A202265 -id:A202265 - cp's OEIS frontend

A062115 Numbers with no prime substring in their decimal expansion.

0, 1, 4, 6, 8, 9, 10, 14, 16, 18, 40, 44, 46, 48, 49, 60, 64, 66, 68, 69, 80, 81, 84, 86, 88, 90, 91, 94, 96, 98, 99, 100, 104, 106, 108, 140, 144, 146, 148, 160, 164, 166, 168, 169, 180, 184, 186, 188, 400, 404, 406, 408, 440, 444, 446, 448, 460, 464, 466

Offset: 1

Erich Friedman, Jun 28 2001

This is a 10-automatic sequence, a consequence of the finitude of A071062. - Charles R Greathouse IV, Sep 27 2011

Subsequence of A202259 (right-truncatable nonprimes). Supersequence of A202262 (composite numbers in which all substrings are composite), A202265 (nonprime numbers in which all substrings and reversal substrings are nonprimes). - Jaroslav Krizek, Jan 28 2012

			25 is not included because 5 is prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jeffrey Shallit, Minimal primes, Journal of Recreational Mathematics 30:2 (1999-2000), pp. 113-117.
Index entries for 10-automatic sequences.

Subsequence of A084984. [Arkadiusz Wesolowski, Jul 05 2011]
Cf. A071062.
Cf. A163753 (complement).

a062115 n = a062115_list !! (n-1)
a062115_list = filter ((== 0) . a039997) a084984_list
-- Reinhard Zumkeller, Jan 31 2012

from sympy import isprime
def ok(n):
    s = str(n)
    ss = (int(s[i:j]) for i in range(len(s)) for j in range(i+1, len(s)+1))
    return not any(isprime(k) for k in ss)
print([k for k in range(500) if ok(k)]) # Michael S. Branicky, May 02 2023

# faster for initial segment of sequence; uses ok, import above
from itertools import chain, count, islice, product
def agen(): # generator of terms
    yield from chain((0,), (int(t) for t in (f+"".join(r) for d in count(1) for f in "14689" for r in product("014689", repeat=d-1)) if ok(t)))
print(list(islice(agen(), 100))) # Michael S. Branicky, May 02 2023

A039997(a(n)) = 0. - Reinhard Zumkeller, Jul 16 2007
From Charles R Greathouse IV, Mar 23 2010: (Start)
a(n) = O(n^(log_4 10)) = O(n^1.661) because numbers containing only 0,4,6,8 are in this sequence.
a(n) = Omega(n^(log_13637 1000000)) = Omega(n^1.451) for similar reasons; this bound can be increased by considering longer sequences of digits. (End)

Offset corrected by Arkadiusz Wesolowski, Jul 27 2011

A202259 Right-truncatable nonprimes: every prefix is nonprime number.

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 40, 42, 44, 45, 46, 48, 49, 60, 62, 63, 64, 65, 66, 68, 69, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 120, 121, 122, 123, 124, 125, 126, 128, 129, 140, 141, 142

Offset: 1

Jaroslav Krizek, Dec 25 2011

Supersequence of A202260, A202265. - Barry Carter, Sep 16 2016

Robert Israel, Table of n, a(n) for n = 1..10000
Barry Carter, Table of n, a(n) for n=1..23498959 (bz2 compressed)
Barry Carter, Mathematica package

Cf. A012883 (right-truncatable noncomposites), A024770 (right-truncatable primes), A202260 (right-truncatable composites).

filter:= proc(n) option remember; not isprime(n) and procname(floor(n/10)) end proc:
for i from 0 to 9 do filter(i):= not isprime(i) od:
select(filter, [$0..1000]); # Robert Israel, Nov 01 2016

A202262 Composite numbers in which all substrings are composite.

