A025475 -id:A025475 - cp's OEIS frontend

A053707 First differences of A025475, powers of a prime but not prime.

3, 4, 1, 7, 9, 2, 5, 17, 15, 17, 40, 4, 3, 41, 74, 13, 33, 54, 18, 151, 17, 96, 104, 112, 120, 63, 307, 38, 312, 168, 199, 139, 10, 12, 192, 408, 316, 356, 240, 375, 393, 424, 128, 288, 912, 320, 298, 30, 1032, 271, 1217, 792, 408, 840, 432, 286, 602, 1872, 984, 504

Offset: 1

Labos Elemer, Feb 14 2000

nonn

In the graph of this sequence, the lowest curve corresponds to differences of squares of twin primes; the next-lowest curve is for squares of adjacent primes differing by 4, etc. - T. D. Noe, Aug 03 2007

			2^0 = 1 is the first number that meets the definition of A025475, the next one is 2^2 = 4, hence a(1) = 4 - 1 = 3.
a(3) = A025475(4) - A025475(3) = 9 - 8 = 1; a(11) = A025475(12) - A025475(11) = 121 - 81 = 40.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A025475.

Differences@ Join[{1}, Select[Range@ 16200, And[! PrimeQ@ #, PrimePowerQ@ #] &]] (* Michael De Vlieger, Jul 04 2016 *)

{k=1; for(n=2,16300,if(matsize(factor(n))[1]==1&&factor(n)[1,2]>1,d=n-k; print1(d,","); k=n))} \\ Klaus Brockhaus, Sep 25 2003

from sympy import primepi, integer_nthroot
def A053707(n):
    if n==1: return 3
    def f(x): return int(n-2+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax)+1 >= kmax:
        kmax <<= 1
    rmin, rmax = 1, kmax
    while True:
        kmid = kmax+kmin>>1
        if f(kmid)+1 < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    while True:
        rmid = rmax+rmin>>1
        if f(rmid) < rmid:
            rmax = rmid
        else:
            rmin = rmid
        if rmax-rmin <= 1:
            break
    return kmax-rmax # Chai Wah Wu, Aug 13 2024

a(n) = A025475(n+1) - A025475(n).

Edited by Klaus Brockhaus, Sep 25 2003

A195942 Zeroless prime powers (excluding primes): Intersection of A025475 and A052382.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 3125, 3481, 3721, 4489, 4913, 5329, 6241, 6561, 6859, 6889, 7921, 8192

Offset: 1

M. F. Hasler, Sep 25 2011

Reinhard Zumkeller and T. D. Noe, Table of n, a(n) for n = 1..4094 (terms < 10^10)
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.

Cf. A195985, A195943, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

a195942 n = a195942_list !! (n-1)
a195942_list = filter (\x -> a010051 x == 0 && a010055 x == 1) a052382_list
-- Reinhard Zumkeller, Sep 27 2011

mx = 10^10; t = {1}; p = 2; While[pw = 2; While[n = p^pw; n <= mx, If[Union[IntegerDigits[n]][[1]] > 0, AppendTo[t, n]]; pw++]; pw > 2, p = NextPrime[p]]; t = Sort[t] (* T. D. Noe, Sep 27 2011 *)

for( n=1,9999, is_A025475(n) && is_A052382(n) && print1(n","))

A195942 = A025475 intersect A052382.

A010055(a(n)) * (1 - A010051(a(n))) * A168046(a(n)) = 1. - Reinhard Zumkeller, Sep 27 2011

A053706 Primes p such that between p and the next prime, 2 prime powers (A025475) occur.

Original entry on oeis.org

7, 23, 113, 2179, 32749

Offset: 1

Labos Elemer, Feb 14 2000

No other terms < 4290000000. - Jud McCranie, Jun 20 2000

There are no other terms < 2^63. - Donovan Johnson, Mar 11 2013

			Between 7 and 11 the 2 prime powers are 8 and 9, between 23 and 29 the 2 prime powers are 25 and 27, between 113 and 127 the 2 prime powers are 121 and 125, while between 32749 and 32771 the 2 prime powers are 32761 = 181^2 and 32768 = 2^15.

Cf. A025475, A053607.

nn = 2^20; Prime /@ Keys@ Select[PositionIndex[PrimePi /@ Union@ Flatten@ Table[Array[p^# &, Floor@ Log[p, nn] - 1, 2], {p, Prime@ Range@ PrimePi@ Sqrt@ nn}]], Length[#] > 1 &] (* Michael De Vlieger, Mar 21 2024 *)

isok(p) = isprime(p) && (q=nextprime(p+1)) && (v=vector(q-p, x, p+x)) && (#select(x->(isprimepower(x) && !isprime(x)), v) == 2);
lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, Jul 15 2017

Corrected by James Sellers, Feb 22 2000

A025476 Prime root of n-th nontrivial prime power (A025475, A246547).

