A007605 -id:A007605 - cp's OEIS frontend

A347702 Prime numbers that give a remainder of 1 when divided by the sum of their digits.

11, 13, 17, 41, 43, 97, 101, 131, 157, 181, 233, 239, 271, 311, 353, 401, 421, 491, 521, 541, 599, 617, 631, 647, 673, 743, 811, 859, 953, 1021, 1031, 1051, 1093, 1171, 1201, 1249, 1259, 1301, 1303, 1327, 1373, 1531, 1601, 1621, 1801, 1871, 2029, 2111, 2129, 2161

Offset: 1

Burak Muslu, Sep 10 2021

			97 is a term since its sum of digits is 9+7 = 16, and 97 mod 16 = 1.

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

Cf. A000040, A007605, A136251.
Subsequence of A209871.
A259866 \ {31}, and the primes associated with A056804 \ {1, 2} and A056797 are subsequences.

select(t -> isprime(t) and t mod convert(convert(t,base,10),`+`) = 1, [seq(i,i=3..10000,2)]); # Robert Israel, Mar 05 2024

Select[Range[2000], PrimeQ[#] && Mod[#, Plus @@ IntegerDigits[#]] == 1 &] (* Amiram Eldar, Sep 10 2021 *)

isok(p) = isprime(p) && ((p % sumdigits(p)) == 1); \\ Michel Marcus, Sep 10 2021

from sympy import primerange
def ok(p): return p%sum(map(int, str(p))) == 1
print(list(filter(ok, primerange(1, 2130)))) # Michael S. Branicky, Sep 10 2021

A370848 Lesser of two consecutive primes such that the product of its digits is also prime and the sum of the digits of the other is composite.

Original entry on oeis.org

13, 17, 31, 71, 113, 1151, 11131, 112111, 113111, 131111, 1111211, 1111711, 11111117, 11111171, 71111111, 115111111, 1111111121, 1111115111, 1115111111, 1117111111, 1151111111, 1711111111, 11111111113, 11113111111, 31111111111, 111113111111, 111511111111, 1111171111111

Offset: 1

Claude H. R. Dequatre, Mar 03 2024

			13 is a term because 13 is prime, the product of its digits is 3 which is also prime and the sum of the digits of 17, the next prime to 13, is 8 which is composite.
23 is not a term because the product of its digits is 6 which is not prime.
131 is not a term because although it is prime and the product of its digits is 3 which is also prime, the sum of the digits of 137, the next prime to 131, is 11 which is not composite.

Hugo Pfoertner, Table of n, a(n) for n = 1..3701

Cf. A000040, A002808, A007605, A053666.
Cf. A370850.
Except for the first, all terms of this sequence are in A370851.

Select[Prime[Range[5*10^6]],PrimeQ[Apply[Times,IntegerDigits[#]]]&&CompositeQ[Total[IntegerDigits[NextPrime[#]]]]&] (* James C. McMahon, Mar 03 2024 *)

isok(p)=my(x=vecprod(digits(p)),y=sumdigits(nextprime(p+1)));isprime(x) && !isprime(y);
forprime(p=2,20000,if(isok(p),print1(p", ")))

a370848(maxdigits=20) = {my (L=List()); for (n=2, maxdigits, my (r=(10^n-1)/9, d=digits(r)); foreach ([2,3,5,7], s, for (k=1, #d, my (dd=d); dd[k]=s; my(q=fromdigits(dd)); if (ispseudoprime(q) && ! isprime(sumdigits(nextprime(q+1))), listput(L,q))))); vecsort(Vec(L))};
a370848() \\ Hugo Pfoertner, Mar 03 2024

from itertools import count, islice
from sympy import isprime, nextprime
def A370848_gen(): # generator of terms
    for l in count(1):
        k = (10**l-1)//9
        for m in range(l):
            a = 10**m
            for j in (1,2,4,6):
                p = k+a*j
                if isprime(p) and not isprime(sum(map(int,str(nextprime(p))))):
                    yield p
A370848_list = list(islice(A370848_gen(),20)) # Chai Wah Wu, Mar 25 2024

a(17)-a(21) from Michel Marcus, Mar 03 2024

a(22)-a(28) from Hugo Pfoertner, Mar 03 2024

A007513 a(n) = initial prime of n consecutive primes such that first and last have same digit sum.

