A069567 -id:A069567 - cp's OEIS frontend

A166681 Primes p which have at least one prime anagram larger than p.

13, 17, 37, 79, 107, 113, 127, 131, 137, 139, 149, 157, 163, 167, 173, 179, 181, 191, 197, 199, 239, 241, 251, 277, 281, 283, 313, 337, 347, 349, 359, 367, 373, 379, 389, 397, 419, 457, 461, 463, 467, 479, 491, 563, 569, 571, 577, 587, 593, 613

Offset: 1

Pierre CAMI, Oct 18 2009

Primes like 113, 137, 149, 157 etc have more than one such larger anagram, but are only listed once.

			13 is the first with 31 as prime anagram.
17 is the second with 71 as prime anagram.
31 has one anagram 13 but this is not >31 so 31 is not in the sequence.

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A055387, A109308, A069567.

filter:= proc(p) local L,Lp,q,i;
  if not isprime(p) then return false fi;
  L:= convert(p,base,10);
  for Lp in combinat:-permute(L) do
    q:= add(Lp[i]*10^(i-1),i=1..nops(L));
    if q > p and isprime(q) then return true fi
  od;
false
end proc:
select(filter, [seq(i,i=13..1000,2)]); # Robert Israel, Jan 18 2023

paQ[n_]:=Length[Select[FromDigits/@Permutations[IntegerDigits[n]],#>n && PrimeQ[#]&]]>0; Select[Prime[Range[200]],paQ] (* Harvey P. Dale, Sep 23 2013 *)

from itertools import islice
from sympy.utilities.iterables import multiset_permutations
from sympy import isprime, nextprime
def A166681_gen(): # generator of terms
    p = 13
    while True:
        for q in multiset_permutations(str(p)):
            if (r:=int(''.join(q)))>p and isprime(r):
                yield p
                break
        p = nextprime(p)
A166681_list = list(islice(A166681_gen(),20)) # Chai Wah Wu, Jan 17 2023

Definition clarified, sequence extended. - R. J. Mathar, Oct 12 2012

A227692 Smaller of two consecutive squares which are anagrams (permutations) of each other.

Original entry on oeis.org

169, 24649, 833569, 20367169, 214534609, 368678401, 372142681, 392554969, 407676481, 771617284, 1013021584, 1212780625, 1404075841, 1567051396, 1623848209, 2538748996, 2866103296, 2898960964, 3015437569, 3967236196, 4098688441, 4937451289, 5854239169

Offset: 1

Michel Lagneau, Aug 12 2013

Given the n-th square, it is occasionally possible to form the (n+1)-th square using the same digits in a different order.

"Anagram" means that both squares must not only use the same digits but must use each digit the same number of times.

			169 and 196 are two successive squares.

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

Cf. A069567, A247305.

with(numtheory):for n from 1 to 80000 do:p1:=n^2:p2:= (n+1)^2:pp1:=convert(p1,base,10): pp2:=convert(p2,base,10):n1:=sort(pp1):n2:=sort(pp2): if n1=n2 then printf(`%d, `,p1):else fi:od:

lst = {}; k = 1; s = t = 0; ss = {0}; While[k < 155001, s = t; t += k; st = Sort@IntegerDigits@ t; If[ss == st, AppendTo[lst, s]]; ss = st; k += 2]; lst (* Robert G. Wilson v, Oct 24 2014 *)

from itertools import count, islice
def agen(): # generator of terms
    ip, sp, hp = 0, 0, "0"
    for i in count(1):
        s = i*i
        h = "".join(sorted(str(s)))
        if h == hp: yield sp
        ip, sp, hp = i, s, h
print(list(islice(agen(), 23))) # Michael S. Branicky, Feb 18 2024

A339080 Smaller members of binary Ormiston prime pairs: two consecutive primes whose binary representations are anagrams of each other.

