A131835 -id:A131835 - cp's OEIS frontend

A062332 Primes starting and ending with 1.

11, 101, 131, 151, 181, 191, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231, 1291, 1301, 1321, 1361, 1381, 1451, 1471, 1481, 1511, 1531, 1571, 1601, 1621, 1721, 1741, 1801, 1811, 1831, 1861, 1871, 1901, 1931, 1951, 10061, 10091, 10111, 10141

Offset: 1

Amarnath Murthy, Jun 21 2001

Complement of A208261 (nonprime numbers with all divisors starting and ending with digit 1) with respect to A208262 (numbers with all divisors starting and ending with digit 1). - Jaroslav Krizek, Mar 04 2012

Intersection of A030430 and A045707. - Michel Marcus, Jun 08 2013

			102701 is a member as it is a prime and the first and the last digits are both 1.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A208259 (Numbers starting and ending with digit 1).

Cf. A010051, A017281, A131835, A000030.

a062332 n = a062332_list !! (n-1)
a062332_list = filter ((== 1) . a010051') a208259_list
-- Reinhard Zumkeller, Jul 16 2014

fl1Q[n_]:=Module[{idn=IntegerDigits[n]},First[idn]==Last[idn]==1]; Select[ Prime[Range[1300]],fl1Q] (* Harvey P. Dale, Apr 30 2012 *)

{ n=-1; t=log(10); forprime (p=2, 5*10^5, if ((p-10*(p\10)) == 1 && (p\10^(log(p)\t)) == 1, write("b062332.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 05 2009

A010051(a(n)) * A000030(a(n)) * (a(n) mod 10) = 1. - Reinhard Zumkeller, Jul 16 2014

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 29 2001

Missing term a(36)=1901 added by Harry J. Smith, Aug 05 2009

A208259 Numbers starting and ending with digit 1.

Original entry on oeis.org

1, 11, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 1001, 1011, 1021, 1031, 1041, 1051, 1061, 1071, 1081, 1091, 1101, 1111, 1121, 1131, 1141, 1151, 1161, 1171, 1181, 1191, 1201, 1211, 1221, 1231, 1241, 1251, 1261, 1271, 1281, 1291, 1301, 1311, 1321, 1331

Offset: 1

Jaroslav Krizek, Feb 24 2012

A000030(a(n)) = a(n) mod 10 = 1. - Reinhard Zumkeller, Jul 16 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Intersection of A017281 and A131835. Union of A062332 and A208260.
Supersequence of A208262 (numbers with all divisors starting and ending with digit 1).
Cf. A062332 (primes starting and ending with a digit 1), A208260 (nonprime numbers starting and ending with a digit 1).
Cf. A017281, A131835, A000030.

a208259 n = a208259_list !! (n-1)
a208259_list = 1 : map ((+ 1) . (* 10)) a131835_list
-- Reinhard Zumkeller, Jul 16 2014

Select[Range[2000], First[IntegerDigits[#]] == 1 && Last[IntegerDigits[#]] == 1 &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)

A098174 a(n) is the smallest e > 0 such that the initial digit of n^e = 1 in decimal representation.

Original entry on oeis.org

1, 4, 9, 2, 3, 4, 5, 8, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 3, 3, 3, 3, 3, 3, 5, 7, 9, 25, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 11, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10, 11, 12, 13, 14, 16, 18, 20, 23, 27, 32, 40, 53

Offset: 1

Reinhard Zumkeller, Aug 30 2004

A000030(n^a(n)) = 1; A098175(n) = n^a(n).
From Rémy Sigrist, Jun 25 2018: (Start)
We can extend this sequence to every Gaussian integers as follows:
- for any Gaussian integer z, let f(z) be the least k > 0 such that the initial decimal digit of the real part of z^k equals 1, or -1 if no such k exists,
- naturally f(n) = a(n) for any n > 0,
- apparently f(z) = -1 iff z = 0,
- see Links section for the color plot of f.
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Color plot of f(x + i*y) for x = -200..1000 and y = -200..500

Cf. A000030, A098175, A131835.

a(n, base=10) = my (nk=n); for (k=1, oo, my (z); logint(nk, base, &z); if (nk\z==1, return (k), nk*=n)) \\ Rémy Sigrist, Jun 21 2018

A175252 Numbers whose digit representation is equal to the digit representation of the initial terms of their sets of divisors in increasing order.

