A050150 -id:A050150 - cp's OEIS frontend

A062090 a(1) = 1, a(n) = smallest odd number that does not divide the product of all previous terms.

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Amarnath Murthy, Jun 16 2001

In A050150 but not here: [729, 15625, 59049, 117649, 531441]; here but not in A050150: [1, 6561, 390625]. - Klaus Brockhaus, Nov 01 2001

If "a(1) = 1," is removed from the definition, the subsequent terms remain the same, since 1 is the empty product. The resulting sequence then comprises the odd terms of A050376. - Peter Munn, Nov 03 2020

			After 13 the next term is 17 (not 15) as 15 = 3*5 divides the product of all the previous terms.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A026477, A062091, A050150 (a different sequence).
Odd terms of {1} U A050376.
Subsequence of A336882.

a062090 n = a062090_list !! (n-1)
a062090_list = f [1, 3 ..] [] where
   f (x:xs) ys = g x ys where
     g _ []     = x : f xs (x : ys)
     g 1 _      = f xs ys
     g z (v:vs) = g (z `div` gcd z v) vs
-- Reinhard Zumkeller, Aug 16 2013

a = {1}; Do[b = Apply[ Times, a]; k = 1; While[ IntegerQ[b/k], k += 2]; a = Append[a, k], { n, 2, 60} ]; a
nxt[{p_,on_}]:=Module[{c=on+2},While[Divisible[p,c],c+=2];{p*c,c}]; NestList[ nxt,{1,1},60][[All,2]] (* Harvey P. Dale, Jul 29 2021 *)

1 together with numbers of the form p^(2^k) where p is an odd prime and k is a nonnegative integer. [Corrected by Peter Munn, Nov 03 2020]

For n >= 2, a(n) = A336882(2^(n-2)). - Peter Munn, Nov 03 2020

Corrected and extended by Dean Hickerson, Jul 10 2001

A235479 Primes whose base-2 representation also is the base-9 representation of a prime.

Original entry on oeis.org

11, 13, 19, 41, 79, 109, 137, 151, 167, 191, 193, 199, 227, 239, 271, 307, 313, 421, 431, 433, 457, 487, 491, 521, 563, 613, 617, 659, 677, 709, 727, 757, 929, 947, 1009, 1033, 1051, 1249, 1483, 1693, 1697, 1709, 1721, 1831, 1951, 1979, 1987, 1993

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
A subsequence of A027697, A050150, A062090 and A176620.

			11 = 1011_2 and 1011_9 = 6571 are both prime, so 11 is a term.

Robert Price, Table of n, a(n) for n = 1..1958
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235466 ⊂ A077723, A235266, A152079, A235475 - A235478, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

is(p,b=9)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

A278911 Odd numbers with prime sum of divisors.

Original entry on oeis.org

9, 25, 289, 729, 1681, 2401, 3481, 5041, 7921, 10201, 15625, 17161, 27889, 28561, 29929, 83521, 85849, 146689, 279841, 458329, 491401, 531441, 552049, 579121, 597529, 683929, 703921, 707281, 734449, 829921, 1190281, 1203409, 1352569, 1394761, 1423249, 1481089

Offset: 1

Jaroslav Krizek, Nov 30 2016

nonn

Also odd numbers with prime number and sum of divisors; if the sum of divisors is prime, then the number of divisors is prime.
Values of prime sums are sorted in A247837.
Subsequence of A050150 (odd numbers with prime number of divisors).
Odd terms of A023194.
All terms are squares of the form p^e such that p is odd prime and e+1 is a prime.

			sigma(9) = 13 (prime).

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

Cf. A000203, A005408, A023194, A050150, A193070, A247837, A278914.

[n: n in[2..10^7] | IsOdd(n) and IsPrime(SumOfDivisors(n)) and IsPrime(NumberOfDivisors(n))];

N:= 10^7: # to get all terms <= N
Ps:= select(isprime, [seq(i,i=3..floor(N^(1/2)),2)]):
es:= map(`-`,select(isprime, [seq(i,i=3..floor(log[3](N))+1,2)]),1):
Pes:= [seq(seq([p,e],p=Ps),e=es)]:
filter:= proc(pe) local v; v:= (pe[1]^(pe[2]+1)-1)/(pe[1]-1); pe[1]^pe[2] <= N and isprime(v) end proc:
sort(map(pe -> pe[1]^pe[2], select(filter, Pes))); # Robert Israel, Jan 22 2019

Select[Range[1, 2*10^6, 2], PrimeQ@DivisorSigma[1, #] &] (* Michael De Vlieger, Dec 01 2016 *)

isok(n) = (n % 2) && isprime(sigma(n)); \\ Michel Marcus, Dec 01 2016

a(n) = A193070(n)^2. - Michel Marcus, Dec 01 2016

A235478 Primes whose base-2 representation also is the base-8 representation of a prime.