Original entry on oeis.org

11, 23, 37, 59, 83, 103, 107, 131, 139, 151, 167, 173, 179, 199, 227, 229, 263, 277, 347, 409, 419, 439, 487, 491, 503, 557, 563, 613, 647, 653, 659, 683, 719, 727, 757, 811, 823, 827, 839, 853, 911, 941, 947, 953, 967, 997, 1019, 1063, 1091, 1093, 1123, 1163

Offset: 1

Amiram Eldar, Nov 22 2020

Equivalently, the smaller of two consecutive primes with the same length of binary representation (A070939) and the same binary weight (A000120).

			11 is a term since 11 and 13 are consecutive primes whose binary representations, 1011 and 1101, are anagrams of each other.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jens Kruse Andersen, Ormiston Tuples.
Andy Edwards, Ormiston Pairs, Australian Mathematics Teacher, Vol. 58, No. 2 (2002), pp. 12-13.
Giovanni Resta, Ormiston pairs.
Eric Weisstein's World of Mathematics, Rearrangement Prime Pair.

Cf. A000120, A069567 (decimal analog), A070939, A072274.

Transpose[Select[Partition[Prime[Range[200]], 2, 1], Sort[IntegerDigits[First[#],2]] == Sort[IntegerDigits[Last[#],2]]&]][[1]] (* after Harvey P. Dale at A069567 *)

from sympy import nextprime
from itertools import islice
def hash(n): return "".join(sorted(bin(n)[2:]))
def agen(start=2): # generator of terms
    p = nextprime(start-1); q=nextprime(p)
    hp, hq = list(map(hash, [p, q]))
    while True:
        if hp == hq: yield p
        p, q = q, nextprime(q)
        hp, hq = hq, hash(q)
print(list(islice(agen(), 52))) # Michael S. Branicky, Feb 19 2024

A353346 Numbers k such that the elements of the continued fraction of the abundancy index of k and k+1 are anagrams of each other.

Original entry on oeis.org

2084564, 11784194, 13667268, 52820326, 68397891, 101183694, 128247668, 135641787, 137681487, 170542955, 266319572, 284966486, 384109196, 386860419, 482419526, 483785771, 546800667, 579468939, 606809224, 622241109, 703636544, 737703005, 829965829, 830993564, 834224684, 973986250

Offset: 1

Amiram Eldar, Apr 15 2022

nonn

			2084564 is a term since the sequences of elements of the continued fractions of sigma(2084564)/2084564 = 941472/521141 and sigma(2084565)/2084565 = 1270656/694855, {1, 1, 4, 5, 1, 8, 1, 5, 2, 4, 1, 7, 3} and {1, 1, 4, 1, 5, 8, 1, 2, 4, 7, 3, 1, 5} respectively, are anagrams of each other.

Amiram Eldar, Table of n, a(n) for n = 1..108

Cf. A000203 (sigma), A069567, A017665, A017666, A349685, A353345, A353347.

ab[n_] := Sort[ContinuedFraction[DivisorSigma[-1, n]]]; seq[max_] := Module[{s = {}, n = 2, c = 0, ab1 = ab[1], ab2}, While[n < max, ab2 = ab[n]; If[ab1 == ab2, AppendTo[s, n - 1]]; ab1 = ab2; n++]; s]; seq[1.4*10^7]

A163679 Smaller prime p in Ormiston pairs (p, q) with q - p = 36.

Original entry on oeis.org

98737, 116293, 187237, 240437, 276781, 343337, 357437, 447137, 454637, 456293, 465337, 508037, 542837, 565937, 586237, 623071, 802037, 817237, 820837, 836071, 837737, 839837, 843137, 850637, 884537, 897781, 903037, 913337, 1032071

Offset: 1

Klaus Brockhaus, Aug 03 2009

An Ormiston pair (or rearrangement prime pair) is a pair of consecutive primes that use the same digits in a different order.

			(187237, 187273) is an Ormiston pair with gap 36, so 187237 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jens Kruse Andersen, Ormiston Tuples
Eric Weisstein's World of Mathematics, Rearrangement Prime Pair

Subsequence of A069567.

Cf. A072274, A163863.

