A007605 -id:A007605 - cp's OEIS frontend

A172035 Smallest exponent k > 1 that sum of digits of k-th power of the n-th prime is a prime (n=1,2,...) or 0 if no such k exists.

5, 0, 2, 2, 9, 3, 2, 5, 3, 2, 7, 2, 4, 5, 2, 2, 5, 2, 3, 6, 2, 2, 2, 2, 4, 8, 4, 2, 2, 4, 2, 8, 2, 3, 2, 2, 4, 4, 6, 2, 4, 2, 10, 3, 4, 2, 3, 2, 4, 3, 5, 6, 3, 4, 4, 2, 2, 2, 2, 2, 3, 4, 3, 3, 3, 5, 3, 3, 8, 2, 3, 12, 2, 3, 2, 5, 2, 3, 8, 16, 8, 3, 4, 2, 3, 2, 4, 2, 2, 5, 7, 4, 3, 8, 3, 2, 6, 2, 3, 6, 2, 2, 10

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jan 23 2010

k = 1 is the "trivial" case: sod(prime(n)) = prime(m)

n = 2, prime(2) = 3: 3^k is for k > 1 a multiple of 3^2.

			sod(2^5)=5, sod(5^2)=7, sod(7^2)=13, sod(11^9)=53, sod(13^3)=19, sod(17^2)=19, sod(19^5)=37, sod(23^3)=17, sod(29^2)=13, sod(31^7)=31, sod(37^2)=19, sod(41^4)=31, sod(43^5)=31, sod(47^2)=13, sod(53^2)=19, sod(59^5)=47, sod(61^2)=13, sod(67^3)=19, sod(71^6)=37, sod(73^2)=19, sod(79^2)=13, sod(83^2)=31, sod(89^2)=19, sod(97^4)=43, sod(101^8)=67, sod(103^4)=31, sod(107^2)=19, sod(109^2)=19, sod(113^4)=31, sod(127^2)=19, sod(131^8)=61, sod(137^2)=31, sod(139^3)=37, sod(149^2)=7, sod(151^2)=13, sod(157^4)=31, sod(163^4)=37, sod(167^6)=73, sod(173^2)=31, sod(179^4)=37, sod(181^2)=19, sod(191^10)=97, sod(193^3)=37, sod(197^4)=37, sod(199^2)=19, sod(211^3)=37, sod(223^2)=31, sod(227^4)=43, sod(229^3)=37.

M. Fujiwara, Y. Ogawa: Introduction to truly beautiful Mathematics, Chikuma Shobo, Tokyo 2005.
Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005.
Hans Schubart: Einfuehrung in die klassische und moderne Zahlentheorie, Vieweg, Braunschweig 1974.

Cf. A046704, A007605

Cf. A172216. - Klaus Brockhaus, Jan 29 2010

S:=[ 5, 0 ]; for n in [3..103] do j:=2; while not IsPrime(&+Intseq(NthPrime(n)^j)) do j+:=1; end while; Append(~S, j); end for; S; // Klaus Brockhaus, Jan 29 2010

More terms from Klaus Brockhaus, Jan 29 2010

Edited by Charles R Greathouse IV, Aug 02 2010

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

A230222 Smallest of four consecutive palindromic primes with equal digital sum.

Original entry on oeis.org

185595581, 317565713, 10832723801, 10875857801, 16831813861, 16832623861, 33396769333, 36215951263, 39003830093, 1069319139601, 1075309035701, 1181969691811, 1221739371221, 1269056509621, 1270668660721, 1292808082921, 1320348430231, 1385647465831

Offset: 1

Shyam Sunder Gupta, Oct 12 2013

			185595581 is in the sequence because 185595581, 185676581, 185757581 and 185838581 are consecutive palindromic primes and the sum of the digits of each = 47.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..3924

Cf. A002385, A007953, A007605, A210629, A230199

A239694 Base 8 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 4, 6, 3, 5, 9, 8, 10, 9, 6, 8, 12, 11, 10, 12, 4, 8, 3, 9, 6, 5, 6, 10, 12, 9, 11, 8, 15, 5, 4, 6, 9, 11, 10, 9, 13, 12, 11, 13, 16, 4, 8, 10, 8, 13, 10, 12, 9, 15, 10, 13, 5, 11, 10, 12, 11, 8, 10, 13, 13, 17, 12, 16, 9, 8, 11, 13, 10, 16, 17, 16

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-8 dominance order on the natural numbers.

