Anton Deynega - cp's OEIS frontend

A321281 a(n) is the number of primes of the form p*10^n + q, where p, q are the digits from 1 to 9.

21, 15, 13, 8, 9, 5, 3, 8, 8, 2, 2, 3, 2, 0, 2, 2, 2, 3, 2, 5, 1, 4, 0, 3, 1, 1, 1, 2, 2, 0, 2, 0, 0, 0, 2, 2, 1, 1, 3, 1, 0, 2, 0, 0, 3, 2, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 2, 1, 0, 2, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1

Offset: 1

Anton Deynega, Nov 10 2018

			a(6) = 5 because there are five primes of the form p*10^6 + q where p, q are the digits from 1 to 9: 1000003, 2000003, 7000003, 7000009, 8000009.

Robert Israel, Table of n, a(n) for n = 1..3000
Sabin Tabirca and Kieran Reynolds, Lacunary Prime Numbers.

Cf. A000040.

f:= n -> nops(select(isprime,[seq(seq(p*10^n+q,p=1..9),q=[1,3,7, 9])])):
map(f, [$1..100]); # Robert Israel, Nov 14 2018

a[n_]:=(c=0; Do[ Do[ If[PrimeQ[i*10^n+j], c++], {i,1,9}], {j,1,9,2}]; c); Array[a, 20] (* Amiram Eldar, Nov 14 2018 *)

a(n)={my(t=10^n); sum(i=1, 9, sum(j=1, 5, isprime(2*j-1+i*t)))} \\ Andrew Howroyd, Nov 10 2018

a(16)-a(86) from Andrew Howroyd, Nov 10 2018

A320292 Zerofree numbers k such that the product (m+n)*p, where m,n are the first and the last digits of k, and p is the number which is the part of k between m and n, is a divisor of k.

Original entry on oeis.org

126, 162, 212, 216, 234, 413, 432, 672, 864, 891, 918, 2112, 2132, 2176, 2691, 2772, 2871, 2912, 3168, 4144, 4199, 4224, 4455, 5184, 6336, 8448, 21372, 21771, 23391, 43673, 53768, 55328, 64116, 171432, 228177, 316764, 412272, 515484, 594342, 638715, 663832, 824544, 1588248, 5136248, 7222932

Offset: 1

Anton Deynega, Oct 09 2018

This sequence is infinite since it contains all the terms of the form 6*(10^(6*t)+20)/35 and 33*(10^(6*t)*75+2)/7 for t > 0. The first pattern corresponds to terms 171432, 171428571432, 171428571428571432, ..., the second to terms 353571438, 353571428571438, 353571428571428571438,... . - Giovanni Resta, Oct 10 2018

			234 is divisible by 3*(2+4).
4199 is divisible by 19*(4+9).
7222932 is divisible by 22293*(7+2).

Intersection of A052382 and A320121.

Select[Range[100, 10^6], And[FreeQ[#2, 0], Mod[#1, If[#2 == 0, #1 - 1, #2] & @@ {#1, (First@ #2 + Last@ #2) FromDigits@ Most@ Rest@ #2}] == 0] & @@ {#, IntegerDigits@ #} &] (* Michael De Vlieger, Oct 11 2018 *)

isok(n) = {d = digits(n); if ((#d >= 3) && vecmin(d), x = d[1]; y = d[#d]; w = vector(#d-2, k, d[k+1]); z = fromdigits(w); if (z, return (!(n % (z*(x+y))))); ); return (0); } \\ Michel Marcus, Oct 10 2018

A320121 Numbers k such that the product (m+n)*p, where m,n are the first and the last digits of k, and p is the number which is the part of k between m and n, is a divisor of k.

Original entry on oeis.org

110, 120, 126, 140, 150, 162, 210, 212, 216, 220, 234, 240, 250, 360, 413, 432, 480, 510, 520, 540, 550, 630, 672, 864, 891, 918, 1010, 1020, 1040, 1050, 1062, 1080, 1100, 1200, 1250, 1400, 1500, 2010, 2012, 2016, 2020, 2034, 2040, 2050, 2072, 2079, 2080, 2100, 2112, 2132, 2176, 2200, 2250, 2400, 2500

Offset: 1

Anton Deynega, Oct 06 2018

The sequence is infinite if one considers numbers like 10........010 (with N zeros between ones, N = 0, 1, 2, ...). The problem of looking for odd terms and zerofree terms remains. Still unclear is whether the sequence contains infinitely many zerofree terms. The proportions between numbers of N-digit numbers included in the sequence need to be explored.

