A038615 -id:A038615 - cp's OEIS frontend

A155006 Primes p such that (p-2)(p+2)-+2p are primes.

5, 7, 13, 23, 37, 43, 73, 167, 233, 263, 433, 557, 587, 593, 607, 727, 857, 1153, 1597, 1627, 1753, 2143, 2663, 2713, 3433, 3607, 3863, 3947, 4027, 4363, 4423, 4673, 5147, 5477, 5623, 5807, 5903, 6277, 7237, 7333, 7577, 8287, 8647, 8837, 8887, 9043, 10067

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

3*7-10=11, 3*7+10=31,...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939

lst={};Do[p=Prime[n];If[PrimeQ[(p-2)*(p+2)-2*p]&&PrimeQ[(p-2)*(p+2)+2*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]],AllTrue[(#-2)(#+2)+{2#,-2#},PrimeQ]&] (* Harvey P. Dale, Jan 01 2025 *)

A329760 Primes without {2, 7} as digits.

Original entry on oeis.org

3, 5, 11, 13, 19, 31, 41, 43, 53, 59, 61, 83, 89, 101, 103, 109, 113, 131, 139, 149, 151, 163, 181, 191, 193, 199, 311, 313, 331, 349, 353, 359, 383, 389, 401, 409, 419, 431, 433, 439, 443, 449, 461, 463, 491, 499, 503, 509, 541, 563, 569, 593, 599, 601, 613

Offset: 1

Alois P. Heinz, Nov 20 2019

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Marianne Freiberger, Primes without 7s.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)
Index to entries for primes with digits in a given set

Cf. A000040, A020459, A038604, A038615.

[p: p in PrimesUpTo(700) | not 2 in Intseq(p) and not 7 in Intseq(p) ]; // Vincenzo Librandi, Jan 02 2020

Select[Prime[Range[120]], DigitCount[#, 10, 2]==0 && DigitCount[#, 10, 7]==0 &] (* Vincenzo Librandi, Jan 02 2020 *)

{ A038604 } intersect { A038615 }.

A076805 Triskaidekaphobic or 13-free primes: primes that do not contain the number 13.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 127, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307

Offset: 1

Cino Hilliard, Nov 18 2002

			The PARI program will mask out a sequence containing k or mask in a sequence containing k. The program is limited to primes < 400000000.
The PARI program will generate the following for input as shown: kprimes(2,100,7,0) = 2 3 5 11 13 19 23 29 31 41 43 53 59 61 83 89; kprimes(2,1000,13,1) = 13 113 131 137 139 313 613; kprimes(300000,4000000,314159,1) = 314159 3314159 5314159

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triskaidekaphobia
Wikipedia, Triskaidekaphobia

A generalization of the examples A038603, A038615 etc. for k = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 by Vasiliy Danilov.
Cf. A038615 A038603.
Complement of A166573 with respect to A000040; cf. A011760

import Data.List (isInfixOf)
a076805 n = a076805_list !! (n-1)
a076805_list = filter (not . ("13" `isInfixOf`) . show) a000040_list
-- Reinhard Zumkeller, Nov 09 2011

Select[Prime[Range[90]],!MemberQ[Partition[IntegerDigits[#],2,1],{1,3}]&] (* Harvey P. Dale, Mar 25 2012 *)
Select[Prime[Range[100]],SequenceCount[IntegerDigits[#],{1,3}]==0&] (* Harvey P. Dale, Aug 11 2023 *)