Original entry on oeis.org

2, 523, 109, 79, 2, 13, 5, 127, 47, 17, 5, 127, 53, 17, 7, 67, 31, 37, 47, 37, 83, 11, 43, 19, 157, 2, 37, 5, 47, 5, 19, 67, 7, 29, 19, 53, 31, 73, 53, 29, 139, 13, 67, 83, 7, 47, 29, 17, 79, 7, 19, 37, 59, 43, 271, 19, 29, 181, 167, 59, 97, 5, 149, 7, 59, 337, 41, 53, 43, 127

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

			523 and 541 are first pair of consecutive primes with same sum of digits (10).

J. R. Smart, A new function from a table of primes, J. Rec. Math., 7 (No. 4, 1974), 293-294.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. R. Smart, A new function from a table of primes, J. Recreational Mathematics 7.4 (1974), 293-294. (Annotated scanned copy)

import Data.Function (on)
import Data.List (elemIndex)
import Data.Maybe (fromJust)
a007513 n = a000040_list !! (fromJust $ elemIndex 0 $
   zipWith ((-) `on` a007953) a000040_list $ drop (n-1) a000040_list)
-- Reinhard Zumkeller, Aug 17 2011

A007953 := proc(n) local d; add(d,d=convert(n,base,10)) ; end proc: A007605 := proc(n) A007953(ithprime(n)) ; end proc: A007513 := proc(n) for i from 1 do if A007605(i) = A007605(i+n-1) then return ithprime(i) ; end if; end do ; end proc: seq(A007513(n),n=1..120) ; # R. J. Mathar, Dec 09 2009

prms=Prime[Range[2000]];First/@Table[First[Select[Partition[prms,n,1], Total[ IntegerDigits[ First[#]]]==Total[IntegerDigits[Last[#]]]&]], {n,75}] (* Harvey P. Dale, May 20 2011 *)

Terms beyond a(57) by R. J. Mathar, Dec 09 2009

A058049 Numbers k such that the sum of the digits of the first k primes is a prime.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 11, 12, 14, 23, 33, 43, 45, 48, 64, 69, 72, 73, 77, 87, 94, 95, 96, 98, 110, 118, 124, 130, 133, 140, 148, 152, 154, 157, 162, 171, 174, 178, 181, 196, 200, 201, 206, 210, 212, 219, 232, 241, 244, 253, 257, 267, 269, 272, 277, 299, 304, 306

Offset: 1

Robert G. Wilson v, Nov 18 2000

What is intriguing about this sequence is that the number of primes less than 10^m is of the same magnitude as A006880. Here they begin 7, 25, 122, 934.

			5 is a term because sum of digits of first 5 primes, 2+3+5+7+(1+1)=19, is prime.
a(5) = 6 because in A051351(6) = 2 + 3 + 5 + 7 + 2 (sum of eleven's digits) + 4 (sum of thirteen's digits) which equals the sum of the digits through the sixth prime = 23 which itself is a prime.

Z. Stankova-Frenkel and J. West, Explicit enumeration of 321,hexagon-avoiding permutations, arXiv:math/0106073 [math.CO], 2001.

Corresponding primes: A104247. Primes: A000040, sum of digits of primes: A007605.

