A052085 -id:A052085 - cp's OEIS frontend

A052081 Numbers k such that k^2 is a square whose decimal expansion's digits are grouped together.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 78, 79, 80, 82, 83

Offset: 1

Patrick De Geest, Jan 15 2000

			81 is not a term since in 81^2 = 6561, the two 6's are separated by the 5.
88 is a term since in 88^2 = 7744, the 7 and 4 are grouped.

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

Cf. A052082, A052085, A052086, A052083, A052084.

filter:= proc(n) local L,i,v,V;
  L:= convert(n^2,base,10);
  for i from 1 to nops(L) do
    v:= L[i];
    if not assigned(V[v]) then V[v]:= i
    elif V[v] < i-1 then return false
    else V[v]:= i
    fi
  od;
  true
end proc:
select(filter, [$0..100]); # Robert Israel, Nov 08 2023

A052082 Squares with at least one of the decimal expansion digits occurring separated.

Original entry on oeis.org

121, 484, 676, 1521, 1681, 2025, 3136, 3969, 4624, 5625, 5929, 6561, 8281, 8464, 9409, 10201, 10404, 10609, 10816, 11881, 12100, 12321, 14161, 14641, 14884, 15129, 15625, 16129, 16641, 17161, 17424, 19321, 19881, 21025, 21316, 24025

Offset: 1

Patrick De Geest, Jan 15 2000

			10609 is a term since the two digits 0 are separated by the 6.
7744 is not a term since all the digits 7 and 4 are grouped.

Cf. A052081, A052085, A052086, A052083, A052084.

A052083 a(n)^3 is a cube whose decimal expansion's digits are grouped together.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 27, 29, 30, 32, 35, 38, 40, 41, 42, 45, 46, 48, 49, 50, 53, 55, 59, 60, 61, 63, 66, 69, 71, 73, 75, 76, 80, 82, 84, 88, 90, 93, 96, 97, 100, 107, 113, 120, 124, 126, 129, 130, 131, 132, 135, 140

Offset: 1

Patrick De Geest, Jan 15 2000

			81 is not a term because in 81^3 = 531441 the two digits 1 are separated by the two 4's.
15 is a term because in 15^3 = 3375 the digits 3, 7 and 5 are grouped.

Cf. A052084, A052085, A052086, A052081, A052082.

Edited, and offset changed to 1, by Jon E. Schoenfield, Oct 17 2019

A052084 Cubes with at least one of the decimal digits occurring separated.

Original entry on oeis.org

343, 1331, 12167, 15625, 17576, 21952, 29791, 35937, 39304, 46656, 50653, 59319, 79507, 85184, 103823, 132651, 140608, 157464, 175616, 185193, 195112, 238328, 262144, 274625, 300763, 314432, 343000, 373248, 405224, 456533, 474552, 493039

Offset: 1

Patrick De Geest, Jan 15 2000

			The cube 300763 is a term since the digits 3 are separated by the substring 0076.
The cube 97336 is not a term since its 3-digits are grouped.

Cf. A052083, A052085, A052086, A052081, A052082.

Offset changed to 1 by Jon E. Schoenfield, Oct 17 2019

A052086 Primes with at least one of the decimal digits occurring separated.

Original entry on oeis.org

101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 1013, 1019, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1213, 1217, 1231, 1291, 1301, 1303, 1319, 1321, 1361, 1373, 1381, 1451, 1471, 1481, 1511, 1531, 1571, 1601, 1613

Offset: 1

Patrick De Geest, Jan 15 2000

The first 15 terms are palindromic.

			1471 is a term since the two digits 1 are separated by the substring 47.
Prime 277 is not a term since the only repeated digits (77) are grouped.

Cf. A052085, A052081, A052082, A052083, A052084.

Offset changed to 1 by Jon E. Schoenfield, Oct 17 2019

A334321 Non-palindromic primes.

Original entry on oeis.org

13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 103, 107, 109, 113, 127, 137, 139, 149, 157, 163, 167, 173, 179, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 317, 331, 337, 347, 349

Offset: 1

Bernard Schott, Apr 23 2020

Not the same as A052085, primes whose decimal digits are grouped together: 2, 3, 5, 7, 11 are not terms of this sequence, then the next difference occurs for prime 1013 that belongs to this sequence but not to A052085.

			97 is prime and is not a palindrome, hence 97 belongs to this sequence.

Equals A000040 \ A002385.

Intersection of A029742 and A000040.

Select[Range[350], PrimeQ[#] && !PalindromeQ[#] &] (* Amiram Eldar, Apr 23 2020 *)

isok(p) = if (isprime(p), my(d=digits(p)); d != Vecrev(d)); \\ Michel Marcus, Apr 23 2020

Formula : A087999(a(n)) = 1.

A235480 Primes whose base-3 representation is also the base-9 representation of a prime.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 23, 31, 37, 41, 43, 53, 67, 71, 73, 83, 89, 97, 103, 149, 157, 199, 239, 251, 257, 271, 277, 293, 307, 313, 331, 337, 359, 383, 397, 421, 431, 433, 499, 541, 557, 571, 587, 599, 601, 613, 631, 653, 659, 661, 683, 691, 709, 727, 751, 769, 823, 887, 911, 983, 1009, 1021, 1031, 1049, 1051, 1063, 1129, 1163, 1217

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.

Appears to be a subsequence of A015919, A045344, A052085, A064555 and A143578.

			5 = 12_3 and 12_9 = 11 are both prime, so 5 is a term.

Giovanni Resta, Table of n, a(n) for n = 1..10000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235265, A235473 - A235479, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime@ Range@ 500, PrimeQ@ FromDigits[ IntegerDigits[#, 3], 9] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=9,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: Code only valid for b > c.

cp's OEIS Frontend

A052081 Numbers k such that k^2 is a square whose decimal expansion's digits are grouped together.

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A052082 Squares with at least one of the decimal expansion digits occurring separated.

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A052083 a(n)^3 is a cube whose decimal expansion's digits are grouped together.

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A052084 Cubes with at least one of the decimal digits occurring separated.

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A052086 Primes with at least one of the decimal digits occurring separated.

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A334321 Non-palindromic primes.

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Mathematica

PARI

Formula

A235480 Primes whose base-3 representation is also the base-9 representation of a prime.

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Programs

Mathematica

PARI