A088135 -id:A088135 - cp's OEIS frontend

A088136 Primes such that sum of first and last digits is prime.

11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 131, 151, 181, 191, 211, 223, 229, 233, 239, 241, 251, 263, 269, 271, 281, 283, 293, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 601, 607, 617, 631, 641, 647, 661, 677, 691

Offset: 1

Zak Seidov, Sep 20 2003

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A008040 (primes), A010051 (isprime), A000030 (first digit of n), A010879 (last digit of n).

Cf. A046704, A007953, A088133, A088134, A088135.

Select[Prime[Range[400]],PrimeQ[First[IntegerDigits[#]]+ Last[ IntegerDigits[ #]]]&] (* Harvey P. Dale, Jun 23 2017 *)

select( {is_A088136(p)=isprime(p\10^logint(p,10)+p%10)&&isprime(p)}, primes(99)) \\ M. F. Hasler, Apr 23 2024

from sympy import isprime, primerange
def ok(p): s = str(p); return isprime(int(s[0]) + int(s[-1]))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(691)) # Michael S. Branicky, Nov 23 2021

A088133 Sum of first and last digits of n. Different from A115299.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14

Offset: 0

Zak Seidov, Sep 20 2003

Cf. A000030 (first digit of n), A010879 (last digit of n).

Cf. A046704, A007953, A088134, A088135, A088136.

Total[{First[IntegerDigits[#]],Last[IntegerDigits[#]]}]&/@Range[90] (* Harvey P. Dale, Aug 21 2018 *)

apply( {A088133(n)=n\10^logint(n+!n, 10)+n%10}, [0..99]) \\ M. F. Hasler, Apr 22 2024

list(map(A088133 := lambda n: int(str(n)[0])+n%10, range(99))) # M. F. Hasler, Apr 22 2024

a(n) = A000030(n) + A010879(n). - M. F. Hasler, Apr 22 2024

Extended to a(0) = 0 by M. F. Hasler, Apr 22 2024

A088134 Numbers n such that sum of first and last digits is prime.

Original entry on oeis.org

1, 11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 41, 43, 47, 49, 50, 52, 56, 58, 61, 65, 67, 70, 74, 76, 83, 85, 89, 92, 94, 98, 101, 102, 104, 106, 111, 112, 114, 116, 121, 122, 124, 126, 131, 132, 134, 136, 141, 142, 144, 146, 151, 152, 154, 156, 161, 162, 164

Offset: 1

Zak Seidov, Sep 20 2003

Cf. A046704, A007953, A088133, A088135, A088136.

A108660 Square-loop primes.

Original entry on oeis.org

2, 13, 31, 79, 97, 227, 881, 1013, 2797, 3181, 3631, 8101, 22727, 81001, 101363, 109013, 131363, 181813, 272227, 310181, 310901, 318181, 318881, 631013, 636313, 810401, 818101, 901097, 904097, 972227, 1018813, 1090013, 1810013, 2272727

Offset: 1

Zak Seidov, Jun 16 2005

Primes such that each pair of adjacent digits (and also the first and the last ones) sums up to a square. First term is arguable since there is 'no pair of adjacent digits', but there are the "first" and "last" digits.

Harvey P. Dale, Table of n, a(n) for n = 1..120

Cf. A046704, A007953, A088133, A088134, A088135, A088136, A108659.

Select[Prime[Range[200000]],And@@(IntegerQ[Sqrt[#]]&/@(Total/@Partition[ IntegerDigits[#],2,1,1]))&] (* Harvey P. Dale, Mar 03 2014 *)

A108659 Square-chain primes (including square-loop primes).

Original entry on oeis.org

2, 13, 31, 79, 97, 101, 109, 131, 181, 227, 313, 401, 409, 631, 727, 797, 881, 1009, 1013, 1097, 2797, 3109, 3181, 3631, 4001, 4013, 7901, 8101, 9001, 9013, 10009, 10181, 10909, 10979, 13109, 18131, 18181, 22279, 22727, 27901, 31013, 36313

Offset: 1

Zak Seidov, Jun 16 2005

Primes such that each pair of adjacent digits sums up to a square. First term is a square-loop prime, cf. A108660.

Cf. A046704, A007953, A088133, A088134, A088135, A088136, A108660.

Join[{2},Select[Prime[Range[5,4000]],PrimeQ[#]&&AllTrue[Sqrt[#]&/@(Total/@Partition[ IntegerDigits[ #],2,1]),IntegerQ]&]] (* Harvey P. Dale, Jun 02 2024 *)

A086924 Primes such that sum of the first and last digits is a square.

Original entry on oeis.org

2, 13, 31, 79, 97, 103, 113, 163, 173, 193, 227, 257, 277, 311, 331, 613, 643, 653, 673, 683, 709, 719, 739, 769, 811, 821, 881, 907, 937, 947, 967, 977, 997, 1013, 1033, 1063, 1093, 1103, 1123, 1153, 1163, 1193, 1213, 1223

Offset: 1

Zak Seidov, Sep 20 2003

Cf. A088133, A088134, A088135, A088136.

fldsQ[n_]:=Module[{idn=IntegerDigits[n]},IntegerQ[Sqrt[ idn[[1]] + idn[[-1]]]]]; Select[Prime[Range[200]],fldsQ] (* Harvey P. Dale, Aug 12 2017 *)

okdigs(n) = digs = digits(n); issquare(digs[1]+digs[#digs]);
isok(n) = isprime(n) && okdigs(n); \\ Michel Marcus, Oct 05 2013

cp's OEIS Frontend

A088136 Primes such that sum of first and last digits is prime.

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A088133 Sum of first and last digits of n. Different from A115299.

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A088134 Numbers n such that sum of first and last digits is prime.

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A108660 Square-loop primes.

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Mathematica

A108659 Square-chain primes (including square-loop primes).

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Mathematica

A086924 Primes such that sum of the first and last digits is a square.

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Mathematica

PARI