/* k primes - kprimes.gp. Primes containing or not containing digit k=1,2,3...9,10,11... PARI does not have a good string manipulation capability. This program circumvents that by using the % modulo and floor operators. Also commented out is a log implementation which is slower than string apps. The program either masks out prime numbers k or masks them in.*/

log10(z) = if(z>0,floor(log(z)/log(10))+1,1);\\integer function for log(z) base 10 + 1
{ kprimes(n1,n2,k,t) = \\n1,n2=range,k=mask,t=0 mask out t=1 mask in
ct=0; pct=0; forprime(p=n1,n2,x=p; f=0;\\x=temp variable to diminish p
ln = length(Str(p));\\get length of the prime p using strings
lk = length(Str(k));\\get length of mask integer k using strings
ln = log10(p);\\get length of the prime p using logs
lk = log10(k);\\get length of mask integer k using strings
r = 10^lk; \\set the remainder length = length of k
for(j=1,ln-lk+1, \\permute through the digits
d = x % r;\\get lk digits
if(d==k,f=1; break);\\break for loop if match and set flag
x = floor(x/10);\\diminish x for next test for k
); if(f==t,print1(p" "); ct+=1);\\if no k string of digits,print
); print(); print(ct); }

hasNo13(n)=n=digits(n); for(i=2,#n, if(n[i]==3&&n[i-1]==1, return(0))); 1
select(hasNo13, primes(10^4)) \\ Charles R Greathouse IV, Dec 02 2013

is_A076805(n,t=13)=!until(t>n\=10,t==n%100&&return) \\ M. F. Hasler, Dec 02 2013

a(n) >> n^1.0044, where the exponent is log(r)/log(10) with r the larger root of x^2 - 10x + 1. [Charles R Greathouse IV, Nov 09 2011]

kfreep(n, k) = true if for primes p over the range n and integer k, p mod 10^(floor(log_10(k))+1) <> k for p = p/10 mod 10^(floor(log_10(k))+1) <> k over the range floor(log_10(k))+1. This is a mathematical definition of the recurrence. In practice it is convenient to use string operations. E.g., If Not Instr(Str(p), Str(k)) then true or k is not a substring of p so list p.

Original PARI code restored by M. F. Hasler, Dec 02 2013

A155007 Primes p such that (p-3)(p+3)-+3p are primes.

Original entry on oeis.org

7, 17, 37, 113, 157, 227, 283, 293, 313, 347, 443, 587, 787, 883, 1063, 1097, 1237, 1303, 1327, 1427, 1567, 1723, 1933, 1973, 2087, 2347, 2467, 2687, 2777, 3457, 3593, 4447, 4703, 4793, 4967, 5737, 5827, 6317, 6607, 6793, 6857, 8297, 8563, 8803, 9433

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

4*10-3*7=19, 4*10+3*7=61, ...

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006

lst={};Do[p=Prime[n];If[PrimeQ[(p-3)*(p+3)-3*p]&&PrimeQ[(p-3)*(p+3)+3*p],AppendTo[lst,p]],{n,7!}];lst

A254315 Number of distinct digits in the prime factorization of n (counting terms of the form p^1 as p).

Original entry on oeis.org

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

Offset: 2

Michel Lagneau, Jan 28 2015

Write n as product of primes raised to powers; then a(n) is the total number of distinct digits in product representation (number of distinct digits in all the primes and number of distinct digits in all the exponents that are greater than 1).
a(n)<=10. The least n such that a(n)=10 is n = 41701690 = 2*5*47*83*1069.
Property: a(p) = A043537(p), for p prime.
From Michel Marcus, Feb 21 2015: (Start)
For p in A038604, a(p^2) = A043537(p) + 1.
For p in A038611, a(p^3) = A043537(p) + 1.
For p in A038612, a(p^4) = A043537(p) + 1.
For p in A038613, a(p^5) = A043537(p) + 1.
For p in A038614, a(p^6) = A043537(p) + 1.
For p in A038615, a(p^7) = A043537(p) + 1.
For p in A038616, a(p^8) = A043537(p) + 1.
For p in A038617, a(p^9) = A043537(p) + 1.
(End)

			a(36)=2 because 36 = 2^2 * 3^2 => 2 distinct digits.
a(414)=2 because 414 = 2 * 3^2 * 23 => 2 distinct digits.

