A025475 -id:A025475 - cp's OEIS frontend

A301630 a(n) = distance of n-th prime to nearest prime power p^k, k=0 and k >= 2 (A025475).

1, 1, 1, 1, 2, 3, 1, 3, 2, 2, 1, 5, 8, 6, 2, 4, 5, 3, 3, 7, 8, 2, 2, 8, 16, 20, 18, 14, 12, 8, 1, 3, 9, 11, 20, 18, 12, 6, 2, 4, 10, 12, 22, 24, 28, 30, 32, 20, 16, 14, 10, 4, 2, 5, 1, 7, 13, 15, 12, 8, 6, 4, 18, 22, 24, 26, 12, 6, 4, 6, 8, 2, 6, 12, 18, 22, 28, 36, 40, 48, 58, 60, 70, 72, 73, 69, 63, 55

Offset: 1

Altug Alkan, Mar 24 2018

			a(9) = a(10) = 2 because 5^2 is the nearest prime power (A025475) to prime(9) = 23 and 3^3 is the nearest prime power (A025475) to prime(10) = 29.

Altug Alkan, Table of n, a(n) for n = 1..10000

Cf. A047972, A047973, A061670.

There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2).

Primes:= select(isprime, [2,seq(i,i=3..1000,2)]):
Ppows:= sort([1,seq(seq(p^j, j=2..floor(log[p](1000))),p=Primes)]):
for n from 1 while Primes[n] < Ppows[-1] do
  i:= ListTools:-BinaryPlace(Ppows,Primes[n]);
  A[n]:= min(Primes[n]-Ppows[i],Ppows[i+1]-Primes[n])
od:
seq(A[i],i=1..n-1); # Robert Israel, Mar 26 2018

isA025475(n) = {isprimepower(n) && !isprime(n) || n==1}
a(n) = {my(k=1, p=prime(n)); while(!isA025475(p+k) && !isA025475(p-k), k++); k; }

a(n) = A061670(A000040(n)).

A331701 Prime powers (A025475) that can be represented as a sum of two prime powers.

Original entry on oeis.org

8, 9, 16, 25, 32, 64, 81, 125, 128, 256, 512, 1024, 2048, 4096, 5041, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 1

Alex Ratushnyak, Jan 25 2020

nonn

A000079 is a subsequence, starting from the 4th term, 2^3.

The subsequence of odd terms begins: 9, 25, 81, 125, 5041.

			9 = 8 + 1.
25 = 16 + 9.
81 = 49 + 32.
125 = 121 + 4.
5041 = 71^2 = 4913 + 128 = 17^3 + 2^7.

Cf. A025475, A000079.

Select[#, Last@ # == 1 &][[All, 1]] &@ Fold[Function[{s, k}, Append[s, If[And[! PrimeQ@ k, DivisorSigma[1, k]*EulerPhi[k] > (k - 1)^2], {k, If[AnyTrue[IntegerPartitions[k, {2}], SubsetQ[s[[All, 1]], #] &], 1, 0]}, Nothing]]], {}, Range[10^4]] (* Michael De Vlieger, Jan 31 2020 *)

from sympy import isprime
TOP = 10**5
primePowers={}
primePowers[1]=1
for x in range(2,TOP):
    if isprime(x):
        p = pp = x
        while pp < TOP**2:
            pp *= p
            primePowers[pp] = 1
a=[]
pps = sorted(primePowers.keys())[:]
for pp in pps:
    for p in pps:
        if p*2 > pp: break
        if (pp-p) in primePowers:
            print(pp)
            a.append(pp)
            break
print(sorted(a))

A062559 A B2 sequence consisting of powers of primes but not primes: a(n) = least value from A025475 such that sequence increases and pairwise sums of distinct elements are all distinct.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 49, 64, 121, 243, 256, 343, 512, 729, 961, 1024, 1331, 1369, 1849, 2048, 2187, 2197, 2401, 3125, 3481, 4096, 4913, 5329, 6561, 6859, 6889, 8192, 10201, 12769, 14641, 15625, 16384, 16807, 19683, 22201, 22801, 27889, 28561, 29791, 32768

Offset: 0

Labos Elemer, Jul 03 2001

nonn

32 is the first prime power > 1 not in this sequence.

Cf. A000961, A025475, A010672, A011185, A025582, A005282.

from itertools import count, islice
from sympy import factorint
def A062559_gen(): # generator of terms
    aset2, alist = set(), []
    for k in count(0):
        if len(f:=factorint(k).values()) == 1 and max(f) > 1:
            bset2 = set()
            for a in alist:
                if (b:=a+k) in aset2:
                    break
                bset2.add(b)
            else:
                yield k
                alist.append(k)
                aset2.update(bset2)
A062559_list = list(islice(A062559_gen(),30)) # Chai Wah Wu, Sep 11 2023

More terms from Sean A. Irvine, Apr 03 2023

A065746 Number of divisors of squares of all true powers of primes: a(n) = A000005(A025475(n+1)^2).

Original entry on oeis.org

5, 7, 5, 9, 5, 7, 11, 5, 13, 9, 5, 7, 15, 5, 11, 17, 5, 7, 5, 19, 5, 9, 13, 5, 5, 21, 7, 5, 5, 5, 23, 15, 7, 5, 9, 5, 11, 5, 5, 25, 5, 7, 5, 5, 5, 17, 7, 5, 5, 27, 5, 5, 5, 5, 5, 7, 5, 9, 13, 5, 29, 11, 5, 5, 5, 19, 5, 5, 7, 5, 5, 5, 9, 7, 5, 5, 5, 31, 5, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 5, 21, 5, 33

Offset: 1

Labos Elemer, Nov 16 2001

nonn

			tau(p^(2c)) = 2c+1 is prime if c = (odd prime -1)/2 = 1, 2, 3, 5, 6, 8, ... = A005097.

Cf. A000005, A025475, A065403, A065404, A065405, A005097.

DivisorSigma[0, Select[Range[60000], ! PrimeQ[#] && PrimePowerQ[#] &]^2] (* Amiram Eldar, Apr 13 2024 *)

tau(p^(2c)), where tau is the number of divisors, c > 1 and p is prime.

Name corrected by Amiram Eldar, Apr 13 2024

A068343 a(n) is a prime power and sum of all prime powers <= a(n) is A025475.

Original entry on oeis.org

1, 2111, 43997, 69499

Offset: 1

Naohiro Nomoto, Mar 08 2002

No other terms up to 10^9. - Michel Marcus, Mar 26 2020

a(5) > 10^12, if it exists. - Giovanni Resta, Apr 12 2020

			2111 is a term because 1+2+3+4+5+7+8+..+2111 = 323761 = 569^2.

Cf. A000961, A024918, A025475.

ispp1(n) = isprimepower(n) || (n==1); \\ A000961
ispp2(n) = ispower(n, , &p) && isprime(p) || n==1; \\ A025475
lista(nn) = {my(s=0); for (n=1, nn, if (ispp1(n), s += n; if (ispp2(s), print1(n, ", "));););} \\ Michel Marcus, Mar 26 2020

A077274 Differences between two successive powers of a prime but not a prime (A025475) in more than one way.

Original entry on oeis.org

4, 7, 17, 408, 792, 912, 1608, 1848, 2472, 2912, 3192, 3288, 3432, 3528, 4008, 4080, 4920, 5160, 5208, 5928, 6072, 6792, 6888, 7080, 7512, 7728, 7800, 8520, 8832, 10632, 10848, 11400, 11880, 11928, 12792, 13200, 13440, 13560, 14280, 14640

Offset: 1

Robert G. Wilson v, Oct 31 2002

nonn

			4 = 8 - 4 = 125 - 121.
7 = 16 - 9 = 32768 - 32761.
17 = 49 - 32 = 81 - 64 = 529 - 512.

Cf. A025475.

pp = Sort[ Flatten[ Table[ Prime[n]^i, {n, 1, PrimePi[ Sqrt[10^10]]}, {i, 2, Log[ Prime[n], 10^10]}]]]; l = Length[pp]; b = Sort[ Take[pp, -l + 1] - Take[pp, l - 1]]; Union[ b[[ Select[ Range[375], b[[ # ]] == b[[ # + 1]] &]]]]

A136335 Indices of the primes p for which the number of representations of p as the sum of a perfect prime power (A025475: q^e with e>1) and an integer k which is less than q exceeds one.

