A003592 -id:A003592 - cp's OEIS frontend

A107326 Numbers of the form (2^i)*(13^j).

1, 2, 4, 8, 13, 16, 26, 32, 52, 64, 104, 128, 169, 208, 256, 338, 416, 512, 676, 832, 1024, 1352, 1664, 2048, 2197, 2704, 3328, 4096, 4394, 5408, 6656, 8192, 8788, 10816, 13312, 16384, 17576, 21632, 26624, 28561, 32768, 35152, 43264, 53248, 57122

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), May 21 2005

A204455(13*a(n)) = 13, and only for these numbers. - Wolfdieter Lang, Feb 04 2012

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Vincenzo Librandi)

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599.

fQ[n_] := PowerMod[26,n,n]==0; Select[Range[60000],fQ] (* Vincenzo Librandi, Feb 04 2012 *)
mx = 60000; Sort@ Flatten@ Table[2^i*13^j, {i, 0, Log[2, mx]}, {j, 0, Log[13, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

list(lim)=my(v=List(),N); for(n=0,log(lim)\log(13),N=13^n; while(N<=lim,listput(v,N);N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

Sum_{n>=1} 1/a(n) = (2*13)/((2-1)*(13-1)) = 13/6. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(2)*log(13)*n)) / sqrt(26). - Vaclav Kotesovec, Sep 23 2020

A107364 Numbers of the form (3^i)*(13^j).

Original entry on oeis.org

1, 3, 9, 13, 27, 39, 81, 117, 169, 243, 351, 507, 729, 1053, 1521, 2187, 2197, 3159, 4563, 6561, 6591, 9477, 13689, 19683, 19773, 28431, 28561, 41067, 59049, 59319, 85293, 85683, 123201, 177147, 177957, 255879, 257049, 369603, 371293, 531441

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), May 23 2005

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

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326.

[n: n in [1..10^7] | PrimeDivisors(n) subset [3, 13]]; // Vincenzo Librandi, Jun 27 2016

mx = 540000; Sort@ Flatten@ Table[3^i*13^j, {i, 0, Log[3, mx]}, {j, 0, Log[13, mx/3^i]}] (* Robert G. Wilson v, Aug 17 2012 *)
fQ[n_]:=PowerMod[39, n, n] == 0; Select[Range[2 10^7], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(13),N=13^n;while(N<=lim,listput(v,N);N*=3));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

Sum_{n>=1} 1/a(n) = (3*13)/((3-1)*(13-1)) = 13/8. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(3)*log(13)*n)) / sqrt(39). - Vaclav Kotesovec, Sep 23 2020

A077497 Primes of the form 2^r*5^s + 1.

Original entry on oeis.org

2, 3, 5, 11, 17, 41, 101, 251, 257, 401, 641, 1601, 4001, 16001, 25601, 40961, 62501, 65537, 160001, 163841, 16384001, 26214401, 40960001, 62500001, 104857601, 167772161, 256000001, 409600001, 655360001, 2441406251, 2500000001, 4194304001, 10485760001

Offset: 1

Amarnath Murthy, Nov 07 2002

nonn

These are also the prime numbers p for which there is an integer solution x to the equation p*x = p*10^p + x, or equivalently, the prime numbers p for which (p*10^p)/(p-1) is an integer. - Vicente Izquierdo Gomez, Feb 20 2013

For n > 2, all terms are congruent to 5 (mod 6). - Muniru A Asiru, Sep 03 2017

			101 is in the sequence, since 101 = 2^2*5^2 + 1 and 101 is prime.

Ray Chandler, Table of n, a(n) for n = 1..3150

Cf. A005109, A077497, A077498, A077500, A003592, A077313, A019434.