Michel Lagneau, Table of n, a(n) for n = 2..10000

Cf. A027748, A043537, A050252, A254317.

with(ListTools):
nn:=100:
  for n from 2 to nn do:
    n0:=length(n):lst:={}:x0:=ifactors(n):
    y:=Flatten(x0[2]):z:=convert(y,set):
    z1:=z minus {1}:nn0:=nops(z1):
     for k from 1 to nn0 do :
      t1:=convert(z1[k],base,10):z2:=convert(t1,set):
      lst:=lst union z2:
     od:
     nn1:=nops(lst):printf(`%d, `,nn1):
     od :

f[n_] := Block[{pf = FactorInteger@ n, i}, Length@ DeleteDuplicates@ Flatten@ IntegerDigits@ Rest@ Flatten@ Reap@ Do[If[Last[pf[[i]]] == 1, Sow@ First@ pf[[i]], Sow@ FromDigits@ Flatten[IntegerDigits /@ pf[[i]]]], {i, Length@ pf}]]; Array[f,100] (* Michael De Vlieger, Jan 29 2015 *)

print1(1,", ");for(k=2,100,s=[];F=factor(k);for(i=1,#F[,1],s=concat(s,digits(F[i,1]));if(F[i,2]>1,s=concat(s,digits(F[i,2]))));print1(#vecsort(s,,8),", ")) \\ Derek Orr, Jan 30 2015

from sympy import factorint
def A254315(n):
    return len(set([x for l in [[d for d in str(p)]+[d for d in str(e) if d != '1'] for p,e in factorint(n).items()] for x in l]))
# Chai Wah Wu, Feb 24 2015

A386358 Primes without {7, 9} as digits.

Original entry on oeis.org

2, 3, 5, 11, 13, 23, 31, 41, 43, 53, 61, 83, 101, 103, 113, 131, 151, 163, 181, 211, 223, 233, 241, 251, 263, 281, 283, 311, 313, 331, 353, 383, 401, 421, 431, 433, 443, 461, 463, 503, 521, 523, 541, 563, 601, 613, 631, 641, 643, 653, 661, 683, 811, 821, 823

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038615 and A038617.

Cf. A000040, A020471, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 4, 5, 6, 8]];

Select[Prime[Range[120]], DigitCount[#, 10, 7] == 0 && DigitCount[#, 10, 9] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 4, 5, 6, 8]) \\ uses function in A385776

print(list(islice(primes_with("01234568"), 41))) # uses function/imports in A385776

A155008 Primes p such that (p-a)(p+a)-+ap are primes,a=4.

Original entry on oeis.org

3, 5, 7, 11, 19, 29, 31, 59, 101, 139, 239, 271, 829, 1031, 1201, 1439, 1511, 1531, 2251, 2609, 3929, 4349, 4969, 5449, 5639, 5711, 5801, 5881, 5981, 6521, 6569, 6701, 6949, 6959, 8221, 8831, 9001, 9181, 9209, 9419, 9511, 9929, 10139, 10711, 11839, 11981

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

3*11-28=5, 3*11+28=61, ...

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006, A155007

lst={};Do[p=Prime[n];If[PrimeQ[(p-4)*(p+4)-4*p]&&PrimeQ[(p-4)*(p+4)+4*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]],AllTrue[(#-4)(#+4)+{4#,-4#},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 27 2020 *)

A155009 Primes p such that (p-a)(p+a)-+ap are primes,a=5.

Original entry on oeis.org

2, 7, 11, 17, 19, 23, 41, 43, 61, 67, 107, 109, 131, 137, 179, 197, 263, 269, 331, 353, 397, 641, 677, 743, 859, 941, 1163, 1171, 1213, 1303, 1319, 1433, 1447, 1453, 1543, 1601, 1783, 2221, 2351, 2371, 2417, 2503, 2657, 2689, 2791, 2797, 2909, 3037, 3301

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

1*12-35=-23, 1*12+35=47; 6*16-55=96-55=41, 6*16-55=96+55=151, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006, A155007, A155008

lst={};Do[p=Prime[n];If[PrimeQ[(p-5)*(p+5)-5*p]&&PrimeQ[(p-5)*(p+5)+5*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[500]],AllTrue[(#-5)(#+5)+{5#,-5#},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 01 2016 *)

A224321 Primes without "7" as a digit that remain prime when any single digit is replaced with "7".

