A093170 -id:A093170 - cp's OEIS frontend

A096507 Numbers k such that 6*R_k + 1 is a prime, where R_k = 11...1 is the repunit (A002275) of length k.

1, 2, 6, 8, 9, 11, 20, 23, 41, 63, 66, 119, 122, 149, 252, 284, 305, 592, 746, 875, 1204, 1364, 2240, 2403, 5106, 5776, 5813, 12456, 14235, 39606, 55544, 84239, 275922

Offset: 1

Labos Elemer, Jul 12 2004

nonn

Also numbers k such that (2*10^k + 1)/3 is prime.
These numbers form a near-repdigit sequence (6)w7.
All the terms from k = 2403 through 14235 correspond to primes. - Joao da Silva (zxawyh66(AT)yahoo.com), Oct 03 2005

			k = 9 gives 2000000001/3 = 666666667, which is prime.
k = 20 gives 66666666666666666667, which is prime.

Makoto Kamada, Prime numbers of the form 66...667.
Index entries for primes involving repunits

Cf. A002275, A056657, A093170, A096503, A096504, A096505, A096506, A096508.

Select[Range@ 2500, PrimeQ[FromDigits@ Table[6, {#}] + 1] &] (* or *)
Select[Range@ 2500, PrimeQ[2 (10^# - 1)/3 + 1] &] (* Michael De Vlieger, Jul 04 2016 *)

a(n) = A056657(n) + 1.

More terms from Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004
39606 and 55544 from Serge Batalov, Jun 2009
84239 from Serge Batalov, Jul 06 2009 confirmed as next term by Ray Chandler, Feb 23 2012
a(33) from Kamada data by Tyler Busby, Apr 14 2024

A056657 Numbers k such that 60*R_k + 7 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

0, 1, 5, 7, 8, 10, 19, 22, 40, 62, 65, 118, 121, 148, 251, 283, 304, 591, 745, 874, 1203, 1363, 2239, 2402, 5105, 5775, 5812, 12455, 14234, 39605, 55543, 84238, 275921

Offset: 1

Robert G. Wilson v, Aug 09 2000

nonn

Also numbers k such that (20*10^k+1)/3 is prime.

			7, 67, 666667, 66666667, 666666667, 66666666667, etc. are primes.

Makoto Kamada, Prime numbers of the form 66...667.
Index entries for primes involving repunits

Cf. A002275, A093170, A096507.

Do[ If[ PrimeQ[ 60*(10^n - 1)/9 + 7 ], Print[n]], {n, 25556}]

a(n) = A096507(n) - 1.

More terms from Robert G. Wilson v, Oct 22 2003
2239,2402,5105,5775 from Farideh Firoozbakht, Dec 23 2003
39605 and 55543 from Serge Batalov, Jun 2009
84238 from Serge Batalov, Jul 06 2009 confirmed as next term by Ray Chandler, Feb 23 2012
a(33) derived from A096507 by Robert Price, Jul 07 2024

A105324 Numbers n such that 2*reversal(n)=sigma(n).

Original entry on oeis.org

6, 73, 483, 4074, 4473, 4623, 7993, 42813, 69855, 253782, 799993, 7999993, 46000023, 426000213, 4600000023, 6718967838, 42600000213, 46000000023, 79999999993, 426000000213

Offset: 1

Farideh Firoozbakht, Apr 16 2005

I. If p=8*10^n-7 is a prime then p is in the sequence because reversal(p)=4*10^n-3 & sigma(p)=8*10^n-6 so 2*reversal(p) =sigma(p). 73,7993,799993 & 7999993 are such terms.
II. If q=(2*10^n+1)/3 is a prime then (a): 69*q is in the sequence because 69*q=46*10^n+23; reversal (69*q)=32*10^n+64 & sigma(69*q)=96*q+96=64*10^n+128 so 2*reversal (69*q)=sigma(69*q). 483,4623 & 46000023 are such terms. (b): 639*q is in the sequence because 639*q=426*10^n+213; reversal (639*q)=312*10^n+624 & sigma(639*q)=936*q+936=624*10^n+1248 so 2*reversal(639*q)=sigma(639*q). 42813 & 426000213 are such terms.
a(21) > 10^12. - Giovanni Resta, Oct 28 2012

			253782 is in the sequence because reversal(253782)=287352; sigma(253782)=574704 & 2*287352=574704.