K:=10^7;; # to get all terms <= K.
A:=Filtered(Filtered([1..K],i-> i mod 6=5),IsPrime);;
B:=List(A,i->Factors(i-1));;
C:=[];;  for i in B do if Elements(i)=[2] or Elements(i)=[2,5]  then Add(C,Position(B,i)); fi; od;
A077497:=Concatenation([2,3],List(C,i->A[i])); # Muniru A Asiru, Sep 03 2017

Do[p=Prime[k];s=FindInstance[p x == p 10^p+x,x,Integers];If[s!={},Print[p]],{k,10000}] (* Vicente Izquierdo Gomez, Feb 20 2013 *)

list(lim)=my(v=List(),t);for(r=0,log(lim)\log(5),t=5^r;while(t<=lim,if(isprime(t+1),listput(v,t+1)); t<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 29 2013

Corrected and extended by Reinhard Zumkeller, Nov 19 2002

More terms from Ray Chandler, Aug 02 2003

A132741 Largest divisor of n having the form 2^i*5^j.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 1, 2, 5, 16, 1, 2, 1, 20, 1, 2, 1, 8, 25, 2, 1, 4, 1, 10, 1, 32, 1, 2, 5, 4, 1, 2, 1, 40, 1, 2, 1, 4, 5, 2, 1, 16, 1, 50, 1, 4, 1, 2, 5, 8, 1, 2, 1, 20, 1, 2, 1, 64, 5, 2, 1, 4, 1, 10, 1, 8, 1, 2, 25, 4, 1, 2, 1, 80, 1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 1, 2, 5, 32, 1, 2

Offset: 1

Reinhard Zumkeller, Aug 27 2007

The range of this sequence, { a(n); n>=0 }, is equal to A003592. - M. F. Hasler, Dec 28 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003592, A006519, A007732, A007814, A051626, A060904, A112765, A132740.
Cf. A001620, A082633.
Cf. A379003 (ordinal transform), A379004 (rgs-transform).
Cf. also A355582.

a132741 = f 2 1 where
   f p y x | r == 0    = f p (y * p) x'
           | otherwise = if p == 2 then f 5 y x else y
           where (x', r) = divMod x p
-- Reinhard Zumkeller, Nov 19 2015

A132741 := proc(n) local f,a; f := ifactors(n)[2] ; a := 1; for f in ifactors(n)[2] do if op(1,f) =2 then a := a*2^op(2,f) ; elif op(1,f) =5 then a := a*5^op(2,f) ; end if; end do;a; end proc: # R. J. Mathar, Sep 06 2011

a[n_] := SelectFirst[Reverse[Divisors[n]], MatchQ[FactorInteger[#], {{1, 1}} | {{2, }} | {{5, }} | {{2, }, {5, }}]&]; Array[a, 100] (* Jean-François Alcover, Feb 02 2018 *)
a[n_] := Times @@ ({2, 5}^IntegerExponent[n, {2, 5}]); Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

A132741(n)=5^valuation(n,5)<M. F. Hasler, Dec 28 2015

a(n) = n / A132740(n).
a(A003592(n)) = A003592(n).
A051626(a(n)) = 0.
A007732(a(n)) = 1.
From R. J. Mathar, Sep 06 2011: (Start)
Multiplicative with a(2^e)=2^e, a(5^e)=5^e and a(p^e)=1 for p=3 or p>=7.
Dirichlet g.f. zeta(s)*(2^s-1)*(5^s-1)/((2^s-2)*(5^s-5)). (End)
a(n) = A006519(n)*A060904(n) = 2^A007814(n)*5^A112765(n). - M. F. Hasler, Dec 28 2015
Sum_{k=1..n} a(k) ~ n*(12*log(n)^2 + (24*gamma + 36*log(2) - 24)*log(n) + 24 - 24*gamma - 36*log(2) + 36*gamma*log(2) + 2*log(2)^2 - 18*log(5) + 18*gamma*log(5) + 27*log(2)*log(5) + 2*log(5)^2 + 18*log(5)*log(n) - 24*gamma_1)/(60*log(2)*log(5)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Jan 26 2023

A107466 Numbers of the form (5^i)*(13^j).

Original entry on oeis.org

1, 5, 13, 25, 65, 125, 169, 325, 625, 845, 1625, 2197, 3125, 4225, 8125, 10985, 15625, 21125, 28561, 40625, 54925, 78125, 105625, 142805, 203125, 274625, 371293, 390625, 528125, 714025, 1015625, 1373125, 1856465, 1953125, 2640625

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), May 27 2005

nonn

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

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364.

mx = 2700000; Sort@ Flatten@ Table[5^i*13^j, {i, 0, Log[5, mx]}, {j, 0, Log[13, mx/5^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(13),N=13^n;while(N<=lim,listput(v,N);N*=5));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

from sympy import integer_log
def A107466(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(integer_log(x//13**i,5)[0]+1 for i in range(integer_log(x,13)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 25 2025

