A377051 -id:A377051 - cp's OEIS frontend

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

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.

A378366 Difference between n and the greatest non prime power <= n (allowing 1).

Original entry on oeis.org

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

Sequences obtained by subtracting each term from n are placed in parentheses below.
For nonprime we almost have A010051 (A179278).
For prime we have A064722 (A007917).
For perfect power we have A069584 (A081676).
For squarefree we have (A070321).
For prime power we have A378457 = A276781-1 (A031218).
For nonsquarefree we have (A378033).
For non perfect power we almost have A075802 (A378363).
Subtracting from n gives (A378367).
The opposite is A378371, adding n A378372.
A000015 gives the least prime power >= n (cf. A378370 = A377282 - 1).
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. A007916, A007920, A013632, A065514, A074984, A113646, A345531, A377051, A377054, A377281, A377289, A378357.

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

a(n) = n - A378367(n).

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

A379542 Second term of the n-th differences of the prime numbers.

Original entry on oeis.org

3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280

Offset: 0

Gus Wiseman, Jan 12 2025

sign

Also the inverse zero-based binomial transform of the odd prime numbers.

For all primes (not just odd) we have A007442.
Including 1 in the primes gives A030016.
Column n=2 of A095195.
The version for partitions is A320590 (first column A281425), see A175804, A053445.
For nonprime instead of prime we have A377036, see A377034-A377037.
Arrays of differences: A095195, A376682, A377033, A377038, A377046, A377051.
A000040 lists the primes, differences A001223, A036263.
A002808 lists the composite numbers, differences A073783, A073445.
A008578 lists the noncomposite numbers, differences A075526.
Cf. A064113, A065890, A084758, A140119, A173390, A258025, A258026, A293467, A333214, A333254, A377041.

nn=40;Table[Differences[Prime[Range[nn+2]],n][[2]],{n,0,nn}]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * prime(k+2)); \\ Michel Marcus, Jan 12 2025

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * prime(k+2).

cp's OEIS Frontend

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

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A378366 Difference between n and the greatest non prime power <= n (allowing 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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A377044 The n-th perfect-power A001597(n) minus the n-th prime-power A246655(n).

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A378252 Least prime power > 2^n.

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A379542 Second term of the n-th differences of the prime numbers.

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Mathematica

PARI

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