Original entry on oeis.org

1, 12, 124, 135, 1525, 13515, 124816, 12356910, 1243162124, 1525125625, 12478141928, 12510254150, 1234689111216, 1351553159265, 1597717414885, 12356910151830, 13791121336377, 123561015253050, 124510202550100, 135152575125375, 1236103206309618, 123456101215203060, 123569101518304590

Offset: 1

Jaroslav Krizek, Mar 14 2010

From Michel Marcus, Sep 25 2022: (Start)
The term 124 (2^2*31) corresponds to the term of A077352 that is a prime.
The terms 135 (5*3^3), 1525 (5^2*61) and 1525125625 (5^4*2440201) correspond to the terms of A077353 that are powers of primes. (End)
The term 1597717414885 = 5 * 977 * 1741 * 187861, found by David A. Corneth, is especially remarkable for the magnitude of its 2nd smallest prime factor (counting repetitions). - Peter Munn, Oct 10 2022
Define g(n) to be the LCM of the divisors of a(n) that appear in the digit string of a(n) as specified in the definition, and let f(n) = log(g(n))/log(a(n)). Are there are only finitely many n for which f(n) >= f(4) = log(15)/log(135) = 0.55206901...? - Peter Munn, Oct 19 2022
a(26) > 10^23 (there are no terms with 23 digits). - Tim Peters, Dec 21 2022

			a(1) = 1: d(1) = {1}.
a(2) = 12: d(12) = {1, 2, 3, 4, 6, 12}.
a(3) = 124: d(124) = {1, 2, 4, 31, 62, 124}.
a(4) = 135: d(135) = {1, 3, 5, 9, 15, 27, 45, 135}.

Tim Peters, Table of n, a(n) for n = 1..25
David A. Corneth and Michel Marcus, Some terms found in search for terms for A357692
David A. Corneth and Michel Marcus, Some terms <= 10^500

Cf. A037278, A357692. Subsequence of A131835.

Cf. A077352, A077353.

isok(k) = my(s=""); fordiv(k, d, s=concat(s, Str(d)); if (eval(s)==k, return(1)); if (eval(s)> k, return(0))); \\ Michel Marcus, Sep 22 2022

is(n, {u = 10^5}) = { my(oldu = u, s, d, fe); s = ""; u = min(u, n); fe = factor(n, u); d = divisors(fe); if(#fe~ > 0 && fe[#fe~, 1] > u, d = select(x -> x < fe[#fe~, 1], d); ); for(i = 1, #d, if(d[i] > u, return(is(n, 10*oldu)); ); s=concat(s, Str(d[i])); if(eval(s) == n, return(1)); if(eval(s) > n, return(0)); ); is(n, 10*oldu); } \\ David A. Corneth, Oct 12 2022, Nov 07 2022

from sympy import divisors
def ok(n):
    target, s = str(n), ""
    if target[0] != "1": return False
    for d in divisors(n):
        s += str(d)
        if len(s) >= len(target): return s == target
        elif not target.startswith(s): return False
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Sep 22 2022

a(9)-a(10) from Michel Marcus, Sep 22 2022
a(11)-a(12) from Michel Marcus, Oct 02 2022
a(13)-a(15) from Tim Peters, Oct 17 2022
a(16)-a(17) from Giovanni Resta, Oct 20 2022
a(18)-a(20) from Tim Peters, Oct 27 2022
a(21) from Tim Peters, Oct 30 2022
a(22)-a(23) from Tim Peters, Nov 04 2022

A000865 Numbers beginning with letter 'o' in English.

Original entry on oeis.org

1, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135

Offset: 1

N. J. A. Sloane

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A000852.

Subsequence of A131835.

Select[Range[1000],First[Characters[IntegerName[#,"Words"]]]=="o"&] (* James C. McMahon, Dec 11 2023 *)

A357299 a(n) is the number of divisors of n whose first digit equals the first digit of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3

Offset: 1

Bernard Schott, Sep 23 2022

Similar to A330348, but with last digit.
a(n) >= 1 because there is always a divisor that fits: n.
a(n) >= 2 for n>1 in A131835.

			The divisors of 26 that start in 2 are 2 and 26, so a(26) = 2.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000030, A131835, A330348, A356549, A357300.

f[n_] := IntegerDigits[n][[1]]; a[n_] := DivisorSum[n, 1 &, f[#] == f[n] &]; Array[a, 100] (* Amiram Eldar, Sep 23 2022 *)

a(n) = my(fd=digits(n)[1]); sumdiv(n, d, digits(d)[1] == fd); \\ Michel Marcus, Sep 23 2022

from sympy import divisors
def a(n): f = str(n)[0]; return sum(1 for d in divisors(n) if str(d)[0]==f)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Sep 23 2022

A206288 Nonprime numbers with all divisors starting with digit 1.

Original entry on oeis.org

1, 121, 143, 169, 187, 1111, 1133, 1177, 1199, 1243, 1313, 1331, 1339, 1391, 1397, 1417, 1441, 1469, 1507, 1529, 1573, 1639, 1651, 1661, 1703, 1717, 1727, 1751, 1781, 1793, 1807, 1819, 1837, 1853, 1859, 1903, 1919, 1921, 1937, 1957, 1963, 1969, 1991

Offset: 1

Jaroslav Krizek, Feb 12 2012

Subsequence of A206286, A131835.