			234 is divisible by 3*(2+4).
1020 is divisible by 2*(1+0).
1062 is divisible by 6*(1+2).
1250 is divisible by 25*(1+0).

Cf. A320292 (zeroless terms).

Select[Range[100, 2500], Mod[#1, If[#2 == 0, #1 - 1, #2] & @@ {#1, (First@ #2 + Last@ #2) FromDigits@ Most@ Rest@ #2}] == 0 & @@ {#, IntegerDigits@ #} &] (* Michael De Vlieger, Oct 11 2018 *)

isok(n) = {d = digits(n); if (#d >= 3, x = d[1]; y = d[#d]; w = vector(#d-2, k, d[k+1]); z = fromdigits(w); if (z, return (!(n % (z*(x+y)))));); return (0);} \\ Michel Marcus, Oct 07 2018

A319858 a(n) is the number of values of m in the interval [2,10] such that the base-m expansion of n, interpreted as a base-10 number, yields a prime.

Original entry on oeis.org

0, 8, 8, 1, 7, 1, 5, 2, 2, 3, 3, 1, 3, 0, 2, 2, 3, 0, 4, 0, 2, 2, 4, 0, 2, 1, 1, 2, 3, 0, 3, 0, 3, 2, 1, 0, 4, 1, 3, 0, 2, 0, 5, 0, 1, 1, 3, 1, 1, 1, 0, 1, 2, 0, 4, 1, 1, 1, 3, 0, 3, 0, 1, 1, 1, 1, 4, 1, 0, 0, 5, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 1, 3, 1, 1, 0, 2, 1, 3, 0, 2, 0, 2, 2, 1, 0, 3, 0, 0, 0

Offset: 1

Anton Deynega, Sep 29 2018

			a(31)=3 because 31 yields primes for 3 bases in [2,10]:  31 = 11111_2 = 1011_3 = 133_4 = 111_5 = 51_6 = 43_7 = 37_8 = 34_9 = 31_10, and of the decimal numbers 11111, 1011, 133, 111, 51, 43, 37, 34, and 31, the 3 primes are 43, 37, and 31.

Cf. A052026 (where a(n) is 0).

Array[Count[FromDigits /@ IntegerDigits[#, Range[2, 10]], ?PrimeQ] &, 105] (* _Michael De Vlieger, Oct 11 2018 *)

a(n) = sum(b=2, 10, isprime(fromdigits(digits(n, b), 10))); \\ Michel Marcus, Sep 30 2018

A319809 Numbers k that are equal to 2*k when expressed in base 9 and then interpreted in base 10.

Original entry on oeis.org

7794416, 7816758, 8131871, 8132006, 8152638, 8152650

Offset: 1

Anton Deynega, Sep 28 2018

			8132006 expressed in base 9 is equal to 16264012; 16264012 = 2*8132006.

Cf. A139285.

Select[Range[10^6, 10^7], FromDigits@ IntegerDigits[#, 9] == 2 # &] (* Michael De Vlieger, Oct 11 2018 *)

isok(n) = fromdigits(digits(n,9), 10) == 2*n; \\ Michel Marcus, Sep 28 2018

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User: Anton Deynega

A321281 a(n) is the number of primes of the form p*10^n + q, where p, q are the digits from 1 to 9.

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A320292 Zerofree numbers k such that the product (m+n)*p, where m,n are the first and the last digits of k, and p is the number which is the part of k between m and n, is a divisor of k.

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A320121 Numbers k such that the product (m+n)*p, where m,n are the first and the last digits of k, and p is the number which is the part of k between m and n, is a divisor of k.

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A319858 a(n) is the number of values of m in the interval [2,10] such that the base-m expansion of n, interpreted as a base-10 number, yields a prime.

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A319809 Numbers k that are equal to 2*k when expressed in base 9 and then interpreted in base 10.

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