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 2, 7, 2, 3, 11, 5, 2, 13, 3, 2, 17, 7, 19, 2, 23, 5, 3, 29, 31, 2, 11, 37, 41, 43, 2, 3, 13, 47, 7, 53, 5, 59, 61, 2, 67, 17, 71, 73, 79, 3, 19, 83, 89, 2, 97, 101, 103, 107, 109, 23, 113, 11, 5, 127, 2, 7, 131, 137, 139, 3, 149, 151, 29, 157, 163, 167, 13, 31, 173, 179

Offset: 1

David W. Wilson

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A025473, A025475, A048148, A246547.

cvm := proc(n, level) local f,opf; if n < 2 then RETURN() fi;
f := ifactors(n); opf := op(1,op(2,f)); if nops(op(2,f)) > 1 or
op(2,opf) <= level then RETURN() fi; op(1,opf) end:
A025476_list := n -> seq(cvm(i,1),i=1..n); # n is search limit
A025476_list(30000);  # Peter Luschny, Sep 21 2011
# Alternative:
isA246547 := n -> n > 1 and not isprime(n) and type(n, 'primepower'):
seq(ifactors(p)[2][1][1], p in select(isA246547, [$1..30000])); # Peter Luschny, Jul 15 2023

Transpose[ Flatten[ FactorInteger[ Select[ Range[2, 30000], !PrimeQ[ # ] && Mod[ #, # - EulerPhi[ # ]] == 0 &]], 1]][[1]] (* Robert G. Wilson v *)

forcomposite(n=4,10^5,if( ispower(n, , &p) && isprime(p), print1(p,", "))) \\ Joerg Arndt, Sep 11 2021

from sympy import primepi, integer_nthroot, primefactors
def A025476(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return primefactors(kmax)[0] # Chai Wah Wu, Aug 15 2024

A061670 Distance to nearest prime power p^k, k=0 and k >= 2 (A025475).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

Offset: 1

Michel ten Voorde, Jun 16 2001

			a(12)=3 because 9=3^2 is the nearest power to 12 (12-9=3).

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).

N:= 1000: # to get a(1)..a(M) where M is the greatest prime power <= N.
Primes:= select(isprime, [2,seq(i,i=3..floor(sqrt(N)))]):
Pows:= sort(convert({1,seq(seq(p^e,e=2..floor(log[p](N))),p=Primes)},list)):
nP:= nops(Pows):
M:= Pows[nP]:
V:= Vector(M):
V[2]:= 1:
for i from 2 to nP-1 do
  for x from ceil((Pows[i]+Pows[i-1])/2) to floor((Pows[i]+Pows[i+1])/2) do
    V[x]:= abs(x - Pows[i])
od od:
for x from ceil((M+Pows[nP-1])/2) to M do V[x]:= M - x od:
convert(V,list); # Robert Israel, Mar 23 2018

isA025475(n) = {isprimepower(n) && !isprime(n) || n==1}
a(n) = {my(k=0); while(!isA025475(n+k) && !isA025475(n-k), k++); k; } \\ Altug Alkan, Mar 23 2018

Definition corrected, and more terms from Robert Israel, Mar 23 2018

A025477 a(n) = exponent of the n-th nontrivial prime power A025475(n).

Original entry on oeis.org

0, 2, 3, 2, 4, 2, 3, 5, 2, 6, 4, 2, 3, 7, 2, 5, 8, 2, 3, 2, 9, 2, 4, 6, 2, 2, 10, 3, 2, 2, 2, 11, 7, 3, 2, 4, 2, 5, 2, 2, 12, 2, 3, 2, 2, 2, 8, 3, 2, 2, 13, 2, 2, 2, 2, 2, 3, 2, 4, 6, 2, 14, 5, 2, 2, 2, 9, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 2, 15, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 10, 2, 16, 2, 3, 2, 2, 2, 2, 7, 2, 3, 2

Offset: 1

David W. Wilson

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001222, A025475, A051903.

With[{nn = 2^20}, {0}~Join~Map[FactorInteger[#][[1, -1]] &, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] ] (* Michael De Vlieger, Oct 23 2023 *)

a(n) = A051903(A025475(n)) = A001222(A025475(n)). - Reinhard Zumkeller, Mar 10 2003

Edited by N. J. A. Sloane, Jan 17 2009 at the suggestion of R. J. Mathar

Name edited by Michael De Vlieger, Oct 23 2023

A072037 Palindromic powers (with positive exponents) of a prime but not a prime (A025475).

Original entry on oeis.org

4, 8, 9, 121, 343, 1331, 10201, 14641, 94249, 1030301, 104060401, 900075181570009, 10022212521222001, 12124434743442121, 12323244744232321, 12341234943214321, 1022321210249420121232201, 1210024420147410244200121, 1210222232227222322220121

Offset: 1

Labos Elemer, Jun 07 2002

			E.g. 94249=307*307

Cf. A025475, A002385.

a = {}; Do[pp = Prime[n]^i; d = IntegerDigits[pp]; If[d == Reverse[d], a = Append[a, pp]], {n, 1, PrimePi[ Sqrt[10^21]]}, {i, 2, Floor[ Log[ Prime[n], 10^21]]}]; Sort[a] (Robert G. Wilson v)

{a=10^15; v=[]; m=sqrt(a); forprime(p=2,m,q=p; while((q=q*p)0,d=divrem(n,10); n=d[1]; rev=10*rev+d[2]); if(q==rev,v=concat(v,q)))); v=vecsort(v); for(j=1,matsize(v)[2],print1(v[j],","))}

Two more term from Klaus Brockhaus, Jun 07 2002
Four more terms from Robert G. Wilson v, Oct 31 2002
Added a(17)-a(19), clarified definition, Donovan Johnson, Sep 01 2012

A088363 Local minima of A053707 (first differences of A025475, powers of a prime but not prime).

