaps between nonprime prime powers: A167186.
Gaps between nonprime prime powers: A053707.
List of nonprime prime powers: A025475.

isA025475(n) = (omega(n) == 1 & !isprime(n)) || (n == 1)
d_max=0;n_prev=1;for(n=2,32e6,if(isA025475(n),d=n-n_prev;if(d>d_max,print(n_prev);d_max=d);n_prev=n))

A167189 Larger prime power associated with record gap in A167186.

Original entry on oeis.org

4, 8, 16, 25, 49, 121, 169, 243, 512, 1331, 1681, 2809, 4913, 6241, 7921, 9409, 14641, 22201, 36481, 44521, 49729, 94249, 185761, 226981, 271441, 292681, 452929, 619369, 769129, 822649, 1181569, 1852321, 5442889, 6265009, 8994001, 10883401, 17648401, 18722929

Offset: 1

Michael B. Porter, Nov 01 2009

nonn

			49 is in the sequence since the gap between it the previous prime power, 32, is greater than any previous gap.

Donovan Johnson, Table of n, a(n) for n = 1..200

Smaller prime power is in A167188.
Record gaps between nonprime prime powers: A167186.
Gaps between nonprime prime powers: A053707.
List of nonprime prime powers: A025475.

isA025475(n) = (omega(n) == 1 & !isprime(n)) || (n == 1)
d_max=0;n_prev=1;for(n=2,32e6,if(isA025475(n),d=n-n_prev;if(d>d_max,print(n);d_max=d);n_prev=n))

A358173 First differences of A286708.

Original entry on oeis.org

36, 28, 8, 36, 52, 4, 16, 9, 63, 36, 68, 8, 32, 9, 43, 16, 76, 72, 27, 1, 108, 16, 64, 36, 68, 4, 28, 89, 36, 27, 4, 69, 71, 27, 29, 20, 72, 77, 47, 32, 128, 36, 36, 136, 8, 56, 25, 91, 188, 8, 188, 92, 9, 99, 4, 40, 144, 28, 109, 62, 49, 64, 49, 18, 97, 11, 81

Offset: 1

Michael De Vlieger, Nov 01 2022

nonn

Consider the sequence of powerful numbers A001694, superset of A246547, the sequence of composite prime powers. Let s = A001694(k) such that omega(s) > 1 be followed by t = A001694(k+1) such that omega(t) = 1.
Since A286708 = A001694 \ A246547, prime powers t are missing in A286708. We consider s = A286708(j) and note that the difference A286708(j+1) - A286708(j) > A001694(k+1) - A001694(k).
Therefore we see a subset S containing s in A286708 that plots "out of place" with respect to the complementary subset R = A286708 \ S; some of this subset S exceeds the maxima of R in the scatterplot of this sequence. The plot of the R resembles the scatterplot of A001694.

			The number 36 is the smallest powerful number that is not a prime power; the next powerful number that is not a prime power is 72, and their difference is 36, hence a(1) = 36.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n), n = 1..2^16, highlighting in red terms s that in A001694 are succeeded by a prime power t.

Cf. A001694, A053707, A076446, A246547, A286708.

With[{nn = 2^25}, Differences@ Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], ! PrimePowerQ[#] &]]

from math import isqrt
from sympy import integer_nthroot, primepi, mobius
def A358173(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    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):
        c, l = n+x+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length())), 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        return c
    return -(a:=bisection(f,n,n))+bisection(lambda x:f(x)+1,a,a) # Chai Wah Wu, Sep 10 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).

A378252 Least prime power > 2^n.

Original entry on oeis.org

2, 3, 5, 9, 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, 4294967311, 8589934609

Offset: 0

Gus Wiseman, Nov 30 2024

nonn

Prime powers are listed by A246655.

Conjecture: All terms except 9 are prime. Hence this is the same as A014210 after 9. Confirmed up to n = 1000.

Subtracting 2^n appears to give A013597 except at term 3.
For prime we have A014210.
For previous we have A014234.
For perfect power we have A357751.
For squarefree we have A372683.
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, diffs A375708 and A375735.
A031218 gives the greatest prime power <= n.
A244508 counts prime powers between powers of 2.
Prime powers between primes are counted by A080101 and A366833.
Cf. A007918, A037035, A053707, A059305, A065514, A069584, A151800, A304521, A377051, A377282, A377283, A377289, A377432, A378249.

Table[NestWhile[#+1&,2^n+1,!PrimePowerQ[#]&],{n,0,20}]

a(n) = my(x=2^n+1); while (!isprimepower(x), x++); x; \\ Michel Marcus, Dec 03 2024

from itertools import count
from sympy import primefactors
def A378252(n): return next(i for i in count(1+(1<Chai Wah Wu, Dec 02 2024

A319187 Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 24, 24, 24, 24, 24, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 72, 72, 72, 72, 72, 72, 72, 72

Offset: 1

Gus Wiseman, Jan 09 2019

nonn

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1. A single number is not considered to be pairwise coprime unless it is equal to 1.

			The a(8) = 3 subsets are {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,7,8}.

Ana Rechtman, Décembre 2020, 4e défi (in French), Images des Mathématiques, CNRS, 2020.

Rightmost terms of A186974 and A320436.
Run lengths are A053707.
Cf. A015614, A036234, A051424, A085945, A186971, A186972, A186994, A276187, A303139, A320423, A320426.
Cf. A000005, A045948.

Table[Length[Select[Subsets[Range[n],{PrimePi[n]+1}],CoprimeQ@@#&]],{n,24}] (* see A186974 for a faster program *)

a(n) = prod(p=1, n, if (isprime(p), logint(n, p), 1)); \\ Michel Marcus, Dec 26 2020

a(n) = Product_{p prime <= n} floor(log_p(n)).

a(n) = A000005(A045948(n)). - Ridouane Oudra, Sep 02 2019

cp's OEIS Frontend

A376519 Positions of terms not appearing for the first time in the first differences (A053289) of perfect-powers (A001597).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A378456 Number of composite numbers between consecutive nonprime prime powers (exclusive).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A167186 Record gaps between nonprime prime powers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A167188 Smaller prime power associated with record gap in A167186.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A167189 Larger prime power associated with record gap in A167186.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A358173 First differences of A286708.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

Formula

A378252 Least prime power > 2^n.

Views

Author