A037010 -id:A037010 - cp's OEIS frontend

A075805 Differences between adjacent palindromic numbers which are products of an even number of distinct primes.

5, 16, 11, 22, 22, 34, 30, 20, 41, 60, 41, 20, 70, 61, 51, 10, 20, 10, 20, 61, 81, 10, 20, 30, 51, 20, 20, 20, 20, 41, 10, 10, 20, 10, 122, 330, 220, 330, 11, 440, 561, 110, 110, 220, 440, 891, 231, 110, 1551, 451, 330, 550, 1122, 110, 220, 552, 100, 300, 400, 100

Offset: 1

Jani Melik, Oct 13 2002

			a(1)=6-1=5, a(2)=22-6=16.

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A037010.

First differences of A075799.

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and numtheory[mobius](n)=1; end; s := []; for n from 1 to 11000 do if test(n) then s := [op(s),n]; end; od; a := [op(2..-1,s)-op(1..-2,s)];

Differences[Select[Range[10000], PalindromeQ[#] && MoebiusMu[#] == 1 &]] (* Paolo Xausa, Mar 10 2025 *)

Edited by Dean Hickerson, Oct 21 2002

A075806 Differences between adjacent palindromic numbers which are products of an odd number of distinct primes.

Original entry on oeis.org

1, 2, 2, 4, 55, 35, 30, 20, 30, 10, 31, 60, 31, 40, 20, 10, 51, 40, 20, 61, 40, 11, 40, 81, 30, 20, 10, 10, 122, 10, 40, 32, 220, 330, 220, 451, 660, 451, 220, 781, 660, 341, 220, 110, 220, 110, 11, 220, 220, 440, 451, 220, 110, 110, 110, 451, 220, 220, 220, 341

Offset: 1

Jani Melik, Oct 13 2002

			a(1)=3-2=1, a(2)=5-3=2, a(5)=66-11=55.

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A037010.

First differences of A075800.

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and numtheory[mobius](n)=-1; end; s := []; for n from 1 to 11000 do if test(n) then s := [op(s),n]; end; od; a := [op(2..-1,s)-op(1..-2,s)];

Differences[Select[Range[10000], PalindromeQ[#] && MoebiusMu[#] == -1 &]] (* Paolo Xausa, Mar 10 2025 *)

Edited by Dean Hickerson, Oct 21 2002

A075804 Differences between adjacent palindromic nonsquarefree numbers A035132.

Original entry on oeis.org

4, 1, 35, 44, 11, 22, 50, 41, 20, 10, 10, 20, 20, 41, 10, 20, 41, 10, 10, 20, 20, 20, 41, 50, 10, 31, 20, 20, 10, 10, 10, 10, 51, 61, 20, 20, 20, 20, 21, 90, 332, 550, 231, 220, 220, 110, 110, 220, 671, 110, 220, 11, 110, 110, 220, 110, 110, 220, 341, 220, 330, 341

Offset: 1

Jani Melik, Oct 13 2002

			a(1) = 8 - 4 = 4, a(2) = 9 - 8 = 1, a(3) = 44 - 9 = 35.

Cf. A037010, A035132.

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and numtheory[mobius](n)=0; end; s := []; for n from 1 to 7000 do if test(n) then s := [op(s),n]; end; od; a := [op(2..-1,s)-op(1..-2,s)];

Differences[Select[Range[10000],PalindromeQ[#]&&!SquareFreeQ[#]&]] (* Harvey P. Dale, Dec 28 2024 *)

Edited by Dean Hickerson, Oct 21 2002

A309321 The number of primes between two consecutive palindromic primes, bounds excluded.

Original entry on oeis.org

0, 0, 0, 0, 20, 5, 3, 5, 0, 21, 5, 2, 1, 52, 4, 3, 0, 17, 0, 1104, 21, 7, 73, 9, 105, 35, 8, 54, 51, 11, 34, 43, 78, 8, 52, 29, 19, 10, 80, 50, 22, 33, 78, 53, 9, 994, 11, 17, 26, 7, 20, 49, 75, 12, 109, 100, 27, 16, 12, 16, 32, 48, 28, 69, 32, 42, 6, 56, 48

Offset: 1

Hauke Löffler, Jul 23 2019

			a(0): Between the first two palindromic primes (2,3) there are 0 primes.
a(6): Between 101 and 131 there are 5 primes (103, 107, 109, 113, 127).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Hauke Löffler)

Cf. A002385, A037010, A075807, A176559.

#Palindromic primes
def count_primes_between(a,b):
    return len(prime_range(a+1,b))
[count_primes_between(A002385[i],A002385[i+1]) for i in range (len(A002385)-1)]
# Alternative:
def A309321list(bound):
    L = []; p = 2
    while p < bound:
        p = next_prime(p)
        delta = 0
        while not Word(p.digits()).is_palindrome():
            delta += 1
            p = next_prime(p)
        L.append(delta)
    return L
A309321list(18181) # Peter Luschny, Jul 23 2019

a(n) = A075807(n+1) - A075807(n) - 1. - Jinyuan Wang, Jul 24 2019

A075801 Differences between adjacent palindromic nonprime numbers A032350.

Original entry on oeis.org

3, 2, 2, 1, 13, 11, 11, 11, 11, 11, 11, 11, 12, 10, 20, 20, 10, 31, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 20, 10, 10, 20, 30, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 20, 10, 20, 10, 31

Offset: 1

Jani Melik, Oct 13 2002

			a(1)=4-1=3, a(5)=22-9=13.

Cf. A037010, A032350.

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and not isprime(n); end; s := []; for n from 1 to 1000 do if test(n) then s := [op(s),n]; end; od; a := [op(2..-1,s)-op(1..-2,s)];

Edited by Dean Hickerson, Oct 21 2002

A075803 Differences between adjacent palindromic squarefree numbers.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 11, 11, 22, 11, 11, 24, 10, 20, 10, 10, 10, 20, 10, 11, 20, 40, 20, 21, 10, 10, 30, 20, 10, 10, 41, 20, 20, 20, 11, 10, 20, 10, 10, 10, 30, 11, 20, 20, 61, 10, 10, 10, 20, 10, 10, 10, 10, 21, 20, 20, 20, 20, 21, 10, 10, 10, 10, 10, 10, 10, 12, 110, 110, 220

Offset: 1

Jani Melik, Oct 13 2002

			a(1)=2-1=1, a(3)=5-3=2, a(6)=11-7=4.

Cf. A037010.

First differences of A071251.

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and numtheory[mobius](n)<>0; end; s := []; for n from 1 to 1500 do if test(n) then s := [op(s),n]; end; od; a := [op(2..-1,s)-op(1..-2,s)];

Edited by Dean Hickerson, Oct 21 2002

A333648 Bemirp gaps: differences between consecutive bemirps.

Original entry on oeis.org

30, 510, 300, 8160, 30, 5910, 3000, 87860, 3030, 58710, 30300, 907980, 3000, 496200, 199980, 3030, 76920, 3000, 20070, 8897800, 3000, 251930, 30000, 517870, 89010, 117320, 3000, 87970, 61980, 4092720, 36980, 68020, 522380, 191620, 106230, 1621950, 7200, 620

Offset: 1

Metin Sariyar, Mar 31 2020

The smallest gap is 10 = 16611666611 - 16611666601 and all terms are divisible by 10 as a result of the rule that all bemirps have to end with 1. Bemirp pairs with a gap 10 are 16611666601, 16611666611, 19911999901, 19911999911, ... .

			a(1) = 1091 - 1061 = 30.

Cf. A001223, A037010, A048895, A185439.

A048895 = Cases[Import["https://oeis.org/A048895/b048895.txt", "Table"], {, }][[All, 2]];a[n_] :=  A048895[[n+1]]-A048895[[n]];a /@ Range[1,100] (* based on A048895 b-file *)

a(n) = A048895(n+1) - A048895(n).

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A075805 Differences between adjacent palindromic numbers which are products of an even number of distinct primes.

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A075806 Differences between adjacent palindromic numbers which are products of an odd number of distinct primes.

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A075804 Differences between adjacent palindromic nonsquarefree numbers A035132.

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A309321 The number of primes between two consecutive palindromic primes, bounds excluded.

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A075801 Differences between adjacent palindromic nonprime numbers A032350.

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A075803 Differences between adjacent palindromic squarefree numbers.

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A333648 Bemirp gaps: differences between consecutive bemirps.

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