Original entry on oeis.org

7, 11, 13, 29, 37, 43, 47, 53, 61, 67, 71, 73, 107, 139, 149, 199, 211, 227, 263, 293, 307, 311, 317, 331, 347, 383, 389, 421, 461, 467, 541, 593, 601, 619, 641, 643, 739, 811, 863, 907, 937, 1061, 1069, 1093, 1117, 1163, 1223, 1283, 1301, 1319, 1321, 1409, 1433, 1439, 1489, 1499, 1523, 1559, 1619, 1697, 1811, 1861, 1879

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
Appears to be a subsequence of A050150, A062090 and A216285.

			11 = 1011_2 and 1011_8 = 521 are both prime, so 11 is a term.

Robert Price, Table of n, a(n) for n = 1..2265
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235465 ⊂ A077722, A235266, A152079, A235475 - A235479, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[300]],PrimeQ[FromDigits[IntegerDigits[#,2],8]]&] (* Harvey P. Dale, Sep 25 2015 *)

is(p,b=8)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

A235467 Primes whose base-4 representation also is the base-3 representation of a prime.

Original entry on oeis.org

2, 89, 137, 149, 281, 293, 353, 389, 409, 421, 593, 613, 661, 1097, 1109, 1289, 1301, 1321, 1381, 1409, 1601, 1609, 1669, 2069, 2129, 2309, 2377, 2389, 2729, 4133, 4229, 4373, 4441, 4513, 4673, 5153

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
This is a subsequence of A002144, A002313, A003655, A050150, A062090, A141293, A175768, A192592, A226181 (conjectural).

			E.g., 89 = 1121_4 and 1121_3 = 43 both are prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

b4b3Q[n_]:=Module[{b4=IntegerDigits[n,4]},Max[b4]<3&&PrimeQ[ FromDigits[ b4,3]]]; Select[Prime[Range[700]],b4b3Q] (* Harvey P. Dale, Dec 14 2021 *)

is(p,b=3,c=4)=vecmax(d=digits(p,c))

forprime(p=1,1e3,is(p,4,3)&&print1(vector(#d=digits(p,3),i,4^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,3,4)

A155186 Primes in A155171.

Original entry on oeis.org

2, 7, 29, 101, 107, 197, 227, 457, 647, 829, 1549, 1627, 2221, 2309, 2347, 2521, 2677, 2801, 3181, 3299, 3529, 3541, 3557, 3739, 3769, 4231, 4549, 4871, 4987, 5651, 5827, 5881, 6037, 6079, 6637, 6827, 7517, 7639, 7937, 9787, 11621, 12041, 12329, 13009

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

Numbers p (prime numbers only) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, s=a+b+c, s-+1 are primes.

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589, A155185, A019389, A062090, A050150

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[p],AppendTo[lst,p]]],{n,8!}];lst

A165285 Primes which are the sum of at least 2 consecutive pars "Prime and PreviousNumber".

Original entry on oeis.org

17, 43, 59, 73, 79, 101, 109, 139, 163, 191, 197, 233, 239, 283, 317, 331, 379, 419, 433, 439, 443, 463, 467, 499, 521, 569, 571, 599, 617, 619, 641, 739, 743, 787, 811, 863, 911, 919, 941, 967, 971, 1021, 1039, 1061, 1063, 1087, 1097, 1109, 1117, 1229, 1289

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 13 2009

(1+2)+(2+3)+(4+5)=17, (4+5)+(6+7)+(10+11)=43, (6+7)+(10+11)+(12+13)=59,..

Cf. A130097, A062090, A050150, A015919

lst={};Do[s=2*Prime[m]-1;Do[p=Prime[n];s+=(2*p-1);If[PrimeQ[s],If[s<=6793,AppendTo[lst,s]]],{n,m+1,3*5!}],{m,1,3*5!}];lst=Take[Union@lst,200]

A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

Original entry on oeis.org

2, 3, 11, 71, 227, 491, 683, 1103, 1187, 2591, 3923, 4271, 4931, 6737, 7193, 7703, 8093, 8753, 8963, 9173, 9377, 10271, 13043, 13451, 13997, 15233, 15443, 15803, 15887, 17957, 18701, 19961, 20681, 21701, 22031, 22073, 24371, 24473, 24683

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1, q=2(prime), a=3, b=4, c=5, s=12-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589, A155185, A019389, A062090, A050150, A155186, A068231, A129517, A054574, A132281.

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[q],AppendTo[lst,q]]],{n,8!}];lst

cp's OEIS Frontend

A062090 a(1) = 1, a(n) = smallest odd number that does not divide the product of all previous terms.

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A235479 Primes whose base-2 representation also is the base-9 representation of a prime.

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A278911 Odd numbers with prime sum of divisors.

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A235478 Primes whose base-2 representation also is the base-8 representation of a prime.

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A235467 Primes whose base-4 representation also is the base-3 representation of a prime.

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A155186 Primes in A155171.

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A165285 Primes which are the sum of at least 2 consecutive pars "Prime and PreviousNumber".

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A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes.

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A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.