A068395 -id:A068395 - cp's OEIS frontend

A192977 Number of times 9*n occurs in A068395.

4, 4, 2, 2, 3, 2, 2, 3, 2, 1, 0, 4, 1, 1, 3, 1, 2, 2, 2, 1, 4, 0, 0, 1, 3, 2, 1, 2, 2, 2, 2, 1, 0, 1, 3, 0, 2, 2, 2, 1, 2, 2, 1, 0, 2, 1, 1, 3, 2, 1, 3, 1, 1, 2, 0, 2, 0, 2, 0, 2, 1, 2, 2, 1, 2, 0, 2, 3, 0, 1, 3, 2, 1, 2, 1, 1, 0, 2, 1, 1, 2, 1, 2, 2, 1, 1

Offset: 0

Reinhard Zumkeller, Aug 04 2011

			n=1: A068395(k) = 9 = 9*1, for k = 5,6,7,8 therefore a(1) = 4;
n=10: 90 = 9*10 doesn't occur in A068395, therefore a(10) = 0;
n=100: A068395(156) = A068395(157) = 900 = 9*100, therefore a(100) = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

import Data.List (group)
a192977_list = f 0 $ group a068395_list where
   f n xss'@(xs:xss)
     | head xs `div` 9 == n = length xs : f (n+1) xss
     | otherwise            = 0 : f (n+1) xss'

A007605 Sum of digits of n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 2, 4, 8, 10, 5, 11, 4, 10, 5, 7, 11, 8, 14, 7, 13, 8, 10, 16, 11, 17, 16, 2, 4, 8, 10, 5, 10, 5, 11, 13, 14, 7, 13, 10, 14, 11, 17, 10, 11, 13, 17, 19, 4, 7, 11, 13, 8, 14, 7, 8, 14, 11, 17, 10, 16, 11, 13, 14, 10, 5, 7, 11, 7, 13, 14, 16, 11, 17, 16, 13, 19, 14, 20, 19, 5

Offset: 1

N. J. A. Sloane, Mira Bernstein, and Robert G. Wilson v

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
CNRS press release, Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers (French) [On a problem posed by Gelfond: the sum of digits of primes] Ann. of Math. (2) 171(2010), no. 3, 1591--1646. MR2680394 (2011j:11137)
Enrique Navarrete, Distributions of Sums of Digits of Primes
Robert G. Wilson v, Letter to C. A. Pickover, Mar. 1993

Cf. A000040, A007953, A038194, A065073, A068395, A133223, A376714, A380192.

For inverse, see A067180 and A067523.

a007605_list = map a007953 a000040_list -- Reinhard Zumkeller, Aug 04 2011

[ &+Intseq(NthPrime(n), 10): n in [1..80] ]; // Klaus Brockhaus, Jun 13 2009

map(t -> convert(convert(t,base,10),`+`), select(isprime, [2,(2*i+1 $ i=1..1000)])); # Robert Israel, Aug 16 2015

Table[Apply[Plus, RealDigits[Prime[n]][[1]]], {n, 1, 100}]
Plus@@ IntegerDigits[Prime[Range[100]]] (* Zak Seidov *)

dsum(n)=my(s);while(n,s+=n%10;n\=10);s
forprime(p=2,1e3,print1(dsum(p)", ")) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = sumdigits(prime(n)); \\ Michel Marcus, Dec 20 2017

from sympy import prime
def a(n): return sum(map(int, str(prime(n))))
print([a(n) for n in range(1, 80)]) # Michael S. Branicky, Feb 03 2021

a(n) = A007953(A000040(n)) = A007953(prime(n)).

A243442 Primes p such that, in base 2, p - digitsum(p) is also a prime.

Original entry on oeis.org

5, 23, 71, 83, 101, 113, 197, 281, 317, 353, 359, 373, 401, 467, 599, 619, 683, 739, 751, 773, 977, 1091, 1097, 1103, 1217, 1223, 1229, 1237, 1283, 1303, 1307, 1429, 1433, 1489, 1553, 1559, 1601, 1607, 1613, 1619, 1699, 1873, 1879, 2039, 2347, 2357, 2389

Offset: 1

Anthony Sand, Jun 05 2014

In all bases b, x = n - digitsum(n) is always divisible by b-1, therefore x can be prime only in base 2 and bases b for which b-1 is prime. For example, in base 10, n - digitsum(n) is always divisible by 10 - 1 = 9 -- see A066568 and A068395. In base 8, 9 = 11, therefore 11 - digitsum(11) = 9 - 2 = 7 is divisible by 7.

			5 - digitsum(5,base=2) = 5 - digitsum(101) = 5 - 2 = 3.
23 - digitsum(10111) = 23 - 4 = 19.
71 - digitsum(1000111) = 71 - 4 = 67.
83 - digitsum(1010011) = 83 - 4 = 79.
101 - digitsum(1100101) = 101 - 4 = 97.