[ p: p in PrimesUpTo(1050000) | q-p eq 36 and a eq b where a is Sort(Intseq(p)) where b is Sort(Intseq(q)) where q is NextPrime(p) ];

Transpose[Select[Select[Partition[Prime[Range[8000]], 2, 1], Last[#] - First[#] == 36 &], Sort[IntegerDigits[First[#]]] == Sort[IntegerDigits[Last[#]]] &]][[1]] (* G. C. Greubel, Aug 02 2017 *)

Keyword base added by Klaus Brockhaus, Sep 18 2009

A163680 Smaller prime p in Ormiston pairs (p, q) with q - p = 54.

Original entry on oeis.org

35617, 40639, 359783, 502339, 552917, 580417, 668417, 719839, 807017, 824339, 833117, 861239, 909917, 961339, 987739, 1078417, 1145539, 1168639, 1185017, 1196539, 1220839, 1313239, 1479617, 1497439, 1710439, 1710539, 1732139

Offset: 1

Klaus Brockhaus, Aug 03 2009

An Ormiston pair (or rearrangement prime pair) is a pair of consecutive primes that use the same digits in a different order.

			(359783, 359837) is an Ormiston pair with gap 54, so 359783 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Jens Kruse Andersen, Ormiston Tuples
Eric Weisstein's World of Mathematics, Rearrangement Prime Pair

Subsequence of A069567.

Cf. A072274, A163863.

[ p: p in PrimesUpTo(1750000) | q-p eq 54 and a eq b where a is Sort(Intseq(p)) where b is Sort(Intseq(q)) where q is NextPrime(p) ];

op54Q[{a_,b_}]:=Sort[IntegerDigits[a]]==Sort[IntegerDigits[b]] && b-a==54; Transpose[Select[Partition[Prime[Range[150000]],2,1],op54Q]][[1]] (* Harvey P. Dale, Jun 16 2014 *)

Keyword base added by Klaus Brockhaus, Sep 18 2009

A163682 Smaller prime p in Ormiston pairs (p, q) with q - p = 90.

Original entry on oeis.org

2030789, 2542237, 3863017, 4508341, 7001123, 7583341, 8482459, 8547677, 8916239, 9194677, 9470017, 11117123, 11755673, 11999563, 13691563, 13898237, 15906127, 16047673, 16272343, 16299013, 16829563, 17437457, 17604347

Offset: 1

Klaus Brockhaus, Aug 03 2009

An Ormiston pair (or rearrangement prime pair) is a pair of consecutive primes that use the same digits in a different order.

			(3863017, 3863107) is an Ormiston pair with gap 90, so 3863017 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jens Kruse Andersen, Ormiston Tuples
Eric Weisstein's World of Mathematics, Rearrangement Prime Pair

Subsequence of A069567.

Cf. A072274, A163863.

[ p: p in PrimesUpTo(17700000) | q-p eq 90 and a eq b where a is Sort(Intseq(p)) where b is Sort(Intseq(q)) where q is NextPrime(p) ];

Transpose[Select[Select[Partition[Prime[Range[70000]], 2, 1], Last[#] - First[#] == 90 &], Sort[IntegerDigits[First[#]]] == Sort[IntegerDigits[Last[#]]] &]][[1]] (* G. C. Greubel, Aug 02 2017 *)

Keyword base added by Klaus Brockhaus, Sep 18 2009

A247305 The smaller of two consecutive triangular numbers which are permutations of each other.

Original entry on oeis.org

404550, 2653056, 3643650, 5633046, 6413571, 10122750, 10656036, 13762881, 19841850, 26634051, 32800950, 47848653, 56769840, 71634465, 89184690, 103672800, 137108520, 317053971, 345069585, 392714325, 408508236, 440762895, 508948560, 598735710, 718830486, 825215625

Offset: 1

K. D. Bajpai, Sep 11 2014

All the terms in sequence are congruent to 0 mod 9.

It appears that the digital root (repeated sum of digits) of the index +1 of a(n) in A000217 is 9 for each n>=1.o 0 mod 9. For example, 404550 = A000217(899), and 899+1 = 900 has digital root 9.

			a(1) = 404550 is in the sequence because {404550 and 405450} are a pair of consecutive triangular numbers having exactly the same digits.
a(2) = 2653056 is in the sequence because {2653056 and 2655360} are two consecutive triangular numbers having exactly the same digits.

