A088133 -id:A088133 - cp's OEIS frontend

A115299 Greatest digit of n + least digit of n. Different from A088133.

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: 1

Rick L. Shepherd, Jan 20 2006

a(101) = 1 and A088133(101) = 2, but all previous terms match.

			a(1) = 1 + 1 = 2, a(232) = 3 + 2 = 5, a(1889009898) = 9 + 0 = 9.

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

Cf. A037904 (greatest-least), A115300 (greatest*least), A088133 (first+last).

Array[Max[#] + Min[#] &@ IntegerDigits[#] &, 120] (* Michael De Vlieger, Dec 12 2023 *)

def a(n): d = list(map(int, str(n))); return max(d) + min(d)
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Dec 12 2023

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

Original entry on oeis.org

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

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.

A088135 Sum of first and last digits of n-th prime.

Original entry on oeis.org

4, 6, 10, 14, 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, 4, 8, 2, 8, 10, 10, 2, 8, 4, 8, 4, 10, 2, 2, 4, 8, 10, 3, 5, 9, 11, 5, 11, 3, 3, 9, 5, 11, 3, 9, 3, 5, 5, 10, 4, 6, 10, 4, 10, 10, 12, 6, 12, 10, 6, 12, 6, 12, 10, 5, 13, 13, 5, 5, 7, 13, 7, 13

Offset: 1

Zak Seidov, Sep 20 2003

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

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

sfl[p_]:=Module[{idn=IntegerDigits[p]},idn[[1]]+idn[[-1]]]; sfl/@Prime[Range[90]] (* Harvey P. Dale, Jan 31 2023 *)

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 *)

A169669 (first digit of n) * (last digit of n) in decimal representation.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0

Offset: 0

Reinhard Zumkeller, Apr 05 2010

a(n) = A000030(n)*A010879(n);
a(n) = A115300(n) for n<=100, A115300(101) = 0;
a(n) = A111707(n) for n<=109, A111707(110) = 1;
0 <= a(n) <= 81, range = A174995;
a(10*n + n mod 10) = a(n);
a(A008592(n)) = 0;
A000030(a(A017281(n))) = A000030(A017281(n));
a(n) = a(A004086(n))*A168184(n);

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

Cf. A088133.

Cf. A000030, A008592, A010879, A111707, A115300, A174995.

a169669 n = a000030 n * mod n 10
-- Reinhard Zumkeller, Apr 29 2015

def a(n): return int(str(n)[0])*(n%10)
print([a(n) for n in range(81)]) # Michael S. Branicky, Jul 13 2022

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

A115299 Greatest digit of n + least digit of n. Different from A088133.

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

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

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A088135 Sum of first and last digits of n-th prime.

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A108660 Square-loop primes.

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A169669 (first digit of n) * (last digit of n) in decimal representation.

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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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