Original entry on oeis.org

10, 31, 14774, 14775, 65686, 110128, 110129, 110130, 110131, 110132, 110133, 110134, 110135, 110136, 110137, 165952, 165953, 165954, 165955, 165956, 165957, 165958, 304841, 304842, 304843, 304844, 304845, 304846, 304847

Offset: 1

Reinhard Zumkeller & Robert G. Wilson v, Mar 23 2008

nonn

Where A138363 exceeds 1.

			A000040(10)=29; A138363(10) = #{4+5^2, 2+3^3} = 2.

Cf. A138363, A074267.

f[n_] := Block[{c = 0, e, j, k = 1, p = Prime@n, q}, j = 1 + PrimePi@ Sqrt@ p; While[k < j, q = Prime@k; If[p < q + q^Floor@ Log[q, p], c++ ]; k++ ]; c]; lst = {}; Do[ If[f@n > 1, AppendTo[lst, n]; Print[Prime@n]], {n, 25*10^5}]

A221563 Triangular numbers representable as t*p, where t>1 is a triangular number, p>1 is a prime power (A025475).

Original entry on oeis.org

120, 528, 1485, 3240, 58653, 70125, 139128, 609960, 886446, 902496, 1062153, 1844160, 4017195, 8386560, 8759205, 38618866, 39760903, 122062500, 160554240, 703893960, 741105750, 797222415, 6115239936, 10854464130, 23667373395, 82103447700, 131197213890

Offset: 1

Alex Ratushnyak, May 03 2013

nonn

			Triangular(32) = 528 = 66 * 8, because 66 is a triangular number and 8 is a prime power, 528 is in the sequence.
Triangular(54) = 1485 = 55 * 27, because 55 is a triangular number and 27 is a prime power, 1485 is in the sequence.

Cf. A000217, A025475, A029549.

A225300 Terms in A025475 that are palindromes in base 2.

Original entry on oeis.org

1, 9, 27, 20457529, 87463589042698969

Offset: 1

Alex Ratushnyak, May 04 2013

			9 = 1001_2.
27 = 11011_2.
20457529 = 1001110000010100000111001_2.

Intersection of A006995 and A025475.

Cf. A029983.

Definition clarified by Chai Wah Wu, Mar 18 2018

A225675 Numbers n such that triangular(n) is representable as p*w, where p is a prime number and w is a prime power (A025475).

Original entry on oeis.org

2, 7, 9, 16, 17, 18, 25, 26, 31, 53, 81, 97, 121, 127, 162, 241, 250, 256, 337, 361, 486, 577, 625, 841, 1249, 1458, 2186, 2401, 2662, 3361, 3481, 3697, 3721, 4373, 4801, 5041, 6241, 6961, 8191, 10201, 10657, 13121, 14641, 17161, 19321, 23761, 25537, 28561, 31249, 32257

Offset: 1

Alex Ratushnyak, May 12 2013

nonn

Corresponding triangular numbers are in A225674.

The subsequence of even terms begins: 2, 16, 18, 26, 162, 250, 256, 486, 1458, 2186, 2662, 39366, 48778, 65536, 410758, 715822, 1594322, 2450086, 6615898, 12872686, 13935742, 15290746, 23394166, 38930218, 41022298, 86093442, ...

Cf. A000040, A000217, A025475, A225674.

cp's OEIS Frontend

A301630 a(n) = distance of n-th prime to nearest prime power p^k, k=0 and k >= 2 (A025475).

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A331701 Prime powers (A025475) that can be represented as a sum of two prime powers.

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A062559 A B2 sequence consisting of powers of primes but not primes: a(n) = least value from A025475 such that sequence increases and pairwise sums of distinct elements are all distinct.

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A065746 Number of divisors of squares of all true powers of primes: a(n) = A000005(A025475(n+1)^2).

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A068343 a(n) is a prime power and sum of all prime powers <= a(n) is A025475.

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PARI

A077274 Differences between two successive powers of a prime but not a prime (A025475) in more than one way.

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Mathematica

A136335 Indices of the primes p for which the number of representations of p as the sum of a perfect prime power (A025475: q^e with e>1) and an integer k which is less than q exceeds one.

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Mathematica

A221563 Triangular numbers representable as t*p, where t>1 is a triangular number, p>1 is a prime power (A025475).

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A225300 Terms in A025475 that are palindromes in base 2.

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