K. D. Bajpai, Table of n, a(n) for n = 1..13000

Cf. A000217, A069567, A072274, A075093.

A247305 = {}; a = {1}; b = {2}; Do[t1 = n*(n + 1)/2; t2 = (n - 1)*(n - 1 + 1)/2; b = Sort[IntegerDigits[t1]]; If[a == b, AppendTo[A247305, t2]]; a = b, {n, 2, 7*10^4}]; A247305

lista(nn) = {for (n=1, nn, dt = vecsort(digits(t=n*(n+1)/2)); dnt = vecsort(digits((n+1)*(n+2)/2)); if (dt == dnt, print1(t, ", ")););} \\ Michel Marcus, Sep 13 2014

A353345 Numbers k such that the elements of the continued fractions of the harmonic means of the divisors of k and k+1 are anagrams of each other.

Original entry on oeis.org

688126, 29900656, 35217656, 71624168, 154979487, 527560886, 871173148, 1370592266, 2461226804, 3232529461, 3232684430, 3431178214, 3471121856, 3486231973, 3527029430, 5732671200, 6258062402, 8784477355, 9334188311, 12670993089, 12707869077, 15120804392, 16317131894

Offset: 1

Amiram Eldar, Apr 15 2022

nonn

			688126 is a term since sequences of elements of the continued fractions of the harmonic means of the divisors of 688126 and 688127, 688126/70281 and 688127/77304, are {9, 1, 3, 1, 3, 1, 2, 9, 1, 1, 6, 8} and {8, 1, 9, 6, 3, 1, 2, 1, 3, 1, 1, 9} respectively, and they are anagrams of each other.

Cf. A069567, A099377, A099378, A349473, A353346, A353347.

Subsequence of A349499.

h[n_] := Sort[ContinuedFraction[DivisorSigma[0, n]/DivisorSigma[-1, n]]]; seq[max_] := Module[{s = {}, n = 2, c = 0, h1 = h[1], h2}, While[n < max, h2 = h[n]; If[h1 == h2, AppendTo[s, n - 1]]; h1 = h2; n++]; s]; seq[4*10^7]

A353347 Numbers k such that the elements of the continued fraction of phi(k)/k and phi(k+1)/(k+1) are anagrams of each other.

Original entry on oeis.org

1287, 96074, 5600160, 18486908, 41746312, 78700687, 211818591, 346666215, 535185325, 600248114, 617086359, 682116194, 972901517, 1326113558, 1397946770, 1404159416, 1785588903, 2090593128, 2286664100, 2349999964, 2396173329, 3154287487, 4029358361, 5401346573

Offset: 1

Amiram Eldar, Apr 15 2022

nonn

			1287 is a term since the sequences of elements of the continued fractions of phi(1287)/1287 = 80/143 and phi(1288)/1288 = 66/161, {0, 1, 1, 3, 1, 2, 2, 2} and {0, 2, 2, 3, 1, 1, 1, 2} respectively, are anagrams of each other.

Cf. A000010 (phi), A069567, A076512, A109395, A342866, A353345, A353346.

r[n_] := Sort[ContinuedFraction[EulerPhi[n]/n]]; seq[max_] := Module[{s = {}, n = 2, c = 0, r1 = r[1], r2}, While[n < max, r2 = r[n]; If[r1 == r2, AppendTo[s, n - 1]]; r1 = r2; n++]; s]; seq[6*10^6]

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A166681 Primes p which have at least one prime anagram larger than p.

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A227692 Smaller of two consecutive squares which are anagrams (permutations) of each other.

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A339080 Smaller members of binary Ormiston prime pairs: two consecutive primes whose binary representations are anagrams of each other.

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A353346 Numbers k such that the elements of the continued fraction of the abundancy index of k and k+1 are anagrams of each other.

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A163679 Smaller prime p in Ormiston pairs (p, q) with q - p = 36.

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A163680 Smaller prime p in Ormiston pairs (p, q) with q - p = 54.

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A163682 Smaller prime p in Ormiston pairs (p, q) with q - p = 90.

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