A130448 -id:A130448 - cp's OEIS frontend

A238334 Squares that do not contain a shorter substring that is a square.

0, 1, 4, 9, 25, 36, 576, 676, 5776, 27556, 33856, 538756, 586756, 665856, 682276, 763876, 767376, 853776, 872356, 2637376, 2775556, 2835856, 5635876, 6885376, 7376656, 22886656, 23755876, 23775376, 26275876, 26687556, 26873856, 32672656, 32878756, 37527876

Offset: 1

T. D. Noe, Mar 05 2014

Giovanni Resta, Table of n, a(n) for n = 1..10000
Michael S. Branicky, Python program
Wikipedia, Substring

Cf. A130448.

fQ[n_] := Module[{d = IntegerDigits[n], len, ds, sq}, len = Length[d]; ds = FromDigits /@ Flatten[Table[Partition[d, i, 1], {i, len - 1}], 1]; sq = Select[ds, IntegerQ[Sqrt[#]] &]; sq == {}]; Select[Range[0, 10000]^2, fQ]

# see link for faster version for producing b-file
from math import isqrt
from itertools import count, islice
def issquare(n): return isqrt(n)**2 == n
def cond(s):
    if len(s) == 1: return True
    if any(d in s for d in "0149"): return False
    ss = (s[i:i+l] for i in range(len(s)) for l in range(2, len(s)))
    return not any(issquare(int(u)) for u in ss)
def agen(): yield from (k**2 for k in count(0) if cond(str(k**2)))
print(list(islice(agen(), 34))) # Michael S. Branicky, Feb 23 2023

A238903 Integers k such that (k^2 + (k+1)^2) has no square proper substring.

Original entry on oeis.org

0, 1, 3, 6, 11, 18, 36, 43, 56, 61, 106, 136, 168, 181, 206, 411, 431, 511, 518, 536, 606, 613, 1056, 1068, 1388, 1631, 1636, 1668, 1686, 1693, 1806, 1813, 1956, 1981, 2068, 2081, 3363, 3411, 3418, 3631, 3693, 3763, 4106, 4331, 5136, 5318, 5411, 5606, 5868, 6011, 6036, 6236, 6238, 6256, 6431, 6456, 6581, 10568, 10668, 10813, 11581, 11588, 11806, 11888

Offset: 1

Zak Seidov, Mar 07 2014

Inspired by (and program used from) A238334.
Note that (m^2+(m+1)^2), for m>0, always ends with 5. Any other patterns?
From Robert Israel, Dec 09 2024: (Start)
The last two digits of k^2 + (k+1)^2 (if more than 2 digits) are 01, 05, 13, 21, 25, 41, 45, 61, 65, 81, or 85.  The only ones of these that don't contain the squares 0, 1, 4, or 25 are 65 and 85, so all terms k > 3 of this sequence have k^2 + (k+1)^2 ending in 65 or 85. (End)

			1^2 + 2^2 = 5, 3^2 + 4^2 = 25, 6^2 + 7^2 = 85.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A001844, A130448, A238334, A238335.

filter:= proc(m) local n,i,j,S;
  n:= m^2 + (m+1)^2;
S:= {seq(seq(floor((n mod 10^i)/10^j),j=0..i-1),i=1 .. ilog10(n)+1)} minus {n};
  not ormap(issqr,S);
end proc:
select(filter, [$0..20000]); # Robert Israel, Dec 09 2024

A347819 Minimal elements for the base-10 representations of the primes greater than 10.

Original entry on oeis.org

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501

Offset: 1

Eric Chen, Sep 16 2021

Sequence is finite with 77 terms, the largest being 5*10^30 + 27 (which can be written 5(0_28)27, where 0_28 means the string of 28 0's). See text file for proof (this file also has proofs for bases 2, 3, 4, 5, 6, 8, 12).
Minimal elements for the base b representations of the primes > b for other bases b: (see the text file for 9 <= b <= 16) (all written in base b)
b=2: {11}
b=3: {12, 21, 111}
b=4: {11, 13, 23, 31, 221}
b=5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 10^95 + 13}
b=6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}
b=7: {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} (conjectured, not proven)
b=8: {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, (10^220-1)/9*40 + 7}.
Equivalently: primes > 10 such that no proper substring (i.e., deleting any positive number of digits) is again a prime > 10. - M. F. Hasler, May 03 2022

			277 is in this sequence because none of 2, 7, 27, 77 is a prime > 10.
857 is in this sequence because none of 8, 5, 7, 85, 87, 57 is a prime > 10.
991 is in this sequence because none of 9, 1, 99, 91 is a prime > 10.
149 is not in this sequence because 19 is subsequence of 149 and 19 is a prime > 10.
389 is not in this sequence because 89 is subsequence of 389 and 89 is a prime > 10.
439 is not in this sequence because 43 is subsequence of 439 and 43 is a prime > 10.

Jinyuan Wang, Table of n, a(n) for n = 1..77
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, Experimental Mathematics 25 (3) (2016), pp. 321-331. DOI:10.1080/10586458.2015.1064048
Eric Chen, Minimal elements for the base b representations of the primes which are > b
Eric Chen, Data for minimal elements for the base b representations of the primes which are > b for 2 <= b <= 16
Eric Chen, Proof for that the data for bases 2, 3, 4, 5, 6, 8, 10, 12 is complete
Eric Chen, Known values or lower bounds of the largest minimal element for the base b representations of the primes which are > b for 2 <= b <= 36
Eric Chen, Condensed table for the status of the minimal element for the base b representations of the primes which are > b for 2 <= b <= 16
Prime Glossary, Minimal prime
J. Shallit, Minimal primes, J. Recreational Math., 30:2 (2000) pp. 113-117.

Cf. A071062 (primes > 10 are not required).

Minimal sets for other sets: A071070 (for composites), A071071 (powers of 2), A071072 (multiples of 4), A071073 (multiples of 3), A111055 (primes of the form 4*n+1), A111056 (primes of the form 4*n+3), A114835 (palindromic primes), A130448 (minimal set of squares).

a(n, k, b)=v=[]; for(r=1, length(digits(n, b)), if(r+length(digits(k, 2))-length(digits(n, b))>0 && digits(k, 2)[r+length(digits(k, 2))-length(digits(n, b))]==1, v=concat(v, digits(n, b)[r]))); fromdigits(v, b)
iss(n, b)=for(k=1, 2^length(digits(n, b))-2, if(ispseudoprime(a(n, k, b)) && a(n, k, b)>b, return(0))); 1
is(n, b=10)=isprime(n) && n>b && iss(n, b) \\ Test whether n is a minimal element for the base b representations of the primes > b. Default value b = 10 for this sequence.
select( {is_A347819(n,b=10)=for(L=2, #n=digits(n,b), forvec(d=vector(L, i, [1,#n]), n[d[1]]&& isprime(fromdigits(vecextract(n,d),b))&& return(L==#n), 2))}, [1..8888]) \\ Better select among primes([1,N]). - M. F. Hasler, May 03 2022

Edited by M. F. Hasler, May 03 2022

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A131599 Square roots of the numbers in A130448.

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A238334 Squares that do not contain a shorter substring that is a square.

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A238903 Integers k such that (k^2 + (k+1)^2) has no square proper substring.

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A347819 Minimal elements for the base-10 representations of the primes greater than 10.

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