			The sixth prime is 13, 13 in base 8 is (1,5) so a(6)=1+5=6.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.

Cf. A007605, A014499, A053829, A239690, A239691, A239692, A239693.

[&+Intseq(NthPrime(n),8): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 8], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = sumdigits(prime(n), 8); \\ Michel Marcus, Mar 04 2023

[sum(i.digits(base=8)) for i in primes_first_n(200)]

a(n) = A053829(A000040(n)).

A247797 Lexicographically earliest permutation of prime numbers, such that adjacent terms have coprime sums of digits in decimal representation.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 13, 29, 17, 41, 31, 43, 19, 47, 37, 61, 53, 67, 59, 83, 71, 89, 73, 137, 79, 113, 97, 131, 101, 139, 103, 151, 107, 157, 109, 173, 127, 179, 149, 191, 163, 193, 167, 197, 181, 199, 211, 223, 227, 229, 233, 241, 251, 263, 239, 269, 257

Offset: 1

Reinhard Zumkeller, Nov 25 2014

A049084(a(n)) defines a permutation of the positive integers, cf. A250552.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals of Mathematics, Vol. 171, No. 3, 1591-1646, 2010.

Cf. A000040, A007953, A007605, A049084, A250552, A119449, A119450.

import Data.List (delete)
a247797 n = a247797_list !! (n-1)
a247797_list = f 1 $ zip a000040_list a007605_list where
   f q' vws = g vws where
     g  ((p,q):pqs) = if gcd q q' == 1
                         then p : f q (delete (p,q) vws) else g pqs

A376714 Sum of squares of the decimal digits of the n-th prime.

Original entry on oeis.org

4, 9, 25, 49, 2, 10, 50, 82, 13, 85, 10, 58, 17, 25, 65, 34, 106, 37, 85, 50, 58, 130, 73, 145, 130, 2, 10, 50, 82, 11, 54, 11, 59, 91, 98, 27, 75, 46, 86, 59, 131, 66, 83, 91, 131, 163, 6, 17, 57, 89, 22, 94, 21, 30, 78, 49, 121, 54, 102, 69, 77, 94, 58, 11

Offset: 1

Katie Khan, Oct 02 2024

			For n=7, the 7th prime = 17 and those digits 1^2 + 7^2 = 50 = a(7).

Katie Khan, Table of n, a(n) for n = 1..10000

Cf. A000040, A003132, A007605.

a[n_]:=Norm[IntegerDigits[Prime[n]]]^2; Array[a,64] (* Stefano Spezia, Oct 03 2024 *)

a(n) = norml2(digits(prime(n))); \\ Michel Marcus, Oct 03 2024

from sympy import prime
def A376714(n): return sum((0, 1, 4, 9, 16, 25, 36, 49, 64, 81)[int(d)] for d in str(prime(n)) if d>'0') # Chai Wah Wu, Oct 04 2024

a(n) = A003132(A000040(n)).

A380192 Sum mod(10) of digits of n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 2, 4, 8, 0, 5, 1, 4, 0, 5, 7, 1, 8, 4, 7, 3, 8, 0, 6, 1, 7, 6, 2, 4, 8, 0, 5, 0, 5, 1, 3, 4, 7, 3, 0, 4, 1, 7, 0, 1, 3, 7, 9, 4, 7, 1, 3, 8, 4, 7, 8, 4, 1, 7, 0, 6, 1, 3, 4, 0, 5, 7, 1, 7, 3, 4, 6, 1, 7, 6, 3, 9, 4, 0, 9, 5, 3, 4, 7, 8, 0, 6, 1, 7, 6, 1, 3, 7, 0, 9

Offset: 1

Enrique Navarrete, Jan 15 2025

All remainders 0,...,9 occur in this sequence.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Enrique Navarrete, Distributions of Sums of Digits of Primes

Cf. A007605, A010879, A053837, A158293.

Mod[DigitSum[Prime[Range[100]]], 10] (* Paolo Xausa, Feb 06 2025 *)

a(n) = sumdigits(prime(n)) % 10; \\ Michel Marcus, Jan 16 2025

a(n) = A010879(A007605(n)).