Sum_{n>=1} 1/a(n) = (5*13)/((5-1)*(13-1)) = 65/48. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(5)*log(13)*n)) / sqrt(65). - Vaclav Kotesovec, Sep 23 2020

A114522 Numbers n such that sum of distinct prime divisors of n is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 31, 32, 34, 36, 37, 40, 41, 43, 44, 47, 48, 49, 50, 53, 54, 58, 59, 61, 64, 67, 68, 71, 72, 73, 79, 80, 81, 82, 83, 88, 89, 96, 97, 100, 101, 103, 107, 108, 109, 113, 116, 118, 121, 125, 127

Offset: 1

Leroy Quet, Dec 05 2005

nonn

Sequence is the union of the primes and sequence A047820.

			24 = 2^3 * 3 and 2 + 3 = 5, which is prime. So 24 is included.

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

Cf. A000079, A003586, A003592.

Cf. A008472, A047820, A108605, A114518.

[k:k in [2..150]| IsPrime(&+PrimeDivisors(k))]; // Marius A. Burtea, Oct 06 2019

f[n_] := Plus @@ First /@ FactorInteger[n]; Select[Range[130], PrimeQ[f[ # ]] &] (* Ray Chandler, Dec 07 2005 *)
Select[Range@127, PrimeQ[Plus @@ First /@ FactorInteger@# ] &] (* Robert G. Wilson v, Dec 07 2005 *)

for(n=1, 200, v=factor(n); s=0; for(i=1,matsize(v)[1],s+=v[i,1]); if(isprime(s), print1(n, ", "))) \\ Lambert Herrgesell (zero815(AT)googlemail.com), Dec 07 2005

Extended by Robert G. Wilson v, Ray Chandler and Lambert Herrgesell (zero815(AT)googlemail.com), Dec 07 2005

A108056 Numbers of the form (7^i)*(13^j).

Original entry on oeis.org

1, 7, 13, 49, 91, 169, 343, 637, 1183, 2197, 2401, 4459, 8281, 15379, 16807, 28561, 31213, 57967, 107653, 117649, 199927, 218491, 371293, 405769, 753571, 823543, 1399489, 1529437, 2599051, 2840383, 4826809, 5274997, 5764801, 9796423, 10706059, 18193357, 19882681

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 02 2005

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

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364, A107466.

n = 10^7; Flatten[Table[7^i*13^j, {i, 0, Log[7, n]}, {j, 0, Log[13, n/7^i]}]] // Sort (* Amiram Eldar, Sep 23 2020 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(13),N=13^n;while(N<=lim,listput(v,N);N*=7));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

from sympy import integer_log
def A108056(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(integer_log(x//13**i,7)[0]+1 for i in range(integer_log(x,13)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Sum_{n>=1} 1/a(n) = (7*13)/((7-1)*(13-1)) = 91/72. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(7)*log(13)*n)) / sqrt(91). - Vaclav Kotesovec, Sep 23 2020

More terms from Amiram Eldar, Sep 23 2020

A352153 Smallest digit in the decimal expansion of 1/n, ignoring leading and trailing 0's.

Original entry on oeis.org

1, 5, 3, 2, 2, 1, 1, 1, 1, 1, 0, 3, 0, 1, 6, 2, 0, 5, 0, 5, 0, 4, 0, 1, 4, 1, 0, 1, 0, 3, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Bernard Schott and Robert G. Wilson v, Mar 06 2022

Leading 0's are not considered, otherwise a(n) would be 0 when n >= 11 (see examples for 13 and 14).

Trailing 0's are not also considered, otherwise when 1/n is a terminating decimal (A003592), a(n) would be also 0.

			1/13 = 0.076923076923076923... with periodic part = '769230' (or '076923'), hence a(13) = 0.
1/14 = 0.0714285714285714285... with periodic part = '714285', hence a(14) = 1.
1/40 = 0.025 hence a(40) = 2.

Index entries for sequences related to decimal expansion of 1/n

Cf. A003592, A333236 (largest digit).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Array[Min@ f@# &, 105]

a(n) = n iff n = 1 or n = 3.
a(10*n) = a(n).
a(10^n) = 1.