Cf. A093170, A096507, A099190, A105322, A105323, A105325, A105326.

reversal[n_]:= FromDigits[Reverse[IntegerDigits[n]]]; Do[If[2* reversal[n]== DivisorSigma[1, n], Print[n]], {n, 1000000000}]
Select[Range[8*10^6],2*IntegerReverse[#]==DivisorSigma[1,#]&] (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Oct 29 2022 *)

a(15)-a(19) from Donovan Johnson, Dec 21 2008

a(20) from Giovanni Resta, Oct 28 2012

A104907 Numbers n such that d(n)*reversal(n)=sigma(n), where d(n) is number of positive divisors of n.

Original entry on oeis.org

1, 73, 861, 7993, 8241, 799993, 7999993, 44908500, 82000041, 293884500, 6279090751, 8200000041, 62698513951, 79999999993, 82000000041, 374665576800, 597921764310, 7999999999993, 8200000000041

Offset: 1

Farideh Firoozbakht, Apr 16 2005

All primes of the form 8*10^n-7 are in the sequence, so 8*10^A099190-3 is a subsequence of this sequence. A105322 is this subsequence. Also if p=(2*10^n+1)/3 is prime then 123*p is in the sequence, so 123*A093170 is a subsequence of this sequence. A105323 is this subsequence.

a(20) > 10^13. - Giovanni Resta, Jul 13 2015

			Let p=8*10^n-7 be a prime so d(p)=2; reversal(p)=4*10^n-3 and sigma(p)
=8*10^n-6 hence d(p)*reversal(p)=sigma(p) and this shows that p
is in the sequence. 73,7993,799993 and 7999993 are such terms.
Also let q=(2*10^n+1)/3 be a prime, so 123*q=82*10^n+41; reversal
(123*q)=14*10^n+28; d(123*q)=8 and sigma(123*q)=168*q+168=112*10^n
+224 hence d(123*q)*reversal(123*q)=sigma(123*q) and this shows
that 123*q is in the sequence. 861,8241 and 82000041 are such terms.

Cf. A056657, A093170, A096507, A099190, A105322, A105323, A105324.

reversal[n_]:= FromDigits[Reverse[IntegerDigits[n]]]; Do[If[DivisorSigma[0, n]*reversal[n] == DivisorSigma[1, n], Print[n]], {n, 1125000000}]
Select[Range[8*10^6],DivisorSigma[0,#]IntegerReverse[#]==DivisorSigma[1,#]&] (* The program generates the first 7 terms of the sequence. *) (* Harvey P. Dale, Jan 31 2023 *)

a(11)-a(15) from Donovan Johnson, Feb 06 2010
a(16) from Giovanni Resta, Feb 06 2014
a(17)-a(19) from Giovanni Resta, Jul 13 2015

A105323 Numbers of the form 41(210^n+1) where (2*10^n+1)/3 is prime (n is in the sequence A096507).

Original entry on oeis.org

861, 8241, 82000041, 8200000041, 82000000041, 8200000000041, 8200000000000000000041, 8200000000000000000000041, 8200000000000000000000000000000000000000041

Offset: 1

Farideh Firoozbakht, Apr 16 2005

nonn

A105323=41*A093170=41*(2*10^A096507+1)=41*(2*10^(A056657+1)+1). If m is in the sequence then d(m)*reversal(m)=sigma(m) (see A104907). So this sequence is a subsequence of A104907.

			861 is in the sequence because 861=41*(2*10^1+1); (2*10^1+1)/3=7 and 7 is prime.

Cf. A104907, A056657, A096507, A093170.