Cf. A051351.

s = 0; Do[ s = s + Apply[ Plus, RealDigits[ Prime[ n ]] [[1]] ]; If[ PrimeQ[ s ], Print[ n ] ], {n, 1, 1000} ]

isok(n) = isprime(sum(k=1, n, sumdigits(prime(k)))); \\ Michel Marcus, Mar 11 2017

from sympy import isprime, nextprime
def sd(n): return sum(map(int, str(n)))
def aupto(limit):
    alst, k, p, s = [], 1, 2, 2
    while k <= limit:
        if isprime(s): alst.append(k)
        k += 1; p = nextprime(p); s += sd(p)
    return alst
print(aupto(306)) # Michael S. Branicky, Jul 18 2021

Edited by R. J. Mathar, Aug 04 2008

A081653 Greatest common divisor of sums of decimal digits of n and of n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 2, 5, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 1, 1, 2, 11, 1, 1, 2, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 5, 1

Offset: 1

Reinhard Zumkeller, Mar 25 2003

			a(8) = GCD(A007953(8), A007953(A000040(8))) = GCD(8, A007953(19)) = GCD(8,1+9) = GCD(4*2,5*2) = 2.

Cf. A000040, A007953, A007605, A081652, A066750.

A081653[n_Integer]:=GCD[Plus@@IntegerDigits[n],Plus@@IntegerDigits[Prime[n]]]; Table[A081653[n],{n,105}] (* Enrique Pérez Herrero, Aug 07 2010 *)

a(n) = GCD(A007953(n), A007605(n)).

A104260 Sum of odd digits (1,3,5,7,9) of n-th prime.

Original entry on oeis.org

0, 3, 5, 7, 2, 4, 8, 10, 3, 9, 4, 10, 1, 3, 7, 8, 14, 1, 7, 8, 10, 16, 3, 9, 16, 2, 4, 8, 10, 5, 8, 5, 11, 13, 10, 7, 13, 4, 8, 11, 17, 2, 11, 13, 17, 19, 2, 3, 7, 9, 6, 12, 1, 6, 12, 3, 9, 8, 14, 1, 3, 12, 10, 5, 7, 11, 7, 13, 10, 12, 11, 17, 10, 13, 19, 6, 12, 19, 1, 9, 10, 1, 4, 6, 12, 3, 9, 12

Offset: 1

Zak Seidov, Feb 26 2005

Sum of even digits (2, 4, 6, 8) of n-th prime: A104261, sum of prime digits (2, 3, 5, 7) of n-th prime: A104250, sum of composite (nonprime) digits (1, 4, 6, 8, 9) of n-th prime: A104251, sum of digits of primes: A007605, primes: A000040.

sodod={};Do[id=IntegerDigits[Prime[i]];lid=Length[id];sod=Sum[If[OddQ[id[[k]]], id[[k]], 0], {k, lid}];sodod={sodod, sod}, {i, 200}];sodod//Flatten
f[n_] := Plus @@ Select[ IntegerDigits[ Prime[n]], OddQ[ # ] &]; Table[ f[n], {n, 88}] (* Robert G. Wilson v, Nov 03 2005 *)

a(n) = A007605(n) - A104261(n).

A104261 Sum of even digits (0,2,4,6,8) of n-th prime.

Original entry on oeis.org

2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 4, 4, 4, 0, 0, 6, 6, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 6, 6, 0, 0, 8, 0, 0, 0, 0, 2, 4, 4, 4, 2, 2, 6, 2, 2, 8, 8, 2, 2, 10, 10, 2, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 6, 0, 0, 8, 8, 0, 4, 4, 4, 6, 4, 4, 4, 8, 8, 4, 10, 10, 10, 4, 12, 4, 4, 0, 0, 2, 2, 4, 4, 0

Offset: 1

Zak Seidov, Feb 26 2005

Sum of odd digits (1, 3, 5, 7, 9) of n-th prime: A104260, sum of prime digits (2, 3, 5, 7) of n-th prime: A104250, sum of composite (nonprime) digits (1, 4, 6, 8, 9) of n-th prime: A104251, sum of digits of primes: A007605, primes: A000040.

sodev={};Do[id=IntegerDigits[Prime[i]];lid=Length[id];sod=Sum[If[EvenQ[id[[k]]], id[[k]], 0], {k, lid}];sodev={sodev, sod}, {i, 200}];sodev//Flatten
f[n_] := Plus @@ Select[ IntegerDigits[ Prime[n]], EvenQ[ # ] && # > 1 &]; Table[ f[n], {n, 102}] (* Robert G. Wilson v, Nov 03 2005 *)

A104261(n) = A007605(n) - A104260(n).

