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Search: a025475 -id:a025475
Displaying 1-10 of 183 results found. page 1 2 3 4 5 6 7 8 9 10 ... 19
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A195942 Zeroless prime powers (excluding primes): Intersection of A025475 and A052382. +20
21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
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.
FORMULA
A195942 = A025475 intersect A052382.
A010055(a(n)) * (1 - A010051(a(n))) * A168046(a(n)) = 1. - Reinhard Zumkeller, Sep 27 2011
MATHEMATICA
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 *)
PROG
(PARI) for( n=1, 9999, is_A025475(n) && is_A052382(n) && print1(n", "))
(Haskell)
a195942 n = a195942_list !! (n-1)
a195942_list = filter (\x -> a010051 x == 0 && a010055 x == 1) a052382_list
-- Reinhard Zumkeller, Sep 27 2011
CROSSREFS
Cf. A195985, A195943, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Sep 25 2011
STATUS
approved
A053707 First differences of A025475, powers of a prime but not prime. +20
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
T. D. Noe, Table of n, a(n) for n=1..10000
FORMULA
a(n) = A025475(n+1) - A025475(n).
EXAMPLE
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.
MATHEMATICA
Differences@ Join[{1}, Select[Range@ 16200, And[! PrimeQ@ #, PrimePowerQ@ #] &]] (* Michael De Vlieger, Jul 04 2016 *)
PROG
(PARI) {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
(Python)
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
CROSSREFS
Cf. A025475.
KEYWORD
nonn
AUTHOR
Labos Elemer, Feb 14 2000
EXTENSIONS
Edited by Klaus Brockhaus, Sep 25 2003
STATUS
approved
A025476 Prime root of n-th nontrivial prime power (A025475, A246547). +20
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
MAPLE
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
MATHEMATICA
Transpose[ Flatten[ FactorInteger[ Select[ Range[2, 30000], !PrimeQ[ # ] && Mod[ #, # - EulerPhi[ # ]] == 0 &]], 1]][[1]] (* Robert G. Wilson v *)
PROG
(PARI) forcomposite(n=4, 10^5, if( ispower(n, , &p) && isprime(p), print1(p, ", "))) \\ Joerg Arndt, Sep 11 2021
(Python)
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
CROSSREFS
Cf. A025473, A025475, A048148, A246547.
KEYWORD
easy,nonn,changed
AUTHOR
David W. Wilson
STATUS
approved
A061670 Distance to nearest prime power p^k, k=0 and k >= 2 (A025475). +20
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to distance to nearest element of some set
EXAMPLE
a(12)=3 because 9=3^2 is the nearest power to 12 (12-9=3).
MAPLE
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
PROG
(PARI) 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
CROSSREFS
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).
KEYWORD
nonn,look
AUTHOR
Michel ten Voorde, Jun 16 2001
EXTENSIONS
Definition corrected, and more terms from Robert Israel, Mar 23 2018
STATUS
approved
A025477 a(n) = exponent of the n-th nontrivial prime power A025475(n). +20
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = A051903(A025475(n)) = A001222(A025475(n)). - Reinhard Zumkeller, Mar 10 2003
MATHEMATICA
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 *)
CROSSREFS
Cf. A001222, A025475, A051903.
KEYWORD
easy,nonn
AUTHOR
David W. Wilson
EXTENSIONS
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
STATUS
approved
A072037 Palindromic powers (with positive exponents) of a prime but not a prime (A025475). +20
3
4, 8, 9, 121, 343, 1331, 10201, 14641, 94249, 1030301, 104060401, 900075181570009, 10022212521222001, 12124434743442121, 12323244744232321, 12341234943214321, 1022321210249420121232201, 1210024420147410244200121, 1210222232227222322220121 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..19.
EXAMPLE
E.g. 94249=307*307
MATHEMATICA
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)
PROG
(PARI) {a=10^15; v=[]; m=sqrt(a); forprime(p=2, m, q=p; while((q=q*p)<a, n=q; rev=0; while(n>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], ", "))}
CROSSREFS
Cf. A025475, A002385.
KEYWORD
base,easy,nonn
AUTHOR
Labos Elemer, Jun 07 2002
EXTENSIONS
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
STATUS
approved
A088363 Local minima of A053707 (first differences of A025475, powers of a prime but not prime). +20
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
MAPLE
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
PROG
(PARI) {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))}
CROSSREFS
Cf. A025475, A053707, A088364, A088365.
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, Sep 27 2003
STATUS
approved
A113495 Lexicographically earliest subsequence of the perfect powers in A025475 such that first differences are an increasing sequence of primes. +20
3
1, 4, 9, 16, 27, 64, 125, 256, 2187, 16384, 161051, 23945242826029513411849172299223580994042798784118784, 23945249190331908165492143678605499565319109933299901 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..16
EXAMPLE
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.
CROSSREFS
Cf. A113759.
KEYWORD
nonn
AUTHOR
Giovanni Teofilatto, Jan 10 2006
EXTENSIONS
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
STATUS
approved
A239520 Prime powers (A025475) such that the distance to the nearest prime power is a triangular number (A000217). +20
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..32.
CROSSREFS
Cf. A000217, A025475, A239521, A239522, A239523.
KEYWORD
nonn
AUTHOR
Alex Ratushnyak, Mar 20 2014
STATUS
approved
A239523 Prime powers (A025475) such that the distances to the two nearest prime powers are primes. +20
3
1, 25, 27, 32, 128, 512, 2048, 8192, 32768, 562949953421312 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A subsequence of A239522.
a(4)...a(10) are powers of 2, a(10) = 2^49.
LINKS
Table of n, a(n) for n=1..10.
CROSSREFS
Cf. A025475, A239520 - A239522.
KEYWORD
nonn,more
AUTHOR
Alex Ratushnyak, Mar 20 2014
STATUS
approved
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Last modified August 29 11:15 EDT 2024. Contains 375512 sequences. (Running on oeis4.)