A031974 -id:A031974 - cp's OEIS frontend

A093521 Runs of 1's of lengths 1, prime(1), prime(2), prime(3), ... separated by 0's.

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 1

Robert G. Wilson v, Mar 29 2004

nonn

Carl Sagan's "Contact" sequence.

Zeros occur at positions given by 1+A110895(k). - Antti Karttunen, Nov 08 2018

W. A. Dembski and J. M. Kushiner, Signs of Intelligence, Baker Book House Co., Grand Rapids, MI, p30-31, 2001,
Carl Sagan, Contact, Simon and Schuster, Chapter 4 "Prime Numbers," pp. 68-82, NY, 1985.

Antti Karttunen, Table of n, a(n) for n = 1..24235 (first 101 runs)
Robin Dougherty, Robert Zemeckis' Contact, Salon.
Alex Kasman, Mathematical Fiction, Contact (1985).
Alex Kasman, How 'Contact' by Carl Sagan Ends.
Index entries for characteristic functions

Cf. A000040, A000042, A005171, A031974, A055976, A056051, A066247, A091247, A110895, A175851, A175856.

a = Table[1, {100}]; Do[ a[[Sum[Prime[i], {i, n}] + n]] = 0, {n, 1, 8}]; a

up_to = 111;
A093521list(up_to) = { my(v=vector(up_to), i=2, j); v[1] = 1; v[2] = 0; forprime(p=2, oo, j=p; while(j, if(i==up_to, return(v), i++; v[i] = 1; j--)); if(i==up_to, return(v), i++; v[i] = 0)); };
v093521 = A093521list(up_to);
A093521(n) = v093521[n];

Data section extended up to n=111 by Antti Karttunen, Nov 08 2018

A179005 Indices of nonprime repunits: nonnegative numbers n such that 11...111 = (10^n - 1)/9 is composite.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Andrew McFarland, Jan 03 2011

nonn

The complement of A004023 in the set of nonnegative integers.

			3 appears because the 3-digit repunit 111 is composite (37*3).
4 appears because the 4-digit repunit 1111 is composite (101*11).

Complement of A004023. Cf. A002275, A004022, A031974, A259102.

isok(n) = !isprime((10^n - 1)/9); \\ Michel Marcus, Sep 05 2013

A259102 Composite repunits with a prime number of 1's.

Original entry on oeis.org

111, 11111, 1111111, 11111111111, 1111111111111, 11111111111111111, 11111111111111111111111111111, 1111111111111111111111111111111, 1111111111111111111111111111111111111, 11111111111111111111111111111111111111111, 1111111111111111111111111111111111111111111

Offset: 1

N. J. A. Sloane, Jun 23 2015

Alois P. Heinz, Table of n, a(n) for n = 1..164
R. Ondrejka, Letter to N. J. A. Sloane, May 15 1976

Cf. A002275, A004022, A004023, A031974, A179005.

f:=n->(10^n-1)/9; [f(3),f(5),f(7),f(11),f(13),f(17),f(29),f(31),f(37),f(41),f(43),f(47)];  # cf. A004023
# second Maple program:
r:= n-> (10^n-1)/9:
b:= proc(n) option remember; local p;
      p:=`if`(n=1, 1, b(n-1));
      do p:= nextprime(p);
         if not isprime(r(p)) then return p fi
      od
    end:
a:= n-> r(b(n)):
seq(a(n), n=1..15);  # Alois P. Heinz, Jun 25 2015

A323060 a(n) = R_(prime(n)#) / Product_{j=1..n} R_(prime(j)), where prime(n)# is the n-th primorial number A002110(n) and R_k = (10^k - 1)/9.

Original entry on oeis.org

1, 91, 8190090090909099099181

Offset: 1

Patrick A. Thomas, Jan 19 2019

a(4) has 196 digits.

The numbers R_k = 1, 11, 111, ... are sometimes called "Rep-units" or "repunits". The octal versions of a(1) through a(4) may be obtained from the decimal versions by replacing each 6 with a 4, each 7 with a 5, each 8 with a 6, and each 9 with a 7. Similar relations exist for other bases.