Anthony Sand, Table of n, a(n) for n = 1..1000

Cf. A243441.

Select[Prime[Range[400]],PrimeQ[#-Total[IntegerDigits[#,2]]]&] (* Harvey P. Dale, May 15 2019 *)

isok(n) = isprime(n) && isprime(n - hammingweight(n)); \\ Michel Marcus, Jun 05 2014

A184328 Primes whose digital product is a positive square.

Original entry on oeis.org

11, 19, 41, 149, 191, 199, 229, 263, 281, 313, 331, 419, 433, 449, 491, 499, 661, 683, 797, 821, 829, 863, 881, 911, 919, 941, 977, 991, 1229, 1289, 1433, 1499, 1559, 1669, 1747, 1889, 1933, 1949, 1999, 2129, 2383, 2693, 2819, 2833, 2963, 3319, 3391, 3413

Offset: 1

Dario Piazzalunga, Dec 24 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A053666, A007954, A007605, A065073, A068395.

[p: p in PrimesUpTo(4000) | not IsZero(t) and IsSquare(t) where t is &*Intseq(p)]; // Bruno Berselli, Dec 25 2012

fQ[n_] := Module[{d = Times @@ IntegerDigits[n]}, d > 0 && IntegerQ[Sqrt[d]]];Select[Prime[Range[1000]], fQ] (* T. D. Noe, Dec 24 2012 *)

Corrected and extended by T. D. Noe, Dec 24 2012

A206011 The n-th semiprime minus its sum of digits.

Original entry on oeis.org

0, 0, 0, 9, 9, 9, 18, 18, 18, 18, 27, 27, 27, 27, 27, 36, 36, 45, 45, 45, 45, 54, 54, 54, 63, 63, 72, 72, 72, 72, 81, 81, 81, 81, 99, 108, 108, 108, 108, 117, 117, 117, 117, 126, 126, 135, 135, 135, 135, 135, 144, 144, 144, 153, 153, 153, 162, 162, 171, 171

Offset: 1

Jonathan Vos Post, Feb 02 2012

This is to semiprimes A001358 as A068395 is to primes A000040. As with A068395, this is always a multiple of 9, hence cannot be prime. But, as happens first for a(4), a(n) can be semiprime.

			a(4) = 10 - 1 = 9.
a(5) = 14 - 5 = 9.

Cf. A000040, A001358, A007953, A068395.

read("transforms") :
A206011 := proc(n)
    s := A001358(n) ;
    s -digsum(s) ;
end proc: # R. J. Mathar, Sep 14 2012

#-Total[IntegerDigits[#]]&/@Select[Range[200],PrimeOmega[#]==2&] (* Harvey P. Dale, Nov 24 2022 *)

a(n) = A001358(n) - A007953(A001358(n)).

A321150 Primes p such that p minus its digit sum is a square.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 41, 43, 47, 97, 151, 157, 233, 239, 331, 337, 457, 593, 599, 743, 911, 919, 1301, 1303, 1307, 1531, 1783, 1787, 1789, 2039, 2311, 2617, 2939, 3613, 3617, 4373, 4783, 4787, 4789, 5641, 5647, 6581, 7079, 7591, 8111, 8117, 8677, 9239, 9829, 11681, 11689, 13001, 13003, 13007

Offset: 1

Marius A. Burtea, Oct 28 2018

			11 is prime and 11 - (1+1) = 9 = 3^2 is square, so 11 is a term of the sequence.
457 is prime and 457 - (4+5+7) = 441 = 21^2 is square, so 457 is a term of the sequence.
2939 is prime and 2939 - (2+9+3+9) = 2916 = 54^2 is square, so 2939 is a term of the sequence.
101 is prime and 101 - (1+0+1) = 99 is not square, so 101 is not a term of the sequence.

Robert Israel, Table of n, a(n) for n = 1..3000

Cf. A000290, A007605, A007953, A245064, A068395.

Intersection of A000040 and A066566.

select(t -> isprime(t) and issqr(t - convert(convert(t,base,10),`+`)),
[2,seq(i,i=3..20000,2)]); # Robert Israel, Apr 15 2019

Select[Prime@ Range@ 2000, IntegerQ@ Sqrt[# - Total@ IntegerDigits@ #] &] (* Michael De Vlieger, Nov 05 2018 *)

isok(p) = isprime(p) && issquare(p-sumdigits(p)); \\ Michel Marcus, Oct 30 2018

a(26) corrected by Robert Israel, Apr 15 2019

cp's OEIS Frontend

A192977 Number of times 9*n occurs in A068395.

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A007605 Sum of digits of n-th prime.

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A243442 Primes p such that, in base 2, p - digitsum(p) is also a prime.

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A184328 Primes whose digital product is a positive square.

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Magma

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A206011 The n-th semiprime minus its sum of digits.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A321150 Primes p such that p minus its digit sum is a square.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions