A025475 -id:A025475 - cp's OEIS frontend

A239523 Prime powers (A025475) such that the distances to the two nearest prime powers are primes.

Original entry on oeis.org

1, 25, 27, 32, 128, 512, 2048, 8192, 32768, 562949953421312

Offset: 1

Alex Ratushnyak, Mar 20 2014

A subsequence of A239522.

a(4)...a(10) are powers of 2, a(10) = 2^49.

Cf. A025475, A239520 - A239522.

A384783 The number of unordered factorizations of the n-th powerful number into 1 and prime powers p^e where p is prime and e >= 2 (A025475).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 1, 2, 7, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 8, 1, 4, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 2, 1, 12, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 4, 1, 1, 1, 2, 2, 2, 2, 14, 1, 4, 1, 1, 7, 1, 2

Offset: 1

Amiram Eldar, Jun 10 2025

The positive terms in A188585.

			a(5) = 2 since the 5th powerful number, A001694(5) = 16, has 2 factorizations: 2^2 * 2^2 and 2^4.
a(11) = 4 since the 11th powerful number, A001694(11) = 64, has 4 factorizations: 2^2 * 2^2 * 2^2, 2^2 * 2^4, 2^3 * 2^3, and 2^6.

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

Cf. A001694, A025475, A188585, A384784, A384785 (cubefull analog).

f[p_, e_] := PartitionsP[e] - PartitionsP[e-1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq[lim_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}]]]}, Select[s /@ pow, # > 0 &]]; seq[10^4]

s(n) = vecprod(apply(x -> numbpart(x)-numbpart(x-1), factor(n)[, 2]));
pows(lim) = {my(p = List()); for(j = 1, sqrtnint(lim, 3), for(i = 1, sqrtint(lim \ j^3), listput(p, i^2 * j^3))); Set(p); }
list(lim) = {my(p = pows(lim), v = List(), s1); for(k = 1, #p, s1 = s(p[k]); if(s1 > 0, listput(v, s1))); Vec(v);}

a(n) = A188585(A001694(n)).

A384784 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 2 (A025475).

Original entry on oeis.org

1, 16, 64, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

The least term that is not a power of 2 is a(47) = 2^35 * 3^10.
Indices of records in A188585.
All the terms are powerful numbers since A188585(1) = 1 and A188585(n) = 0 if n is a nonpowerful number.
The corresponding record values are 1, 2, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..713
Amiram Eldar, Table of n, a(n), A188585(a(n)) for n = 1..713
Index entries for sequences related to powerful numbers.

Subsequence of A001694 and A025487 (i.e., of A181800).

Cf. A025475, A046055, A188585, A384783, A384786 (cubefull analog).

f[p_, e_] := PartitionsP[e] - PartitionsP[e-1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

A065772 Nontrivial prime powers k from A025475 such that tau(k^2) is prime but sigma(k^2) is a composite number.

Original entry on oeis.org

9, 25, 32, 121, 243, 343, 361, 961, 1331, 1369, 1681, 2048, 2209, 2809, 3481, 3721, 4489, 5041, 6561, 6859, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16384, 16807, 17161, 18769, 19321, 19683, 22201, 22801, 24389, 24649, 26569

Offset: 1

Labos Elemer, Nov 19 2001

nonn

Numbers k = A025475(m) such that A000005(k^2) is prime but A000203(k^2) is composite number.

			For k = 32: k^2 = 1024, tau(1024) = 11, sigma(1024) = 2047 = 23*89.
For k = 243, k^2 = 59049, tau(59049) = 11, sigma(59049) = 88573 = 23*3851.
Up to 10000000, 453 terms were found.

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

Cf. A000005 (tau), A000203 (sigma), A065403, A065404, A065405, A028982, A025475.

Do[ s=DivisorSigma[ 0, n^2 ]; y=DivisorSigma[ 1, n^2 ]; If[ Equal[ Length[ FactorInteger[ n ] ], 1 ]&&!PrimeQ[ n ] &&PrimeQ[ s ]&&!PrimeQ[ y ], Print[ n ] ], {n, 1, 10000000} ]

A067194 Sequence of prime numbers whose reverse is a nontrivial prime power (A025475).