A073342 Average digit (rounded to the nearest integer) in the decimal expansion of n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 1, 2, 4, 5, 3, 6, 2, 5, 3, 4, 6, 4, 7, 4, 7, 4, 5, 8, 6, 9, 8, 1, 1, 3, 3, 2, 3, 2, 4, 4, 5, 2, 4, 3, 5, 4, 6, 3, 4, 4, 6, 6, 1, 2, 4, 4, 3, 5, 2, 3, 5, 4, 6, 3, 5, 4, 4, 5, 3, 2, 2, 4, 2, 4, 5, 5, 4, 6, 5, 4, 6, 5, 7, 6, 2, 4, 5, 2, 3, 3, 5, 4, 6, 5, 4, 4, 6, 7, 6, 5, 7, 3, 5, 3, 3, 3, 5, 6, 5, 7, 4, 6, 7, 6, 8, 2, 4, 3, 5, 5, 3, 4, 4, 6, 5, 7

Offset: 1

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 23 2002

			For the prime 107 we have nearest((1+0+7)/3)=nearest(8/3)=3.

Cf. A000040, A074462.

Floor[Mean[IntegerDigits[#]]+1/2]&/@Prime[Range[120]] (* Harvey P. Dale, Nov 22 2011 *)

a(n)=round(A007605(n)/A097944(n)). - R. J. Mathar, Sep 23 2008

Changed offset to 1, added Cf. to A074462 and extended. - R. J. Mathar, Sep 23 2008

A081652 Greatest common divisor of n and sum of decimal digits of n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 7, 1, 8, 1, 1, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 5, 1, 1, 11, 1, 7, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 11, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 2, 1, 8, 1, 2, 1, 10, 1, 17, 1, 4

Offset: 1

Reinhard Zumkeller, Mar 25 2003

			a(16) = GCD(16, A007953(A000040(16))) = GCD(16, A007953(53)) = GCD(16,5+3) = GCD(8*2,8) = 8.

Cf. A000040, A007953, A081653, A066750.

Table[GCD[n,Total[IntegerDigits[Prime[n]]]],{n,110}] (* Harvey P. Dale, Jul 12 2012 *)

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

A138828 Sum of digits of n-th even perfect number.

Original entry on oeis.org

6, 10, 19, 19, 28, 64, 55, 73, 190, 235, 289, 352, 1405, 1711, 3520, 5833, 5968, 8821, 11548, 11791, 26317, 27298, 30232, 53740, 58960, 62956, 120898, 233722, 299314, 356860, 585478, 2048248, 2329372, 3405232, 3789352, 8056495, 8186041, 18894079, 36485416, 56880973, 65115946, 70334902, 82384129

Offset: 1

Omar E. Pol, Apr 01 2008

Ivan Panchenko, Table of n, a(n) for n = 1..47
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A007605, A066538, A138127.

a[n_] := Block[{e, p, mpe = MersennePrimeExponent@n}, Plus @@ IntegerDigits[(2^mpe - 1) 2^(mpe - 1)]]; Array[a, 44] (* Robert G. Wilson v, Aug 06 2018 *)
Total[IntegerDigits[#]]&/@Table[PerfectNumber[n],{n,43}] (* Requires Mathematica version 10 or later *) (* The program may take a long time to run *) (* Harvey P. Dale, Feb 07 2019 *)

a(n) = A007953(A000396(n)). - R. J. Mathar, May 22 2008 [This assumes that all perfect numbers are even. - Ivan Panchenko, Aug 16 2018]

More terms from R. J. Mathar, May 22 2008
a(15)-a(38) from Donovan Johnson, Nov 09 2010
Definition changed (inserting the word "even") and a(39)-a(43) added by Ivan Panchenko, Aug 06 2018

cp's OEIS Frontend

A172035 Smallest exponent k > 1 that sum of digits of k-th power of the n-th prime is a prime (n=1,2,...) or 0 if no such k exists.

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A184328 Primes whose digital product is a positive square.

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A230222 Smallest of four consecutive palindromic primes with equal digital sum.

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A239694 Base 8 sum of digits of prime(n).

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A247797 Lexicographically earliest permutation of prime numbers, such that adjacent terms have coprime sums of digits in decimal representation.

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Haskell

A376714 Sum of squares of the decimal digits of the n-th prime.

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A380192 Sum mod(10) of digits of n-th prime.

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Mathematica

PARI

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A073342 Average digit (rounded to the nearest integer) in the decimal expansion of n-th prime.

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