A045707 -id:A045707 - 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

A065680 Number of primes <= prime(n) which begin with a 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25

Offset: 1

Reinhard Zumkeller, Nov 13 2001

Considering the frequency of all decimal digits in leading position of prime numbers (A065681 - A065687), we cannot apply Benford's Law. But we observe at 10^e - levels that the frequency for 0 to 9 decreases monotonically, at least in the small range until 10^7.
The "begins with 9" sequence is too dull to include. - N. J. A. Sloane
Note that the primes do not satisfy Benford's law (see A000040). - N. J. A. Sloane, Feb 08 2017

			13 is the second prime beginning with 1: A000040(6) = 13, therefore a(6) = 2. a(664579) = 80020 (A000040(664579) = 9999991 is the largest prime < 10^7).

Harry J. Smith, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Benford's Law
Index entries for sequences related to Benford's law

Cf. A065681, A065682, A065683, A065684, A065685, A065686, A065687, A000040.

For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509.

Accumulate[If[First[IntegerDigits[#]]==1,1,0]&/@Prime[Range[80]]] (* Harvey P. Dale, Jan 22 2013 *)

lista(n) = { my(a=[p\10^logint(p,10)==1 | p<-primes(n)]); for(i=2, #a, a[i]+=a[i-1]); a} \\ Harry J. Smith, Oct 26 2009

A206287 Numbers with all divisors starting with digit 1.

Original entry on oeis.org

1, 11, 13, 17, 19, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1111

Offset: 1

Jaroslav Krizek, Feb 12 2012

Equivalently, integers m with all divisors starting with the same first digit of m; in fact, as 1 divides all the integers, this digit is necessarily 1; also, for these terms m: A357299(m) = A000005(m). - Bernard Schott, Sep 25 2022

			All divisors of 187 (1, 11, 17, 187) start with digit 1.

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

Disjoint union of A045707 and A206288.
Cf. A000005, A355592, A357299, A357300.
Cf. A004615 (with last digit)

filter:= proc(n) andmap(t -> floor(t/10^ilog10(t)) = 1, numtheory:-divisors(n)) end proc:
select(filter, [seq($10^d .. 2*10^d-1, d=0..3)]); # Robert Israel, Dec 25 2024

fQ[n_] := Module[{d = Divisors[n]}, Union[IntegerDigits[#][[1]] & /@ d] == {1}]; Select[Range[1111], fQ] (* T. D. Noe, Feb 13 2012 *)

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 *)

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

A306661 Numbers with chained divisors: Numbers k with divisors such that the last digit of every divisor is the same as the first digit of the next divisor.

Original entry on oeis.org

1, 11, 13, 17, 19, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1111

Offset: 1

Giorgos Kalogeropoulos, May 05 2019

All prime numbers whose first digit is 1 (A045707) have this property.

The first composite numbers having this property are A307858: 121, 1111, 1207, ...

			14641 is such a number because its divisors are 1, 11, 121, 1331, 14641.
Also, 90043 is in the sequence because its divisors are 1, 127, 709, 90043 and the last digit of every divisor is the first digit of the next one.

A307858 and A045707 are subsequences.

Select[Range@1500,And@@(Last@#[[1]]==First@#[[2]]&/@Partition[IntegerDigits/@Divisors@#,2,1])&]

cp's OEIS Frontend

A062332 Primes starting and ending with 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A065680 Number of primes <= prime(n) which begin with a 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A206287 Numbers with all divisors starting with digit 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A206288 Nonprime numbers with all divisors starting with digit 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A206286 Nonprime numbers starting with a digit 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

A306661 Numbers with chained divisors: Numbers k with divisors such that the last digit of every divisor is the same as the first digit of the next divisor.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica