A025475 -id:A025475 - cp's OEIS frontend

A226230 Nontrivial prime powers (A025475) which are a sum of a smaller nontrivial prime power and a perfect square.

8, 9, 25, 32, 81, 125, 128, 169, 289, 343, 512, 625, 1681, 2048, 2197, 3125, 3721, 4913, 8192, 32761, 32768, 50653, 66049, 78125, 97969, 131072, 177241, 357911, 524288, 707281, 1030301, 1419857, 1442401, 1953125, 2097152, 2476099, 3031081, 3463321, 6355441, 7645373

Offset: 1

Alex Ratushnyak, May 31 2013

nonn

			81 = 32 + 7^2, so 81 is in the sequence.
169 = 25 + 12^2, so 169 is in the sequence.

Cf. A025475, A226231, A226232.

#include 
#include 
#include 
#define TOP (1ULL<<32)
typedef unsigned long long U64;
int compare64(const void *p1, const void *p2) {
  if (*(U64*)p1== *(U64*)p2) return 0;
  return (*(U64*)p1 < *(U64*)p2) ? -1 : 1;
}
int main() {
  U64 i, j, p, t, r, n=0, *pp = (U64*)malloc(TOP/2);
  unsigned char *c = (unsigned char *)malloc(TOP/8);
  memset(c, 0, TOP/8);
  for (pp[n++] = i = 1; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0) {
      for (p=i+(i==1), j = p*p; ; j*=p) {
        pp[n++] = j;
        double k = ((double)j) * ((double)p);
        if (k >= ((double)(1ULL<<62)*4.0)) break;
      }
      if(i>1) for (j=i*i>>1; j>3]|= 1<<(j&7);
    }
  qsort(pp, n, 8, compare64);
  for (i = 0; i < n; i++)
    for (p=pp[i], j=0; j < i; j++) {
      t = p - pp[j];
      r = sqrt(t);
      if (r*r==t) { printf("%llu, ", p); break; }
    }
  return 0;
}

for n from 1 do
    if isA025475(n) then
        for j from 1 do
            pj := n-j^2 ;
            if pj < 0 then
                break;
            elif isA025475(pj) then
                printf("%d,\n",n);
                break ;
            end if;
        end do:
    end if:
end do: # R. J. Mathar, Jun 06 2013

A226231 Nontrivial prime powers (A025475) which are a sum of a smaller nontrivial prime power and a perfect square in more than 1 way.

Original entry on oeis.org

25, 125, 512, 3125, 4913, 50653, 78125, 7645373, 48828125, 69343957, 310288733, 410338673, 1220703125, 4103684801, 24820429213, 164296466333, 296709280757, 762939453125, 1772000266301, 10368641602001, 11392143817501, 19073486328125, 29724740184833, 72079590632113

Offset: 1

Alex Ratushnyak, May 31 2013

nonn

Subsequence of A226230.

			125 = 4 + 11^2 = 25 + 10^2 = 121 + 4^2. Because there are three representations, 125 is in the sequence. Note that 100 + 5^2 is not listed, because 100 is not a prime power.

Cf. A025475, A226230, A226232.

#include 
#include 
#include 
#define TOP (1ULL<<32)
typedef unsigned long long U64;
int compare64(const void *p1, const void *p2) {
  if (*(U64*)p1== *(U64*)p2) return 0;
  return (*(U64*)p1 < *(U64*)p2) ? -1 : 1;
}
int main() {
  U64 i, j, k, p, t, r, n=0, *pp = (U64*)malloc(TOP/2);
  unsigned char *c = (unsigned char *)malloc(TOP/8);
  memset(c, 0, TOP/8);
  for (pp[n++] = i = 1; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0) {
      for (p=i+(i==1), j = p*p; ; j*=p) {
        pp[n++] = j;
        double k = ((double)j) * ((double)p);
        if (k >= ((double)(1ULL<<62)*4.0)) break;
      }
      if(i>1) for (j=i*i>>1; j>3]|= 1<<(j&7);
    }
  qsort(pp, n, 8, compare64);
  for (i = 0; i < n; i++)
    for (p=pp[i], j = k = 0; j < i; j++) {
      t = p - pp[j];
      r = sqrt(t);
      if (r*r==t && ++k==2) { printf("%llu, ", p); break; }
    }
  return 0;
}

A226232 Nontrivial prime powers (A025475) which are a sum of a smaller nontrivial prime power and a perfect cube.

Original entry on oeis.org

9, 16, 128, 243, 512, 841, 1024, 1849, 5041, 6561, 8192, 32041, 65536, 76729, 157609, 177147, 187489, 310249, 524288, 734449, 1104601, 4108729, 4194304, 4782969, 5067001, 7778521, 11498881, 12823561, 31956409, 33554432, 38651089, 65302561, 78552769, 90459121, 129140163

Offset: 1

Alex Ratushnyak, May 31 2013

nonn

Cf. A025475, A226230, A226231.

for n from 1 do
    if isA025475(n) then
        for j from 1 do
            pj := n-j^3 ;
            if pj < 0 then
                break;
            elif isA025475(pj) then
                printf("%d,\n",n);
                break ;
            end if;
        end do:
    end if:
end do: # R. J. Mathar, Jun 06 2013

A239522 Prime powers (A025475) such that the distance to the nearest prime power is a prime.

Original entry on oeis.org

1, 4, 16, 25, 27, 32, 81, 125, 128, 169, 243, 256, 512, 529, 2048, 7921, 8192, 32761, 32768, 130321, 131072, 134689, 524288, 2093809, 2097152, 8388608, 8392609, 134217728, 134258569, 536870912, 536987929, 8590138489, 140737625897521, 562949953421312, 562950076511761

Offset: 1

Alex Ratushnyak, Mar 20 2014

nonn

Cf. A025475, A239520 - A239523.

A385380 Partial products of the sequence nonprime powers of primes (A025475).

Original entry on oeis.org

1, 4, 32, 288, 4608, 115200, 3110400, 99532800, 4877107200, 312134860800, 25282923724800, 3059233770700800, 382404221337600000, 48947740331212800000, 8272168115974963200000, 2010136852181916057600000, 514595034158570510745600000, 148717964871826877605478400000

Offset: 1

Amiram Eldar, Jun 27 2025

Indices of records in A385379.

a(n) is the least index k such that A385379(k) = n-1.

Amiram Eldar, Table of n, a(n) for n = 1..217
Index entries for sequences related to powerful numbers.

Cf. A024923, A025475, A385379.

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

FoldList[Times, 1, Select[Range[250], !PrimeQ[#] && PrimePowerQ[#] &]] (* Amiram Eldar, Jun 27 2025 *)

list(lim) = {my(p = 1); print1(p, ", "); for(k = 2, lim, if(isprimepower(k) > 1, p *= k; print1(p, ", ")));}

a(n) = Product_{k=1..n} A025475(k).

A060848 Difference between a nontrivial prime power (A025475) and the next prime.

Original entry on oeis.org

1, 3, 2, 1, 4, 2, 5, 4, 3, 2, 6, 2, 3, 4, 8, 1, 4, 4, 6, 9, 12, 6, 4, 12, 6, 7, 30, 4, 12, 12, 5, 16, 6, 4, 10, 10, 12, 10, 6, 3, 4, 6, 10, 4, 6, 2, 4, 10, 6, 17, 4, 10, 4, 18, 6, 30, 12, 12, 4, 10, 27, 4, 6, 4, 12, 4, 28, 6, 2, 10, 4, 4, 10, 12, 18, 10, 10, 3, 12, 4, 12, 6, 10, 10, 18, 10, 12

Offset: 1

Labos Elemer, May 03 2001

nonn

a(n)=1 only for some powers of 2 corresponding to Fermat primes > 3. - Edited by Robert Israel, Jun 03 2021

			78125=5^7 is followed by 78137, the difference is 12.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A025475, A007918, A013632.

N:= 10^5: # for prime powers <= N
S:= {}:
p:= 1:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  S:= S union {seq(p^i,i=2..floor(log[p](N)))}
od:
map(t -> nextprime(t)-t, sort(convert(S,list))); # Robert Israel, Jun 03 2021

NextPrime[#]-#&/@Select[Range[100000],PrimePowerQ[#]&&!PrimeQ[#]&] (* Harvey P. Dale, Oct 19 2022 *)

a(n) = nextprime(A025475(n)) - A025475(n) = A013632(A025475(n)).

A065523 Numbers n such that sigma(n) is a prime power (A025475).

Original entry on oeis.org

3, 7, 21, 31, 81, 93, 127, 217, 381, 400, 651, 889, 2667, 3937, 8191, 11811, 24573, 27559, 57337, 82677, 131071, 172011, 253921, 393213, 524287, 761763, 917497, 1040257, 1572861, 1777447, 2752491, 3120771, 3670009, 4063201, 5332341

Offset: 1

Robert G. Wilson v, Nov 27 2001

nonn

Donovan Johnson, Table of n, a(n) for n = 1..80

A proper subset of A065496.

s[n_] := DivisorSigma[1, n ]; Select[ Range[2, 10^6], !PrimeQ[ s[ # ]] && Mod[ s[ # ], s[ # ] - EulerPhi[ s[ # ]]] == 0 & ]

a(31)-a(35) from Donovan Johnson, Sep 05 2008

A065743 Smallest number with exactly A025475(n) divisors.

Original entry on oeis.org

1, 6, 24, 36, 120, 1296, 900, 840, 46656, 7560, 44100, 60466176, 810000, 83160, 2176782336, 2822400, 1081080, 2821109907456, 729000000, 101559956668416, 17297280, 131621703842267136, 1944810000, 341510400

Offset: 1

Labos Elemer, Nov 15 2001

nonn

Note that 2^(n-1) has n divisors. - David Wasserman, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..285

Cf. A005179, A025475, A037992, A003680.

a = Table[ 0, {1024} ]; Do[ b = DivisorSigma[ 0, n]; If[ b < 1025 && a[[b]] == 0, a[[b]] = n], {n, 1, 10^8/2} ]; a[[ Select[ Range[2, 1024], !PrimeQ[ # ] && Mod[ #, # - EulerPhi[ # ]] == 0 & ] ]]

a(n) = A005179(A025475(n)).

More terms from David Wasserman, Sep 09 2002

A076047 Primes which are the difference between two successive nontrivial prime powers (A025475).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 41, 139, 151, 199, 271, 307, 751, 1217, 3343, 3617, 4001, 4241, 40841, 97169, 117017, 203897, 746153, 123090449, 137542193, 256534591, 21249911167, 88109383889, 112332648583, 85726065193313, 226411321073393

Offset: 1

Zak Seidov and Robert G. Wilson v, Oct 29 2002

nonn

I have searched through prime powers up to 2^8532. It is very unlikely that there are any other values between the ones listed here, but no prime has been proved to be absent from this sequence. - David Wasserman, Mar 31 2005

			3 = 128 - 125 = 2^7 - 5^3; 7 = 16 - 9 = 32768 - 32761; 17 = 49 - 32 = 81 - 64 = 529 - 512; 4241 = 528529 - 524288 = 727^2 - 2^19.

Cf. A025475, A053810, A075308.

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]]; a = {}; Do[ If[ PrimeQ[ d[[n]]], a = Append[a, d[[n]]]], {n, 1, l - 1}]; Union[a] a = {}; Do[ If[ PrimeQ[ pp[[n + 1]] - pp[[n]]], a = Append[a, pp[[n + 1]] - pp[[n]]]], {n, 1, Length[pp] - 1}]; Union[a]

More terms from David Wasserman, Mar 31 2005

A225388 Prime powers (A025475) representable as (p+q)/2, where p and q are distinct prime powers.

Original entry on oeis.org

25, 125, 169, 243, 256, 289, 625, 841, 1369, 1681, 2809, 3125, 3721, 5329, 9409, 11881, 12769, 18769, 22201, 28561, 37249, 38809, 54289, 76729, 83521, 100489, 113569, 151321, 160801, 196249, 201601, 208849, 292681, 323761, 375769, 390625, 410881, 426409, 452929, 502681

Offset: 1

Alex Ratushnyak, May 05 2013

nonn

			a(1) = 25 = (1 + 49) / 2.
a(2) = 125 = (81 + 169) / 2.
a(3) = 169 = (49 + 289) / 2.
a(4) = 243 = (125 + 361) / 2.
a(5) = 256 = (169 + 343) / 2.
a(6) = 289 = (49 + 529) / 2.

Cf. A025475, A225389.

cp's OEIS Frontend

A226230 Nontrivial prime powers (A025475) which are a sum of a smaller nontrivial prime power and a perfect square.

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A226231 Nontrivial prime powers (A025475) which are a sum of a smaller nontrivial prime power and a perfect square in more than 1 way.

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A226232 Nontrivial prime powers (A025475) which are a sum of a smaller nontrivial prime power and a perfect cube.

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A239522 Prime powers (A025475) such that the distance to the nearest prime power is a prime.

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A385380 Partial products of the sequence nonprime powers of primes (A025475).

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A060848 Difference between a nontrivial prime power (A025475) and the next prime.

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A065523 Numbers n such that sigma(n) is a prime power (A025475).

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A065743 Smallest number with exactly A025475(n) divisors.

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A076047 Primes which are the difference between two successive nontrivial prime powers (A025475).

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