Original entry on oeis.org

23, 61, 163, 521, 821, 1297, 1861, 4201, 9049, 9631, 12391, 14437, 16987, 92767, 94273, 96979, 108061, 123031, 125329, 127291, 142771, 148249, 165901, 180289, 270131, 906421, 906727, 906751, 921931, 942013, 942691, 965443, 969407, 986641

Offset: 1

Shyam Sunder Gupta, Feb 19 2002

			23 is a prime and its reversal is 32 = 2^5.

Chai Wah Wu, Table of n, a(n) for n = 1..16435

Cf. A025475, A007488, A069798.

a = {}; Do[ If[ PrimeQ[n], b = ToExpression[ StringReverse[ ToString[n]]]; If[ !PrimeQ[b] && Mod[b, b - EulerPhi[b]] == 0, a = Append[a, n]]], {n, 1, 10^6} ]; a
Select[Prime[Range[80000]],PrimePowerQ[IntegerReverse[#]]&& CompositeQ[ IntegerReverse[ #]]&] (* Harvey P. Dale, Dec 25 2021 *)

lista(nn) = {forprime(p=2, nn, if (ispower(subst(Polrev(digits(p)), x, 10),,&pp) && isprime(pp), print1(p, ", ")););} \\ Michel Marcus, Jun 03 2016

Edited and extended by Robert G. Wilson v, Feb 19 2002 and Feb 24 2002

A077273 Ordered differences without repetitions between two successive powers of a prime but not a prime (A025475).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 10, 12, 13, 15, 17, 18, 30, 33, 38, 40, 41, 54, 63, 74, 94, 96, 104, 112, 120, 128, 138, 139, 151, 168, 186, 192, 199, 240, 248, 250, 255, 260, 271, 286, 288, 298, 307, 312, 316, 320, 354, 356, 362, 375, 393, 408, 423, 424, 432, 496, 504, 513

Offset: 1

Robert G. Wilson v, Oct 31 2002

nonn

			33 = 546 - 513 = 2568 - 2535 = 6168 - 6135 = 458703 - 458670 = 885465 - 885432 = 1048608 - 1048575 = 6291480 - 6291447 = 37748688 - 37748655 = 125829345 - 125829312.

Cf. A025475.

pp = Sort[ Flatten[ Table[ Prime[n]^i, {n, 1, PrimePi[ Sqrt[10^16]]}, {i, 1, Log[ Prime[n], 10^16]}]]]; l = Length[pp]; d = Sort[Take[pp, -l + 1] - Take[pp, l - 1]]; Union[ Take[d, 10^5]]

A084289 Primes p such that the arithmetic mean of p and the next prime after p is a true prime power from A025475.

Original entry on oeis.org

3, 7, 61, 79, 619, 1669, 4093, 822631, 1324783, 2411797, 2588869, 2778877, 3243589, 3636631, 3736477, 5527189, 6115717, 6405943, 8720191, 9005989, 12752029, 16056031, 16589317, 18087991, 21743551, 25230511, 29343871, 34586131, 37736431, 39150037, 40056229

Offset: 1

Labos Elemer, May 26 2003

nonn

			n = prime(9750374) = 174689077, next prime = 174689101, mean = 174689089 = 13217^2, a prime power. The arithmetic mean of two consecutive primes is never prime, while between two consecutive primes, prime powers occur. These prime powers are in the middle of gap: p+d/2 = q^w. The prime power is most often square and very rarely occurs more than once (see A053706).

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A053706, A000961, A025475.

fi[x_] := FactorInteger[x] ff[x_] := Length[FactorInteger[x]] Do[s=(Prime[n]+Prime[n+1])/2; s1=ff[s]; If[Equal[s1,1],Print[{n,p=Prime[n],s,fi[s],s-p,s1}]], {n,1,10000000}]
Select[Partition[Prime[Range[25*10^5]],2,1],PrimePowerQ[Mean[#]]&][[;;,1]] (* Harvey P. Dale, Oct 15 2023 *)

Primes p(j) such that (p(j)+p(j+1))/2 = q(m)^w, where q(m) is a prime.

A088365 Local maxima of A053707 (first differences of A025475, powers of a prime but not prime).

