A202262 -id:A202262 - cp's OEIS frontend

A217657 Delete the initial digit in decimal representation of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Offset: 0

Reinhard Zumkeller, Oct 10 2012

When n - a(n)*10^[log_10 n] >= 10^[(log_10 n) - 1], where [] denotes floor, or when n < 100 and 10|n, n is the concatenation of A000030(n) and a(n) - corrected by Glen Whitney, Jul 01 2022

a(110) = 10 is the first term > 9. The sequence consists of 10 repetitions of 0 (n = 0..9), then 9 repetitions of {0, ..., 9} (n = 10..99), then 9 repetitions of {0, ..., 99} (n = 100..999), and so on. - M. F. Hasler, Oct 18 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A059995 (drop final digit of n), A000030 (initial digit of n), A202262.

a217657 n | n <= 9    = 0
          | otherwise = 10 * a217657 n' + m where (n', m) = divMod n 10

Array[FromDigits@ Rest@ IntegerDigits@ # &, 121, 0] (* Michael De Vlieger, Dec 22 2019 *)

apply( A217657(n)=n%10^logint(n+!n,10), [0..199]) \\ M. F. Hasler, Oct 18 2017, edited Dec 22 2019

def a(n): return 0 if n < 10 else int(str(n)[1:])
print([a(n) for n in range(121)]) # Michael S. Branicky, Jul 01 2022

a(n) = 0 if n <= 9, otherwise 10*a(floor(n/10)) + n mod 10.

a(n) = n mod 10^floor(log_10(n)), a(0) = 0. - M. F. Hasler, Oct 18 2017

Data extended to include the first terms larger than 9, by M. F. Hasler, Dec 22 2019

A062115 Numbers with no prime substring in their decimal expansion.

Original entry on oeis.org

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

A068669 Noncomposite numbers in which every substring is noncomposite.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 71, 73, 113, 131, 137, 173, 311, 313, 317, 373, 1373, 3137

Offset: 1

Amarnath Murthy, Mar 02 2002

It is easy to see that this sequence is complete - the only potential 5-digit candidate 31373 is not prime. - Tanya Khovanova, Dec 09 2006

			137 is a member as all the substrings, i.e. 1, 3, 7, 13, 37, 137, are noncomposite.
All substrings of 3137 are noncomposite numbers: 1, 3, 7, 13, 37, 137, 313, 3137. - _Jaroslav Krizek_, Dec 25 2011

Cf. A012884, A062115, A202262.

noncompositeQ[n_] := n == 1 || PrimeQ[n]; Reap[ Do[ id = IntegerDigits[n]; lid = Length[id]; test = And @@ noncompositeQ /@ FromDigits[#, 10]& /@ Flatten[ Table[ Take[id, {i, j}], {i, 1, lid}, {j, i, lid}], 1]; If[test, Sow[n]], {n, Join[{1}, Prime /@ Range[10000]]}]][[2, 1]](* Jean-François Alcover, May 09 2012 *)

1 added following a redefinition by Jaroslav Krizek. - R. J. Mathar, Jan 20 2012

A226099 Positive integers that yield a prime when their most significant (i.e., leftmost) decimal digit is removed.

Original entry on oeis.org

12, 13, 15, 17, 22, 23, 25, 27, 32, 33, 35, 37, 42, 43, 45, 47, 52, 53, 55, 57, 62, 63, 65, 67, 72, 73, 75, 77, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 111, 113, 117, 119, 123, 129, 131, 137, 141, 143, 147, 153, 159, 161, 167, 171, 173, 179, 183, 189, 197, 202, 203, 205, 207, 211, 213, 217

Offset: 1

Jonathan Vos Post, May 26 2013

Terms < 110 are the same as in A260181, numbers whose last digit is prime. - M. F. Hasler, Dec 20 2019

These are numbers with decimal expansion of the form k = xp where p is a prime and x is a single digit. Whether or not the number k itself is a prime is irrelevant. - N. J. A. Sloane, Dec 21 2019

			a(1) = 12 because when its most significant (or leftmost) digit (1) is removed, the remaining number 2 is prime, and it is the least such number.
102, 103, 105 and 107 are in the sequence because if the first digit is dropped, what is left is a 1-digit prime with a leading digit '0'.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A217657 (n without initial digit), A000030 (initial digit of n), A260181 (last digit is prime), A202262 (substrings are composite).

[k:k in [1..220]| IsPrime( k-Reverse(Intseq(k))[1]*10^(#Intseq(k)-1 ))]; // Marius A. Burtea, Dec 21 2019

Select[Range@ 300, PrimeQ@ FromDigits@ Rest@ IntegerDigits@ # &] (* Giovanni Resta, Dec 20 2019 *)

select( is(n)=isprime(n%10^logint(n+!n,10)), [0..222]) \\ M. F. Hasler, Dec 20 2019

From M. F. Hasler, Dec 21 2019: (Start)
n in A226099 (this sequence) <=> A217657(n) in A000040 (prime).
a(n) = a(n-4) + 10 for 4 < n < 41, i.e., 20 < a(n) < 110; a(n) = a(n-25) for 61 < n < 287, i.e., 200 < a(n) < 1100, etc. (End)

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).

A225619 Composite numbers that remain composite if any digit is deleted (zero and one are not considered prime).

Original entry on oeis.org

44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 94, 96, 98, 99, 104, 106, 108, 120, 122, 124, 125, 126, 128, 140, 142, 144, 145, 146, 148, 150, 152, 154, 155, 156, 158, 160, 162, 164, 165, 166, 168, 180, 182, 184, 185, 186, 188, 204, 206, 208, 210, 212, 214, 215

Offset: 1

Derek Orr, Aug 04 2013

These are sometimes called "deletable composites".

			142 is composite. If the 1 is deleted, 42 is composite. If the 4 is deleted, 12 is composite. If the 2 is deleted, 14 is composite. Therefore, 142 is included in this sequence.

T. D. Noe, Table of n, a(n) for n = 1..10000
Dave Radcliffe, Python program/code

Cf. A202262 (composite numbers in which all substrings are composite).

prime01Q[n_] := n == 0 || n == 1 || PrimeQ[n]; okQ[n_] := Module[{d = IntegerDigits[n]}, Not[Or @@ prime01Q /@ Table[FromDigits[Delete[d, i]], {i, Length[d]}]]]; Select[Range[215], ! PrimeQ[#] && okQ[#] &] (* T. D. Noe, Aug 14 2013 *)

A330597 Squarefree numbers in which all substrings are squarefree.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 15, 17, 21, 22, 23, 26, 31, 33, 35, 37, 51, 53, 55, 57, 61, 62, 65, 66, 67, 71, 73, 77, 111, 113, 115, 131, 133, 137, 151, 155, 157, 173, 177, 211, 213, 215, 217, 221, 222, 223, 226, 231, 233, 235, 237, 262, 265, 266, 267, 311, 313

Offset: 1

Rémy Sigrist, Dec 19 2019

This sequence is finite, the last term being a(1690) = 77377717.

			1, 3, 7, 13, 37 and 137 are all squarefree, hence 137 is a term.

Rémy Sigrist, Table of n, a(n) for n = 1..1690
Rémy Sigrist, PARI program for A330597

Cf. A005117, A085823 (prime variant), A202262 (composite variant).

sfQ[k_]:=With[{i=IntegerDigits[k]},AllTrue[Flatten[Table[ FromDigits/@ Partition[ i,n,1],{n,IntegerLength[k]}]],SquareFreeQ]]; Select[ Range[ 350],sfQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 09 2020 *)

PARI
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See Links section.
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A366826 Composite numbers whose proper substrings (of their decimal expansions) are all primes.

Original entry on oeis.org

4, 6, 8, 9, 22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 237, 537, 737

Offset: 1

Kalle Siukola, Oct 25 2023

There are no terms greater than 999 because the only three-digit prime whose substrings are all primes is 373 (see A085823) and prepending or appending any prime digit to it would create a different three-digit substring.

			237 is included because it is composite and 2, 3, 7, 23 and 37 are all primes.
4 is included because it is composite and has no proper substrings.

Subsequence of A002808.
Cf. A000040.
Cf. A061371, A062115, A202262.
Cf. A033274, A068669, A085823, A279366.

from itertools import combinations
from sympy import isprime
for n in range(2, 1000):
    if not isprime(n):
        properSubstrings = set(
            int(str(n)[start:end]) for (start, end)
            in combinations(range(len(str(n)) + 1), 2)
        ) - set((n,))
        if all(isprime(s) for s in properSubstrings):
            print(n, end=', ')

cp's OEIS Frontend

A217657 Delete the initial digit in decimal representation of n.

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A062115 Numbers with no prime substring in their decimal expansion.

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A068669 Noncomposite numbers in which every substring is noncomposite.

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A226099 Positive integers that yield a prime when their most significant (i.e., leftmost) decimal digit is removed.

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

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A225619 Composite numbers that remain composite if any digit is deleted (zero and one are not considered prime).

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Mathematica

A330597 Squarefree numbers in which all substrings are squarefree.

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