Original entry on oeis.org

4, 6, 8, 9, 44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 94, 96, 98, 99, 444, 446, 448, 464, 466, 468, 469, 484, 486, 488, 494, 496, 498, 644, 646, 648, 649, 664, 666, 668, 669, 684, 686, 688, 694, 696, 698, 699, 844, 846, 848, 849, 864, 866, 868, 869, 884, 886

Offset: 1

Jaroslav Krizek, Dec 25 2011

Subsequence of A062115, A202260, A029581.
Supersequence of A202265.
This is a 10-automatic sequence, see A071062. - Charles R Greathouse IV, Jan 01 2012

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A202262
Index entries for 10-automatic sequences

Cf. A085823 (primes in which all substrings are primes), A068669 (noncomposite numbers in which all substrings are noncomposite), A062115 (nonprimes in which all substrings are nonprimes).

Cf. A010051.

sub[n_] := Block[{d = IntegerDigits[n]}, Union@ Reap[ Do[Sow@ FromDigits@ Take[d, {i, j}], {j, Length@ d}, {i, j}]][[2, 1]]]; Select[ Range@ 900, Union[{4, 6, 8, 9}, IntegerDigits[#]] == {4, 6, 8, 9} && AllTrue[sub[#], CompositeQ] &] (* Giovanni Resta, Dec 20 2019 *)

PARI
```
See Links section.
```

Data corrected by Reinhard Zumkeller, May 05 2012
Data corrected by Rémy Sigrist, Dec 19 2019
Incorrect Haskell program deleted by M. F. Hasler, Dec 20 2019

A202263 Primes in which all substrings and reversal substrings are primes.

Original entry on oeis.org

2, 3, 5, 7, 37, 73, 373

Offset: 1

Jaroslav Krizek, Dec 25 2011

Sequence is finite with 7 terms.

Subsequence of A085823, A068669, A024770, A012883.

			All substrings and reversal substrings of 373 are primes:3, 7, 37, 73, 373.

Cf. A202264 (noncomposite numbers in which all substrings and reversal substrings are noncomposite), A202265 (nonprimes in which all substrings and reversal substrings are nonprimes), A202266 (composite numbers in which all substrings and reversal substrings are composites).

A202266 Composite numbers in which all substrings and reversal substrings are composites.

Original entry on oeis.org

4, 6, 8, 9, 44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 94, 96, 99, 444, 446, 448, 464, 466, 468, 469, 484, 486, 488, 494, 496, 644, 646, 648, 649, 664, 666, 668, 669, 684, 686, 688, 694, 696, 699, 844, 846, 848, 849, 864, 866, 868, 869, 884, 886, 888, 946, 948

Offset: 1

Jaroslav Krizek, Dec 25 2011

Subsequence of A202265, A202262.

			All substrings and reversal substrings of 446 are composites: 4, 6, 44, 46, 64, 446, 644.

Cf. A202263 (primes in which all substrings and reversal substrings are primes), A202264 (noncomposite numbers in which all substrings and reversal substrings are noncomposite), A202265 (nonprimes in which all substrings and reversal substrings are nonprimes).

A202264 Noncomposite numbers in which all substrings and reversal substrings are noncomposites.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 113, 131, 311, 313, 373

Offset: 1

Jaroslav Krizek, Dec 25 2011

Sequence is finite with 17 terms.
Supersequence of A202263, A085823.
Subsequence of A068669, A012883, A024770, A012883.

			All substrings and reversal substrings of 311 are noncomposites: 1, 3, 11, 13, 31, 113, 311.

Cf. A202263 (primes in which all substrings and reversal substrings are primes), A202265 (nonprimes in which all substrings and reversal substrings are nonprimes), A202266 (composite numbers in which all substrings and reversal substrings are composites).

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A062115 Numbers with no prime substring in their decimal expansion.

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A202259 Right-truncatable nonprimes: every prefix is nonprime number.

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A202262 Composite numbers in which all substrings are composite.

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A202263 Primes in which all substrings and reversal substrings are primes.

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A202266 Composite numbers in which all substrings and reversal substrings are composites.

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A202264 Noncomposite numbers in which all substrings and reversal substrings are noncomposites.

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