Original entry on oeis.org

4, 9, 17, 40, 74, 54, 151, 120, 307, 312, 199, 408, 356, 424, 912, 1032, 1217, 840, 1872, 423, 1608, 2518, 1588, 1920, 1230, 2112, 3713, 1560, 5208, 2832, 3952, 2872, 3192, 3288, 3432, 8400, 5738, 4080, 4272, 4512, 6288, 8280, 8520, 5232, 5352, 3672, 8892

Offset: 1

Klaus Brockhaus, Sep 27 2003

nonn

A053707(k) for k = 1 is a term iff A053707(k) >= A053707(k+1); A053707(k) for k > 1 is a term iff A053707(k-1) < A053707(k) and A053707(k) >= A053707(k+1).

A088366 gives the corresponding indices. Local minima of A053707 are in A088363.

			The first six terms of A053707 are 3,4,1,7,9,2, hence A053707(2) = 4 is the first and A053707(5) = 9 is the second local maximum of A053707.

Cf. A025475, A053707, A088366, A088363.

{m=1; k=0; for(n=2,250000,if(matsize(factor(n))[1]==1&&factor(n)[1,2]>1,d=n-m; if((k<2||b

A225106 Prime powers (A025475) that can be represented as a sum of two distinct prime powers.

Original entry on oeis.org

9, 25, 81, 125, 512, 5041

Offset: 1

Alex Ratushnyak, Apr 28 2013

The next term, if it exists, is bigger than 2^64.

Conjecture: the sequence is finite. - Alex Ratushnyak, Feb 03 2020

			9 = 8 + 1.
25 = 16 + 9.
81 = 32 + 49.
125 = 121 + 4.
512 = 169 + 343.
5041 = 128 + 4913.

Alex Ratushnyak, C program

Cf. A025475, A331701.

C
```
/* See Ratushnyak link. */
```

nn = 10000; p = Sort[Flatten[Table[Prime[n]^i, {n, PrimePi[Sqrt[nn]]}, {i, 2, Log[Prime[n], nn]}]]]; p = Join[{1}, p]; t = Union[Select[Flatten[Table[p[[i]] + p[[j]], {i, Length[p] - 1}, {j, i + 1, Length[p]}]], # <= nn &]]; Intersection[p, t] (* T. D. Noe, Apr 29 2013 *)

A225792 Numbers n such that the sum of first n nontrivial prime powers (A025475 excluding 1) is divisible by n.

Original entry on oeis.org

1, 2, 3, 9, 13, 54, 384, 596, 7289, 18136, 226871, 1577601, 5240465, 13389981, 130707763, 157598052, 617834651, 998713177, 4596574856

Offset: 1

Alex Ratushnyak, May 16 2013

a(16) > 146306912. There are 146306912 nontrivial prime powers <= 2^63. - Donovan Johnson, May 16 2013

a(20) > 41.2*10^9. - Giovanni Resta, May 16 2013

			The sum of first 9 nontrivial prime powers is 4 + 8 + 9 + 16 + 25 + 27 + 32 + 49 + 64 = 234. Because 234 is divisible by 9, the latter is in the sequence.

Cf. A025475, A225791.

a(12)-a(15) from Donovan Johnson, May 16 2013

a(16)-a(19) from Giovanni Resta, May 16 2013

cp's OEIS Frontend

A239523 Prime powers (A025475) such that the distances to the two nearest prime powers are primes.

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A384783 The number of unordered factorizations of the n-th powerful number into 1 and prime powers p^e where p is prime and e >= 2 (A025475).

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A384784 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 2 (A025475).

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A065772 Nontrivial prime powers k from A025475 such that tau(k^2) is prime but sigma(k^2) is a composite number.

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A067194 Sequence of prime numbers whose reverse is a nontrivial prime power (A025475).

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Extensions

A077273 Ordered differences without repetitions between two successive powers of a prime but not a prime (A025475).

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Mathematica

A084289 Primes p such that the arithmetic mean of p and the next prime after p is a true prime power from A025475.

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Programs

Mathematica

Formula

A088365 Local maxima of A053707 (first differences of A025475, powers of a prime but not prime).

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Keywords

Comments

Examples

Crossrefs

Programs

PARI

A225106 Prime powers (A025475) that can be represented as a sum of two distinct prime powers.

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