Original entry on oeis.org

3, 1, 2, 15, 3, 13, 18, 17, 63, 38, 168, 10, 316, 240, 128, 30, 271, 408, 286, 255, 354, 362, 600, 260, 672, 138, 7, 768, 792, 876, 960, 513, 248, 1080, 546, 2328, 1248, 4008, 1392, 751, 2188, 250, 94, 1728, 3528, 3470, 1848, 2460, 3912, 4008, 3063, 2088, 1554

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).

A088364 gives the corresponding indices. Local maxima of A053707 are in A088365.

			The first four terms of A053707 are 3,4,1,7, hence A053707(1) = 3 is the first and A053707(3) = 1 is the second local minimum of A053707.

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

Cf. A025475, A053707, A088364, A088365.

N:= 10^6: # to use values of A025475 up to N
P:= select(isprime, [2,seq(i,i=3..isqrt(N),2)]):
B:= sort([1,seq(seq(p^i,i=2..ilog[p](N)),p=P)]):
DB:= B[2..-1]-B[1..-2]:
T:= select(t -> DB[t] <= DB[t-1] and DB[t] <= DB[t+1], [$2..nops(DB)-1]):
DB[[1,op(T)]]; # Robert Israel, Aug 21 2023

{m=1; k=0; for(n=2,320000,if(matsize(factor(n))[1]==1&&factor(n)[1,2]>1,d=n-m; if((k<2||b>c)&&(!k<1&&d>=c),print1(c,",")); k++; m=n; b=c; c=d))}

A113495 Lexicographically earliest subsequence of the perfect powers in A025475 such that first differences are an increasing sequence of primes.

Original entry on oeis.org

1, 4, 9, 16, 27, 64, 125, 256, 2187, 16384, 161051, 23945242826029513411849172299223580994042798784118784, 23945249190331908165492143678605499565319109933299901

Offset: 1

Giovanni Teofilatto, Jan 10 2006

nonn

Next terms are a(14) = 2^206, a(15) = 590295810335799160457^3, a(16) = 2^754. - Max Alekseyev, May 21 2011

Note: if the definition is changed to refer to the perfect powers in A001597, the sequence becomes A137354. - R. J. Mathar, Mar 07 2008

			a(2) = 4 because 1 + 3 = 4;
a(3) = 9 because 1 + 3 + 5 = 9;
a(4) = 16 because 1 + 3 + 5 + 7 = 16;
a(5) = 27 because 1 + 3 + 5 + 7 + 11 = 27;
a(6) = 64 because 1 + 3 + 5 + 7 + 11 + 37 = 64;
a(7) = 125 because 1 + 3 + 5 + 7 + 11 + 37 + 61 = 125;
a(8) = 256 because 1 + 3 + 5 + 7 + 11 + 37 + 61 + 131 = 256.

Max Alekseyev, Table of n, a(n) for n = 1..16

Cf. A113759.

3 more terms from R. J. Mathar, Mar 07 2008
a(12) from Donovan Johnson, Aug 09 2010
a(13)-a(16) from Max Alekseyev, May 21 2011

A239520 Prime powers (A025475) such that the distance to the nearest prime power is a triangular number (A000217).

Original entry on oeis.org

1, 4, 8, 9, 49, 64, 125, 128, 2187, 2197, 654481, 657721, 1442401, 1442897, 1708249, 7252249, 7946761, 13053769, 13082689, 21557449, 39677401, 39702601, 86136961, 86174089, 106729561, 106770889, 184063489, 253987969, 302516449, 626550961, 626651089, 844425481

Offset: 1

Alex Ratushnyak, Mar 20 2014

nonn

Cf. A000217, A025475, A239521, A239522, A239523.

cp's OEIS Frontend

A053707 First differences of A025475, powers of a prime but not prime.

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A195942 Zeroless prime powers (excluding primes): Intersection of A025475 and A052382.

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A053706 Primes p such that between p and the next prime, 2 prime powers (A025475) occur.

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A025476 Prime root of n-th nontrivial prime power (A025475, A246547).

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Maple

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A061670 Distance to nearest prime power p^k, k=0 and k >= 2 (A025475).

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Maple

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A025477 a(n) = exponent of the n-th nontrivial prime power A025475(n).

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A072037 Palindromic powers (with positive exponents) of a prime but not a prime (A025475).

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Mathematica

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Extensions

A088363 Local minima of A053707 (first differences of A025475, powers of a prime but not prime).