A158293 Primes whose digit sum is a multiple of 10.

Original entry on oeis.org

19, 37, 73, 109, 127, 163, 181, 271, 307, 389, 433, 479, 523, 541, 569, 587, 613, 631, 659, 677, 811, 839, 857, 929, 947, 983, 1009, 1063, 1117, 1153, 1171, 1289, 1423, 1487, 1531, 1559, 1621, 1667, 1801, 1847, 1973, 2017, 2053, 2099, 2143, 2161, 2251

Offset: 1

Parthasarathy Nambi, Mar 15 2009

			The digit sum of 1009 is a multiple of ten.

Chris Caldwell, The First 1,000 Primes

Cf. A007605.

[ p: p in PrimesUpTo(2500) | &+Intseq(p) mod 10 eq 0 ];

Select[Prime[Range[335]], GCD[(Plus@@IntegerDigits[#]), 10] >= 10 &] (* Alonso del Arte, Jan 31 2011 *)
Select[Prime[Range[400]],Divisible[Total[IntegerDigits[#]],10]&] (* Harvey P. Dale, Aug 19 2012 *)

A165439 Primes p such that 3+2*(sum of digits of p) is also a prime.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 43, 53, 59, 61, 67, 71, 73, 89, 101, 103, 107, 109, 113, 127, 131, 139, 149, 151, 157, 163, 167, 179, 181, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 283, 293, 307, 311, 313, 331, 337, 347, 359, 373

Offset: 1

Vincenzo Librandi, Sep 19 2009

			331 is in the sequence because 3+3+1=7 and 2*7+3=17 is prime.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A165440, A007605.

[ p: p in PrimesUpTo(400) | IsPrime(q) where q is 3+2*(&+Intseq(p))];  // Bruno Berselli, Jul 15 2011

Select[Prime[Range[200]],PrimeQ[2 Total[IntegerDigits[#]]+3]&] (* Harvey P. Dale, Jul 14 2011 *)

Keyword:base added by R. J. Mathar, Oct 12 2009

A165440 Primes p such that 3+2*(sum of digits of p) is not a prime.

Original entry on oeis.org

3, 29, 47, 79, 83, 97, 137, 173, 191, 227, 263, 277, 281, 317, 349, 353, 367, 439, 443, 457, 461, 547, 599, 619, 641, 673, 691, 709, 727, 797, 821, 853, 887, 907, 911, 977, 1019, 1069, 1087, 1091, 1109, 1163, 1181, 1217, 1249, 1307, 1361, 1429, 1433, 1447

Offset: 1

Vincenzo Librandi, Sep 19 2009

			1087 is in the sequence because 1+0+8+7=16 and 2*16+3=35 is not prime.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A165439, A007605.

[ p: p in PrimesUpTo(1500) | not IsPrime(q) where q is 3+2*(&+Intseq(p))];  // Bruno Berselli, Jul 15 2011

Select[Prime[Range[250]],!PrimeQ[2*Total[IntegerDigits[#]]+3]&] (* Harvey P. Dale, Jul 15 2014 *)

Keyword:base added by R. J. Mathar, Sep 21 2009

cp's OEIS Frontend

A347702 Prime numbers that give a remainder of 1 when divided by the sum of their digits.

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A370848 Lesser of two consecutive primes such that the product of its digits is also prime and the sum of the digits of the other is composite.

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A007513 a(n) = initial prime of n consecutive primes such that first and last have same digit sum.

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A058049 Numbers k such that the sum of the digits of the first k primes is a prime.

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A081653 Greatest common divisor of sums of decimal digits of n and of n-th prime.

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A104260 Sum of odd digits (1,3,5,7,9) of n-th prime.

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A104261 Sum of even digits (0,2,4,6,8) of n-th prime.

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A158293 Primes whose digit sum is a multiple of 10.

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