A182040 -id:A182040 - cp's OEIS frontend

A182092 Primes in A182040.

100019, 100103, 100109, 100151, 100333, 100411, 100501, 100511, 100801, 100811, 100999, 101009, 101021, 101051, 101081, 101107, 101221, 101333, 101501, 101701, 101771, 101999, 102001, 102101, 102121, 103001, 106661, 107077, 107101, 107171, 107717, 108011

Offset: 1

Jonathan Vos Post, Apr 11 2012

Primes whose decimal representation consists of three distinct digits, one appearing once, one appearing twice, and one appearing three times.
There are 3640 terms.
The subsequence of emirps begins 100411, since 114001  is prime; 100511, since 115001 is prime; 100999, since 999101 is prime;  101333, since 333101 is prime; 101701, since 107101 is prime; 101999, since 999101 is prime.

Jason Kimberley, Table of n, a(n) for n = 1..3640 (complete sequence)

Cf. A000040, A182040, A182051 (primes with a majority of one digit).

A182092 := func;
[n:n in[10^5..10^6]|A182092(n)]; // Jason Kimberley, Nov 02 2012

A000040 INTERSECTION A182040.

A182098 Integers n such that n^2 is a term in A182040.

Original entry on oeis.org

334, 335, 338, 359, 369, 380, 408, 450, 461, 470, 475, 482, 485, 505, 511, 515, 516, 526, 527, 547, 548, 638, 665, 667, 668, 688, 707, 725, 744, 746, 765, 796, 804, 813, 816, 834, 876, 901, 964, 970, 997

Offset: 1

Zak Seidov, Apr 11 2012

There are exactly 41 terms, and primes are: 359, 461, 547, 997.

			334^2 = 111556 = A182040(565),
335^2 = 112225 = A182040(716),
338^2 = 114244 = A182040(916).

A218556 Numbers with d distinct decimal digits (d=1,...,10) such that for each k=1,...,d, some digit occurs exactly k times.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 101, 110, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 133, 141, 144, 151, 155, 161, 166, 171, 177, 181, 188, 191, 199, 200, 202, 211, 212, 220, 221, 223, 224, 225, 226, 227, 228, 229, 232, 233, 242, 244, 252, 255, 262, 266, 272, 277, 282, 288, 292, 299, 300, 303, 311, 313, 322, 323, 330

Offset: 1

M. F. Hasler, Nov 02 2012

For each of the terms, the number of digits is a triangular number A000217.
The number of terms with d = 1,2,3,... different digits is 10, 243, 38880, ... = A218566(10,d) (+ 1 for d=1, accounting for the initial 0).
The sequence is finite, it has N = 1 + sum_{i=1..10} A218566(10,i) = 9083370609101493843078695864582213215764827510991133 terms. The last term is a(N) = 9999999999888888888777777776666666555555444443333222110 (ten "9"s, nine "8"s, ..., one "0").

			The terms a(1)=0 through a(10)=9 have exactly 1 digit occurring exactly once.
The terms a(11)=100 through a(253)=998, listed in A210666, have one digit occurring once and a second, different digit occurring exactly twice.
The terms a(254)=100012 through a(39133)=999887 are listed in A182040.
For d=4, we have the (1+2+3+4 =) 10-digit terms a(39134)=1000011223 through 9999888776 with 4 different digits which occur with frequencies 1,2,3 and 4.

Cf. A071925, A181986, A182040, A182092.

{my(T(n)=n*(n+1)\2); print1(0); for(i=1,2, s=vector(i+1,j,j-1); for(n=10^(T(i)-1),10^T(i)-1,i !=#Set(digits(n)) & next; c=vector(10); for(j=1,#d=digits(n),c[d[j]+1]++); vecsort(c,,8)==s & print1(","n)))}

is_A218556(n)={ my(c=vector(10)); for(i=1,#n=digits(n),c[n[i]+1]++); #(c=vecsort(c,,8))==1+c[#c] && 2*#n==c[#c]*#c }

A218560 Numbers with d distinct ternary digits (d=1,2,3) such that for each k=1,...,d, some digit occurs exactly k times.

Original entry on oeis.org

0, 1, 2, 9, 10, 12, 14, 16, 17, 18, 20, 22, 23, 24, 25, 248, 250, 251, 254, 257, 258, 259, 262, 263, 264, 265, 267, 269, 272, 275, 276, 277, 281, 285, 287, 288, 289, 291, 293, 295, 296, 298, 299, 300, 301, 303, 305, 306, 307, 309, 311, 313, 314, 315, 317, 319, 320, 321, 322, 326, 329, 330, 331, 335

Offset: 1

M. F. Hasler, Nov 02 2012

For each of the terms, the number of ternary (= base 3) digits is a triangular number A000217.
The base 2 analog would have only the 5 terms 0,1,4,5,6. See A218556 for the base 10 analog.
The sequence A167819 is a subsequence containing exactly all terms >= 9.
The sequence is finite, with 255=3+12+240 (= 1 + sum of the 3rd row of A218566) terms.

			The terms a(1)=0 through a(3)=2 have exactly 1 digit occurring exactly once.
The terms a(4)=9=100[3] through a(15)=25=221[3], have one ternary digit occurring once and a second, different digit occurring exactly twice.
The terms a(16)=248=100012[3] through a(255)=714=222110[3] contain each ternary digit at least once. There are no other terms in this sequence.

M. F. Hasler, Table of n, a(n) for n = 1..255 (full sequence).

Cf. A218559, A182040, A218556.

{my(T(n)=n*(n+1)\2); print1(0); for(i=1,3, s=vector(i+1,j,j-1); for(n=3^(T(i)-1),3^T(i)-1,i !=#Set(digits(n,3)) & next; c=vector(4); for(j=1,#d=digits(n,3),c[d[j]+1]++); vecsort(c,,8)==s & print1(","n)))}

is_A218560(n,b=3)={ my(c=vector(b+1)); for(i=1,#n=digits(n,b),c[n[i]+1]++); #(c=vecsort(c,,8))==1+c[#c] && 2*#n==c[#c]*#c }

A218559 Sum_{i=0..n-1} i(n^(i+1)-1)/(n-1)n^(i(i+1)/2).

Original entry on oeis.org

0, 6, 714, 1047188, 30515132780, 21936856591278330, 459986443452971306412268, 324518550895166392891543292552264, 8727963565271662417355532872177263437534624, 9999999999888888888777777776666666555555444443333222110

Offset: 1

M. F. Hasler, Nov 02 2012

Largest number which can be written in base n using d+1 times the digit d, d=0,...,n-1. (Or: such that for each k=1,...,n, some digit is used exactly k times.)

			Written in the respective bases, a(2) = 6 = 110[2], a(3) = 714 = 222110[3], a(4) = 1047188 = 33322110[4], etc.

Cf. A218556, A210666, A182040; A167819, A031948.

a(b)=sum(i=1,b-1,(b^(i+1)-1)\(b-1)*b^(i*(i+1)\2)*i)

cp's OEIS Frontend

A182092 Primes in A182040.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

A182098 Integers n such that n^2 is a term in A182040.

Views

Author

Keywords

Comments

Examples

A218556 Numbers with d distinct decimal digits (d=1,...,10) such that for each k=1,...,d, some digit occurs exactly k times.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

A218560 Numbers with d distinct ternary digits (d=1,2,3) such that for each k=1,...,d, some digit occurs exactly k times.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A218559 Sum_{i=0..n-1} i*(n^(i+1)-1)/(n-1)*n^(i(i+1)/2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A218559 Sum_{i=0..n-1} i(n^(i+1)-1)/(n-1)n^(i(i+1)/2).