			R_30 / (11*111*11111) = 8190090090909099099181.

Author?, "The Ultimate Number Series Challenge", Vidya, Oct 1988, p. 9.

Michel Marcus, Table of n, a(n) for n = 1..4
Patrick A. Thomas, Term a(5) and information on a(6)

Cf. A002110, A002275, A031974.

f[n_] := (10^n - 1)/9; Array[f[Product[Prime@ i, {i, #}]]/Product[f@ Prime@ j, {j, #}] &, 3] (* Michael De Vlieger, Jan 19 2019 *)

R(n) = (10^n-1)/9; \\ A002275
primo(n) = prod(i=1, n, prime(i)); \\ A002110
a(n) = R(primo(n))/prod(j=1, n, R(prime(j))); \\ Michel Marcus, Jan 21 2019

A344825 Integers whose digit sum is prime and whose digit product is a perfect square > 0.

Original entry on oeis.org

11, 14, 41, 49, 94, 111, 119, 122, 128, 133, 155, 166, 182, 188, 191, 199, 212, 218, 221, 229, 236, 263, 281, 289, 292, 298, 313, 326, 331, 362, 368, 386, 449, 494, 515, 551, 559, 595, 616, 623, 632, 638, 661, 683, 779, 797, 812, 818, 821, 829, 836, 863, 881

Offset: 1

Ryan Bresler, May 29 2021

If k is in the sequence then all anagrams of k are in the sequence. - David A. Corneth, May 29 2021

Trivially, this sequence has infinite elements. A031974 is an infinite sequence that is found in this sequence - Ryan Bresler, May 30 2021

			11 is a term because its digit sum is 2 (prime) and its digit product is 1 (perfect square > 0).

David A. Corneth, Table of n, a(n) for n = 1..10000

Intersection of A028834 and A050626.
Subsequence of A052382.
A031974 is a subsequence of this sequence.

q:= n-> (l-> not 0 in l and isprime(add(i, i=l)) and
         issqr(mul(i, i=l)))(convert(n, base, 10)):
select(q, [$0..999])[];  # Alois P. Heinz, May 29 2021

from math import prod
from sympy import isprime, integer_nthroot
def ok(n):
  d = list(map(int, str(n)))
  return 0 not in d and isprime(sum(d)) and integer_nthroot(prod(d), 2)[1]
print(list(filter(ok, range(1000)))) # Michael S. Branicky, May 29 2021

A097708 Sum of prime-length repunits: Sum_{k=1..n} r(prime(k)), where r()=A002275.

Original entry on oeis.org

0, 11, 122, 11233, 1122344, 11112233455, 1122223344566, 11112233334455677, 1122223344445566788, 11112233334455556677899, 11111122223344445566667789010, 1122222233334455556677778900121

Offset: 0

Jason Earls, Aug 21 2004

			a(3)=11233 because 11 + 111 + 11111 = 11233.

Cf. A097709.

Partial sums of A031974.

a(n) = sum(k=1, n, (10^prime(k)-1)/9); \\ Michel Marcus, Jul 17 2022

A178388 Concatenation of the first n primes written in base 3.

Original entry on oeis.org

2, 210, 21012, 2101221, 2101221102, 2101221102111, 2101221102111122, 2101221102111122201, 2101221102111122201212, 21012211021111222012121002, 210122110211112220121210021011, 2101221102111122201212100210111101

Offset: 1

Jonathan Vos Post, May 26 2010

			a(4) = Concatenate[prime(1) base 3, prime(2) base 3, prime(3) base 3, prime(3) base 3] = Concatenate[2 base 3, 3 base 3, 5 base 3, 7 base 3] = Concatenate[2, 10, 12, 21] = 2101221.

Rémy Sigrist, Table of n, a(n) for n = 1..174

Cf. A000040, A001363, A004676, A007088, A007089, A019518, A031974, A100003, A154703.

Module[{nn=15,p3},p3=IntegerDigits[Prime[Range[nn]],3];Table[FromDigits[Flatten[ Take[p3,n]]],{n,nn}]] (* Harvey P. Dale, Aug 25 2022 *)

v = 0; for (n=1, 12, d = digits(prime(n), 3); v = v*10^#d + fromdigits(d); print1 (v ", ")) \\ Rémy Sigrist, Aug 07 2017

Edited by N. J. A. Sloane, Jul 02 2017

A090103 n written in base equal to the number of distinct prime factors of n and a(1)=0.

Original entry on oeis.org

0, 11, 111, 1111, 11111, 110, 1111111, 11111111, 111111111, 1010, 11111111111, 1100, 1111111111111, 1110, 1111, 1111111111111111, 11111111111111111, 10010, 1111111111111111111, 10100, 10101, 10110, 11111111111111111111111, 11000

Offset: 1

Labos Elemer, Dec 16 2003

All primes p are written in number-system of base one so rather long strings of 11...111 arise.

			a(6469693230) = 6469693230.
Symbol A to denote "10" first appears at n = 200560490130 = A002110(11).

Cf. A002110, A001221, A031974 (primes in base one).

tn[x_] := Fold[nd, 0, x]; Do[s=lf[n];If[Equal[s, 1], Print[tn[Table[1, {i, 1, n}]]]]; If[ !Equal[s, 1], Print[tn[IntegerDigits[n, s]]]], {n, 2, 211}]

n in base A001221(n).

A137287 a(n) is the number 2 written (prime(n)-1)/2 times followed by the digit 1; a(1)=2.

Original entry on oeis.org

2, 21, 221, 2221, 222221, 2222221, 222222221, 2222222221, 222222222221, 222222222222221, 2222222222222221, 2222222222222222221, 222222222222222222221, 2222222222222222222221, 222222222222222222222221, 222222222222222222222222221, 222222222222222222222222222221

Offset: 1

Ctibor O. Zizka, Apr 05 2008

Sum of digits of a(n) = prime(n).

			prime(7)=17; (17-1)/2=8; a(7) = 222222221, 8*"2" concatenated with "1".

Cf. A000040, A031974.

a(n) = if (n==1, 2, fromdigits(concat(vector((prime(n)-1)/2, k, 2), 1))); \\ Michel Marcus, Mar 15 2022

A007953(a(n)) = A000040(n). - Michel Marcus, Mar 15 2022

More terms from Michel Marcus, Mar 15 2022

A245359 Largest number k such that d_1^j + d_2^j + … + d_r^j is prime for all j = 1, 2, .. k, or 0 if no such k exists, where d_1, d_2, … d_r are the digits of n. a(n) = -1 if k is infinite.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, -1, 2, 0, 2, 0, 2, 0, 0, 0, 1, 2, 0, 2, 0, 2, 0, 0, 0, 1, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 2, 1, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 2

Offset: 1

Derek Orr, Jul 18 2014

If a(n) = K and reorder the digits of n to make a new number, n'. Thus, a(n') = K.

			1^1 + 2^1 = 3 is prime.
1^2 + 2^2 = 5 is prime.
1^3 + 2^3 = 9 is not prime.
So a(12) and a(21) = 2.

a(n) = for(k=1,10^3,d=digits(n);if(!ispseudoprime(sum(i=1,#d,d[i]^k)),return(k-1)));return(-1)
n=1;while(n<100,print1(a(n),", "); n++)

a(A031974(n)) = -1 for all n.

cp's OEIS Frontend

A093521 Runs of 1's of lengths 1, prime(1), prime(2), prime(3), ... separated by 0's.

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A179005 Indices of nonprime repunits: nonnegative numbers n such that 11...111 = (10^n - 1)/9 is composite.

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A259102 Composite repunits with a prime number of 1's.

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A323060 a(n) = R_(prime(n)#) / Product_{j=1..n} R_(prime(j)), where prime(n)# is the n-th primorial number A002110(n) and R_k = (10^k - 1)/9.

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A344825 Integers whose digit sum is prime and whose digit product is a perfect square > 0.

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A097708 Sum of prime-length repunits: Sum_{k=1..n} r(prime(k)), where r()=A002275.

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PARI

A178388 Concatenation of the first n primes written in base 3.

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Mathematica

PARI

Extensions

A090103 n written in base equal to the number of distinct prime factors of n and a(1)=0.

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