Original entry on oeis.org

2, 3, 5, 11, 13, 19, 31, 41, 43, 61, 109, 139, 251, 643, 4933, 9433, 36493, 191416111, 1304119699

Offset: 1

Arkadiusz Wesolowski, Apr 03 2013

No more terms < 10^13.

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 1304119699
Carlos Rivera, Puzzle 591

Cf. A224319-A224320, A224322. Subsequence of A038615.

lst = {}; n = 7; Do[If[PrimeQ[p], i = IntegerDigits[p]; If[FreeQ[i, n], t = 0; s = IntegerLength[p]; Do[If[PrimeQ@FromDigits@Insert[Drop[i, {d}], n, d], t++, Break[]], {d, s}]; If[t == s, AppendTo[lst, p]]]], {p, 36493}]; lst
Select[Prime[Range[4000]],DigitCount[#,10,7]==0&&AllTrue[FromDigits/@Table[ReplacePart[ IntegerDigits[#],n->7],{n,IntegerLength[#]}],PrimeQ]&] (* The program generates the first 17 terms of the sequence. *) (* Harvey P. Dale, Jun 09 2024 *)

A154942 Primes p such that (p-1)p(p+1)-p-2 and (p-1)p(p+1)+p+2 are primes.

Original entry on oeis.org

3, 5, 29, 71, 113, 263, 1103, 2339, 3203, 3413, 3593, 3659, 3719, 4421, 5939, 6269, 7841, 9011, 9029, 13121, 13841, 14423, 15671, 17033, 19073, 22079, 22811, 26783, 27851, 28949, 29303, 30839, 31973, 32063, 32141, 34301, 38543, 38873, 39119

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

2*3*4=24-3-2=19, 2*3*4=24+3+2=29, ...

Cf. A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)*p*(p+1)-p-2]&&PrimeQ[(p-1)*p*(p+1)+p+2],AppendTo[lst,p]],{n,8!}];lst
prQ[n_]:=Module[{x=n^3-n,y=n+2},And@@PrimeQ[{x+y,x-y}]]; Select[Prime[ Range[4200]],prQ] (* Harvey P. Dale, Jun 21 2012 *)

cp's OEIS Frontend

A155006 Primes p such that (p-2)*(p+2)-+2*p are primes.

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Mathematica

A329760 Primes without {2, 7} as digits.

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Magma

Mathematica

Formula

A076805 Triskaidekaphobic or 13-free primes: primes that do not contain the number 13.

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Haskell

Mathematica

PARI

PARI

PARI

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A155007 Primes p such that (p-3)*(p+3)-+3*p are primes.

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Mathematica

A254315 Number of distinct digits in the prime factorization of n (counting terms of the form p^1 as p).

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Maple

Mathematica

PARI

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A386358 Primes without {7, 9} as digits.

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Magma

Mathematica

PARI

Python

A155008 Primes p such that (p-a)*(p+a)-+a*p are primes,a=4.

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Mathematica

A155009 Primes p such that (p-a)*(p+a)-+a*p are primes,a=5.

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A155006 Primes p such that (p-2)(p+2)-+2p are primes.

A155007 Primes p such that (p-3)(p+3)-+3p are primes.

A155008 Primes p such that (p-a)(p+a)-+ap are primes,a=4.

A155009 Primes p such that (p-a)(p+a)-+ap are primes,a=5.

A154942 Primes p such that (p-1)p(p+1)-p-2 and (p-1)p(p+1)+p+2 are primes.