Complement of A045707 (primes with first digit 1) with respect to A202287 (numbers with all divisors starting with digit 1).

			All divisors of 1859 (1, 11, 13, 169, 1859) start with digit 1.

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

Cf. A045707 (primes with first digit 1), A202287 (numbers with all divisors starting with digit 1).

fd1:= n -> n < 2*10^ilog10(n):
filter:= proc(n) not isprime(n) and andmap(fd1,numtheory:-divisors(n)) end proc:
select(filter, [1,seq(seq(i,i=10^d+1..2*10^d-1,2),d=1..3)]); # Robert Israel, Mar 13 2019

fQ[n_] := Module[{d = Divisors[n]}, Union[IntegerDigits[#][[1]] & /@ d] == {1}]; Select[Range[1991], ! PrimeQ[#] && fQ[#] &] (* T. D. Noe, Feb 13 2012 *)

A262390 Subsequence of terms starting with 1 in A262356.

Original entry on oeis.org

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 1000, 111, 112, 120, 113, 130, 114, 140, 115, 150, 116, 160, 117, 170, 118, 180, 119, 190, 1001, 121, 1100, 1002, 122, 123, 1003, 1101, 1010, 1004, 1005, 1006

Offset: 1

Reinhard Zumkeller, Sep 21 2015

A000030(a(n)) = 1;

A262356(A262393(n)) = a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A262356, A000030, A262393, A131835.

a262390 n = a262390_list !! (n-1)
a262390_list = filter ((== 1) . a000030) a262356_list

Typo in name corrected by Andrey Zabolotskiy, Sep 22 2017

A206286 Nonprime numbers starting with a digit 1.

Original entry on oeis.org

1, 10, 12, 14, 15, 16, 18, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150, 152, 153, 154, 155, 156, 158

Offset: 1

Jaroslav Krizek, Feb 12 2012

Complement of A045707 with respect to A131835. Supersequence of A206288.

Cf. A045707 (primes with first digit 1), A131835 (numbers starting with a digit 1).

Cf. A206288.

Select[Range[200], ! PrimeQ[#] && IntegerDigits[#][[1]] == 1 &] (* T. D. Noe, Feb 13 2012 *)

from sympy import primepi
def A206286(n):
    def f(x): return n-1+x+((m:=10**(l:=len(str(x))-1))-(k:=min((m<<1)-1,x))-primepi(m-1)+primepi(k))-sum((m:=10**i)+primepi(m-1)-primepi((m<<1)-1) for i in range(l))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 10 2024

A208260 Nonprime numbers starting and ending with digit 1.

Original entry on oeis.org

1, 111, 121, 141, 161, 171, 1001, 1011, 1041, 1071, 1081, 1101, 1111, 1121, 1131, 1141, 1161, 1191, 1211, 1221, 1241, 1251, 1261, 1271, 1281, 1311, 1331, 1341, 1351, 1371, 1391, 1401, 1411, 1421, 1431, 1441, 1461, 1491, 1501, 1521, 1541, 1551, 1561, 1581, 1591

Offset: 1

Jaroslav Krizek, Feb 24 2012

Complement of A062332 with respect to A208259. Supersequence of A208261 (nonprime numbers with all divisors starting and ending with digit 1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A208259 (number starting and ending with a number 1), A062332 (primes starting and ending with a number 1).

Cf. A010051, A017281, A131835, A000030.

a208260 n = a208260_list !! (n-1)
a208260_list = filter ((== 0) . a010051') a208259_list
-- Reinhard Zumkeller, Jul 16 2014

Select[Range[2000], ! PrimeQ[#] && First[IntegerDigits[#]] == 1 && Last[IntegerDigits[#]] == 1 &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)
Join[{1},Select[Range[2000],CompositeQ[#]&&NumberDigit[#,0] == NumberDigit[ #,IntegerLength[ #]-1]==1&]] (* Harvey P. Dale, Aug 01 2021 *)

(1 - A010051(a(n))) * A000030(a(n)) * (a(n) mod 10) = 1. - Reinhard Zumkeller, Jul 16 2014

cp's OEIS Frontend

A062332 Primes starting and ending with 1.

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A208259 Numbers starting and ending with digit 1.

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A098174 a(n) is the smallest e > 0 such that the initial digit of n^e = 1 in decimal representation.

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A175252 Numbers whose digit representation is equal to the digit representation of the initial terms of their sets of divisors in increasing order.

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A000865 Numbers beginning with letter 'o' in English.

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A357299 a(n) is the number of divisors of n whose first digit equals the first digit of n.

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A206288 Nonprime numbers with all divisors starting with digit 1.

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A262390 Subsequence of terms starting with 1 in A262356.

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