Do[If[PrimeQ[(2*10^n + 1)/3], Print[41*(2*10^n + 1)]], {n, 63}]

A351975 Numbers k such that A037276(k) == -1 (mod k).

Original entry on oeis.org

1, 6, 14, 18, 48, 124, 134, 284, 3135, 4221, 9594, 16468, 34825, 557096, 711676, 746464, 1333334, 2676977, 6514063, 11280468, 16081252, 35401658, 53879547, 133333334, 198485452, 223856659, 1333333334, 2514095219, 2956260256, 3100811124, 10912946218, 19780160858

Offset: 1

J. M. Bergot and Robert Israel, Feb 26 2022

Numbers k such that the concatenation of prime factors of k is 1 less than a multiple of k.
Contains 2*m for m in A093170.
Terms k where k-1 is prime include 6, 14, 18, 48 and 284. Are there others?

			a(4) = 48 is a term because 48=2*2*2*2*3 and 22223 == -1 (mod 48).

Martin Ehrenstein, Table of n, a(n) for n = 1..38

Cf. A037276, A093170.

tcat:= proc(x,y) x*10^(1+ilog10(y))+y end proc:
filter:= proc(n) local F,t,i;
F:= map(t -> t[1]$t[2], sort(ifactors(n)[2],(a,b)->a[1]

from sympy import factorint
def A037276(n):
    if n == 1: return 1
    return int("".join(str(p)*e for p, e in sorted(factorint(n).items())))
def afind(limit, startk=1):
    for k in range(startk, limit+1):
        if (A037276(k) + 1)%k == 0:
            print(k, end=", ")
afind(10**6) # Michael S. Branicky, Feb 27 2022
# adapted and corrected by Martin Ehrenstein, Mar 06 2022

from itertools import count, islice
from sympy import factorint
def A351975_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        c = 0
        for d in sorted(factorint(k,multiple=True)):
            c = (c*10**len(str(d)) + d) % k
        if c == k-1:
            yield k
A351975_list = list(islice(A351975_gen(),10)) # Chai Wah Wu, Feb 28 2022

a(24)-a(25) from Michael S. Branicky, Feb 27 2022

Prepended 1 and more terms from Martin Ehrenstein, Feb 28 2022

A355970 Primes p such that p^2 is the concatenation of x and 2*x+1 for some x.

Original entry on oeis.org

5, 7, 67, 28573, 666667, 31578949, 64912283, 66666667, 666666667, 66666666667, 29083665338647, 31772053083528493, 50819672131147541, 4299928432854102613, 6811594202898550727, 66666666666666666667, 29136816792745854416111, 46823891622677827205227, 66666666666666666666667

Offset: 1

J. M. Bergot and Robert Israel, Jul 21 2022

			a(3) = 67 is a term because it is prime and 67^2 = 4489 is the concatenation of 44 and 2*44+1=89.

Contains A093170.

Cf. A309808, A309828.

dcat:=proc(a,b) a*10^(1+ilog10(b))+b end proc:
f:= proc(t) local s;
if not issqr(t) then return NULL fi;
s:=sqrt(t);
if isprime(s) then return s fi
end proc:
map(f, [seq(dcat(x,2*x+1), x=1..5*10^7)]);

More terms from Jinyuan Wang, Jul 21 2022

cp's OEIS Frontend

A096507 Numbers k such that 6*R_k + 1 is a prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A056657 Numbers k such that 60*R_k + 7 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A105324 Numbers n such that 2*reversal(n)=sigma(n).

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A104907 Numbers n such that d(n)*reversal(n)=sigma(n), where d(n) is number of positive divisors of n.

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A105323 Numbers of the form 41*(2*10^n+1) where (2*10^n+1)/3 is prime (n is in the sequence A096507).

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A351975 Numbers k such that A037276(k) == -1 (mod k).

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A355970 Primes p such that p^2 is the concatenation of x and 2*x+1 for some x.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Extensions

A105323 Numbers of the form 41(210^n+1) where (2*10^n+1)/3 is prime (n is in the sequence A096507).