A352155 Numbers m such that the smallest digit in the decimal expansion of 1/m is 1, ignoring leading and trailing 0's.

Original entry on oeis.org

1, 6, 7, 8, 9, 10, 14, 24, 26, 28, 32, 35, 54, 55, 56, 60, 64, 65, 66, 70, 72, 74, 75, 80, 82, 88, 90, 100, 104, 112, 128, 140, 175, 176, 224, 240, 260, 280, 320, 350, 432, 448, 468, 504, 512, 528, 540, 548, 550, 560, 572, 576, 584, 592, 600, 616, 625, 640, 650, 660

Offset: 1

Bernard Schott and Robert G. Wilson v, Mar 17 2022

Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 (A003592) would be terms.
If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive terms.
{8, 88, 888, ...} = A002282 \ {0} is a subsequence.

			m = 14 is a term since 1/14 = 0.0714285714285714285... and the smallest term after the leading 0 is 1.
m = 240 is a term since 1/240 = 0.00416666666... and the smallest term after the leading 0's is 1.
m = 888 is a term since 1/888 = 0.001126126126... and the smallest term after the leading 0's is 1.

Cf. A002282, A333402.

Similar with smallest digit k: A352154 (k=0), this sequence (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 1 &]

from itertools import count, islice
from sympy import multiplicity, n_order
def A352155_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,n), multiplicity(5,n)
        k, m = 10**max(m2,m5), 10**(t := n_order(10,n//2**m2//5**m5))-1
        c = k//n
        s = str(m*k//n-c*m).zfill(t)
        if s == '0' and min(str(c)) == '1':
            yield n
        elif '0' not in s and min(str(c).lstrip('0')+s) == '1':
                yield n
A352155_list = list(islice(A352155_gen(),20)) # Chai Wah Wu, Mar 28 2022

A352153(a(n)) = 1.

A352156 Numbers m such that the smallest digit in the decimal expansion of 1/m is 2, ignoring leading and trailing 0's.

Original entry on oeis.org

4, 5, 16, 36, 40, 44, 45, 50, 108, 160, 216, 252, 288, 292, 308, 360, 364, 375, 396, 400, 404, 440, 444, 450, 500, 1024, 1080, 1375, 1600, 2072, 2160, 2368, 2520, 2880, 2920, 3080, 3125, 3375, 3600, 3640, 3750, 3848, 3960, 4000, 4040, 4125, 4224, 4368, 4400, 4440, 4500, 5000

Offset: 1

Bernard Schott and Robert G. Wilson v, Mar 19 2022

Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 (A003592) would be terms.
If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive terms.

			m = 16 is a term since 1/16 = 0.0625 and the smallest term after the leading 0 is 2.
m = 216 is a term since 1/216 = 0.004629629629... and the smallest term after the leading 0's is 2.
m = 4444 is not a term since 1/4444 = 0.00022502250225... and the smallest term after the leading 0's is 0.

Cf. A341383.
Subsequences: A093141 \ {1}, A093143 \ {1}.
Similar with smallest digit k: A352154 (k=0), A352155 (k=1), this sequence (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 2 &]

from itertools import count, islice
from sympy import multiplicity, n_order
def A352156_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,n), multiplicity(5,n)
        k, m = 10**max(m2,m5), 10**(t := n_order(10,n//2**m2//5**m5))-1
        c = k//n
        s = str(m*k//n-c*m).zfill(t)
        if s == '0' and min(str(c)) == '2':
            yield n
        elif '0' not in s and min(str(c).lstrip('0')+s) == '2':
                yield n
A352156_list = list(islice(A352156_gen(),20)) # Chai Wah Wu, Mar 28 2022

A352153(a(n)) = 2.

cp's OEIS Frontend

A107326 Numbers of the form (2^i)*(13^j).

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A107364 Numbers of the form (3^i)*(13^j).

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Magma

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A077497 Primes of the form 2^r*5^s + 1.

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GAP

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A132741 Largest divisor of n having the form 2^i*5^j.

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Haskell

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A107466 Numbers of the form (5^i)*(13^j).

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A114522 Numbers n such that sum of distinct prime divisors of n is prime.

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Magma

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PARI

Extensions

A108056 Numbers of the form (7^i)*(13^j).

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